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1473 y(Cuerna)n(v)-5 b(aca)26 b(Morelos)g(62251,)f(MEXICO.)1542 1572 y(e-mail:)36 b(villegas@matcuer.unam.mx)p eop %%Page: 2 2 2 1 bop 3789 324 a Fw(2)523 614 y Fv(ABSTRA)m(CT)p Fw(.)40 b(This)e(pap)r(er)g(is)h(ab)r(out)f(a)h(relationship)e(b)r(et)n(w)n (een)i(the)g(k)n(ernel)e(of)i(the)g(Bargmann)523 714 y(transform)30 b(\(BT\))h(and)g(the)h(corresp)r(onding)d(canonical)h (transformation.)45 b(W)-7 b(e)31 b(study)h(this)f(fact)g(for)g(a)523 814 y(BT)26 b(\(in)n(tro)r(duced)g(b)n(y)g(L.)g(Thomas)f(and)h(S.)g(W) -7 b(assell\))26 b(when)g(the)h(con\014guration)d(space)h(is)h(the)h (2-sphere)523 913 y Fu(S)579 883 y Ft(2)654 913 y Fw(and)38 b(for)f(a)h(BT)g(that)g(w)n(e)g(in)n(tro)r(duce)f(for)h Fu(S)2096 883 y Ft(3)2133 913 y Fw(.)68 b(It)38 b(is)g(sho)n(wn)g(that) g(the)g(k)n(ernel)f(of)h(the)h(BT)f(is)g(a)523 1013 y(p)r(o)n(w)n(er)e (series)g(of)h(a)g(function)h(whic)n(h)f(is)g(a)f(generating)g (function)i(of)f(the)g(corresp)r(onding)e(canonical)523 1112 y(transformation)30 b(\(CT\))j(\(a)e(classical)g(analog)f(of)h (the)i(BT\).)e(W)-7 b(e)33 b(sho)n(w)e(in)h(eac)n(h)f(case)f(that)j (our)e(CT)g(is)523 1212 y(a)e(comp)r(osition)f(of)h(other)g(t)n(w)n(o)f (CT)h(in)n(v)n(olving)f(the)i(complex)e(quadric)h(in)g Fv(C)2976 1182 y Ft(3)3042 1212 y Fw(or)g Fv(C)3215 1182 y Ft(4)3252 1212 y Fw(.)41 b(W)-7 b(e)30 b(also)e(study)523 1312 y(the)34 b(quan)n(tizations)e(of)h(those)g(other)f(t)n(w)n(o)h(CT) g(b)n(y)g(dealing)g(with)g(spaces)g(of)g(holomorphic)f(functions)523 1411 y(on)g(the)h(men)n(tioned)g(quadric.)51 b(W)-7 b(e)33 b(sho)n(w)f(repro)r(ducing)f(k)n(ernels)h(for)g(those)g(spaces.)51 b(W)-7 b(e)33 b(also)f(relate)523 1511 y(some)e(of)h(the)g(quan)n (tizations)f(with)i(w)n(ork)d(of)i(V.)g(Bargmann)2486 1481 y Ft(20)2586 1511 y Fw(and)f(V.)i(Guillemin)3222 1481 y Ft(21)3293 1511 y Fw(.)47 b(Since)31 b(p)r(o)n(w)n(ers)523 1611 y(of)g(the)h(generating)e(functions)i(are)e(coheren)n(t)h(states)g (for)g Fu(L)2449 1580 y Ft(2)2486 1611 y Fw(\()p Fu(S)2574 1580 y Ft(2)2611 1611 y Fw(\))h(or)e Fu(L)2837 1580 y Ft(2)2874 1611 y Fw(\()p Fu(S)2962 1580 y Ft(3)2999 1611 y Fw(\),)j(w)n(e)e(\014nally)g(sho)n(w)g(that)523 1710 y(the)d(studied)g(BT)g(are)e(actually)h(coheren)n(t)g(states)g (transforms.)p eop %%Page: 3 3 3 2 bop 3789 324 a Fw(3)523 614 y Fv(I.)32 b(INTR)m(ODUCTION)648 761 y Fw(V.)21 b(Bargmann)1133 731 y Ft(1)1189 761 y Fw(in)n(tro)r(duced)f(the)h(so)f(called)g(Bargmann)f(T)-7 b(ransform)19 b(as)h(a)g(unitary)g(transformation)523 861 y(from)i Fu(L)771 831 y Ft(2)808 861 y Fw(\()p Fv(R)912 831 y Fs(n)957 861 y Fw(\))g(on)n(to)g(a)f(space)h Fr(B)1528 873 y Fs(n)1595 861 y Fw(of)g(analytic)f(functions)i(\(of)f Fu(n)g Fw(complex)g(v)-5 b(ariables\))21 b(whic)n(h)h(are)g(square)523 960 y(in)n(tegrable)28 b(with)h(resp)r(ect)g(to)g(a)g(Gaussian)f (measure.)40 b(The)29 b(Bargmann)e(transform)h(is)h(de\014ned)g(as)g (the)523 1060 y(in)n(tegral)d(op)r(erator)1036 1268 y Fv(B)1104 1280 y Fq(R)1160 1264 y Fp(n)1205 1268 y Fw(\011\()p Fv(z)p Fw(\))d Fr(\021)1557 1212 y Fw(1)p 1497 1249 163 4 v 1497 1326 a Fu(\031)1547 1302 y Fs(n=)p Ft(4)1683 1155 y Fo(Z)1729 1343 y Fq(R)1785 1327 y Fp(n)1844 1268 y Fw(exp\()p Fr(\000)2078 1212 y Fw(1)p 2078 1249 42 4 v 2078 1325 a(2)2129 1268 y(\()p Fv(z)2203 1233 y Ft(2)2259 1268 y Fw(+)18 b Fv(x)2392 1233 y Ft(2)2430 1268 y Fw(\))h(+)2564 1195 y Fr(p)p 2633 1195 V 73 x Fw(2)o Fv(z)g Fr(\017)f Fv(x)p Fw(\)\011\()p Fv(x)p Fw(\))p Fu(d)p Fv(x)575 b Fw(\(1\))523 1484 y(\(see)32 b(section)f(I)h(for)g(notation\).)50 b(The)32 b(Bargmann)e(transform)h(is)h(also)f(called)g(the)i (\\Segal-Bargmann)523 1584 y(T)-7 b(ransform".)35 b(\()27 b(see)1165 1553 y Ft(2)1229 1584 y Fw(for)f(a)g(historical)g(remark)f (ab)r(out)i(Bargmann)e(and)i(Segal's)e(w)n(ork)h(on)g(this)h(trans-)523 1683 y(form.)37 b(I.)27 b(Segal)1022 1653 y Ft(3)1086 1683 y Fw(studied)h(the)g(in\014nite)h(dimensional)e(case)g Fu(n)22 b Fw(=)h Fr(1)28 b Fw(\).)648 1783 y(The)35 b(Bargmann)f(space) h Fr(B)1524 1795 y Fs(n)1604 1783 y Fw(pro)n(vides)f(a)h(v)n(ery)f (adequate)h(framew)n(ork)e(to)j(represen)n(t)e(and)h(study)523 1883 y(the)h Fu(n)p Fw(-dimensional)f(quan)n(tum)h(harmonic)f (oscillator)g(\(i.)62 b(e.)g(the)36 b(addition)g(of)g(n)h (one-dimensional)523 1982 y(harmonic)25 b(oscillators)f(with)i(the)g (same)f(frequency\).)36 b(It)27 b(is)e(kno)n(wn)2653 1952 y Ft(4)2716 1982 y Fw(that)h(the)g(Hamiltonian)f(of)h(suc)n(h)g(a) 523 2082 y(oscillator)g(can)h(b)r(e)h(written)g(as)1876 2276 y(^)1855 2297 y Fu(H)h Fw(=)2080 2193 y Fs(n)2041 2218 y Fo(X)2044 2395 y Fs(j)s Ft(=1)2175 2297 y Fu(a)2219 2257 y Fn(y)2219 2320 y Fs(j)2254 2297 y Fu(a)2298 2309 y Fs(j)3724 2297 y Fw(\(2\))523 2574 y(where)e Fu(a)807 2534 y Fn(y)807 2597 y Fs(j)870 2574 y Fw(and)g Fu(a)1075 2586 y Fs(j)1110 2574 y Fw(,)h Fu(j)g Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g(:)g(:)g(:)f(;)h(n)27 b Fw(are)g(creation)f(and)i (annihilation)f(op)r(erators)f(giv)n(en)g(b)n(y)1410 2807 y Fu(a)1454 2767 y Fn(y)1454 2830 y Fs(j)1512 2807 y Fr(\021)1644 2751 y Fw(1)p 1609 2788 111 4 v 1609 2804 a Fr(p)p 1679 2804 42 4 v 1679 2873 a Fw(2)1730 2807 y(\()6 b(^)-48 b Fu(q)1799 2819 y Fs(j)1853 2807 y Fr(\000)18 b Fu({)7 b Fw(^)-49 b Fu(p)2005 2819 y Fs(j)2039 2807 y Fw(\))24 b(=)2227 2751 y(1)p 2192 2788 111 4 v 2192 2804 a Fr(p)p 2262 2804 42 4 v 2262 2873 a Fw(2)2327 2690 y Fo(\022)2388 2807 y Fu(q)2425 2819 y Fs(j)2479 2807 y Fr(\000)2608 2751 y Fu(@)p 2572 2788 121 4 v 2572 2864 a(@)5 b(q)2658 2876 y Fs(j)2702 2690 y Fo(\023)1424 3041 y Fu(a)1468 3053 y Fs(j)1526 3041 y Fr(\021)1658 2985 y Fw(1)p 1623 3022 111 4 v 1623 3039 a Fr(p)p 1693 3039 42 4 v 1693 3107 a Fw(2)1744 3041 y(\()h(^)-48 b Fu(q)1813 3053 y Fs(j)1867 3041 y Fw(+)18 b Fu({)7 b Fw(^)-49 b Fu(p)2019 3053 y Fs(j)2053 3041 y Fw(\))24 b(=)2241 2985 y(1)p 2206 3022 111 4 v 2206 3039 a Fr(p)p 2275 3039 42 4 v 68 x Fw(2)2341 2924 y Fo(\022)2402 3041 y Fu(q)2439 3053 y Fs(j)2492 3041 y Fw(+)2621 2985 y Fu(@)p 2585 3022 121 4 v 2585 3098 a(@)5 b(q)2671 3110 y Fs(j)2716 2924 y Fo(\023)3724 3041 y Fw(\(3\))523 3273 y(with)43 b(^)-57 b Fu(q)749 3285 y Fs(j)812 3273 y Fw(the)28 b(op)r(erator)e(of)h(m)n(ultiplication)h(b)n(y)f(the)h(co)r(ordinate)f Fu(q)2621 3285 y Fs(j)2683 3273 y Fw(and)35 b(^)-49 b Fu(p)23 b Fw(=)f Fr(\000)p Fu({)3129 3241 y Fs(@)p 3099 3255 101 4 v 3099 3302 a(@)t(q)3168 3310 y Fp(j)3209 3273 y Fw(.)648 3407 y(The)38 b(op)r(erators)f(corresp)r(onding)f(to)j Fu(a)1909 3367 y Fn(y)1909 3430 y Fs(j)1944 3407 y Fw(,)i Fu(a)2052 3419 y Fs(j)2126 3407 y Fw(and)2320 3386 y(^)2298 3407 y Fu(H)46 b Fw(in)38 b(the)h(Bargmann)e(space)h Fr(B)3378 3419 y Fs(n)3461 3407 y Fw(are)g(m)n(ulti-)523 3523 y(plication)31 b(b)n(y)f(the)i(complex)e(v)-5 b(ariable)30 b Fu(z)1815 3535 y Fs(j)1850 3523 y Fw(,)i(partial)e(deriv)-5 b(ativ)n(e)30 b(with)h(resp)r(ect)g(to)g Fu(z)3184 3535 y Fs(j)3250 3523 y Fw(and)3414 3460 y Fo(P)3502 3481 y Fs(n)3502 3547 y(j)s Ft(=1)3635 3523 y Fu(z)3674 3535 y Fs(j)3750 3490 y(@)p 3718 3504 102 4 v 3718 3551 a(@)t(z)3789 3559 y Fp(j)523 3634 y Fw(resp)r(ectiv)n(ely)-7 b(.)33 b(In)20 b(particular,)g(the)g(sp)r(ectrum)g(of)f(the)h(n-dimensional)f (quan)n(tum)g(harmonic)g(oscillator)e(is)523 3733 y(v)n(ery)24 b(easy)g(to)h(compute)h(b)r(ecause)e(in)i(the)f(Bargmann)f(represen)n (tation)f(the)j(monomials)e Fu(z)3396 3696 y Fs(a)3432 3704 y Fm(1)3392 3755 y Ft(1)3468 3733 y Fu(z)3511 3696 y Fs(a)3547 3704 y Fm(2)3507 3755 y Ft(2)3596 3733 y Fu(:)14 b(:)g(:)g(z)3750 3703 y Fs(a)3786 3711 y Fp(n)3746 3754 y Fs(n)523 3833 y Fw(\(with)23 b Fu(a)783 3845 y Fs(j)818 3833 y Fu(;)14 b(j)27 b Fw(=)c(1)p Fu(;)14 b Fw(2)p Fu(;)g(:)g(:)g(:)f(;)h(n)p Fw(,)22 b(nonnegativ)n(e)f(in)n (teger)g(n)n(um)n(b)r(ers\))g(are)g(the)h(eigenfunctions)g(of)g(the)g (op)r(erator)523 3870 y Fo(P)611 3891 y Fs(n)611 3957 y(j)s Ft(=1)744 3932 y Fu(z)783 3944 y Fs(j)858 3900 y(@)p 827 3914 V 827 3961 a(@)t(z)898 3969 y Fp(j)939 3932 y Fw(.)648 4043 y(F)-7 b(rom)34 b(the)h(viewp)r(oin)n(t)g(of)g (Classical)e(Mec)n(hanics,)j(the)f(problem)g(of)g(a)f(n-dimensional)g (harmonic)523 4143 y(oscillator)24 b(can)h(b)r(e)h(studied)g(in)g(the)g (phase)f(space)g Fv(R)2184 4113 y Ft(2)p Fs(n)2285 4143 y Fw(=)e Fr(f)p Fw(\()p Fv(q)p Fu(;)14 b Fv(p)p Fw(\))p Fr(j)p Fv(x)p Fu(;)g Fv(p)23 b Fr(2)h Fv(R)2956 4113 y Fs(n)3001 4143 y Fr(g)h Fw(with)h(symplectic)g(form)523 4243 y Fu(!)575 4255 y Fs(n)643 4243 y Fw(=)731 4180 y Fo(P)818 4201 y Fs(n)818 4268 y(j)s Ft(=1)951 4243 y Fu(dq)1031 4255 y Fs(j)1066 4243 y Fr(^)q Fu(dp)1207 4255 y Fs(j)1269 4243 y Fw(and)i(Hamiltonian)1681 4494 y Fu(H)i Fw(=)1878 4438 y(1)p 1878 4475 42 4 v 1878 4551 a(2)1982 4391 y Fs(n)1943 4415 y Fo(X)1946 4592 y Fs(j)s Ft(=1)2077 4494 y Fu(p)2119 4460 y Ft(2)2119 4515 y Fs(j)2174 4494 y Fw(+)18 b Fu(q)2297 4460 y Ft(2)2294 4515 y Fs(j)2353 4494 y Fr(\000)2446 4438 y Fu(n)p 2446 4475 50 4 v 2450 4551 a Fw(2)3724 4494 y(\(4\))648 4758 y(It)31 b(is)g(kno)n(wn)1064 4728 y Ft(4)1132 4758 y Fw(that)g(this)g(problem)g(can)f(also)g(b)r(e)i (describ)r(ed)f(b)n(y)f(in)n(tro)r(ducing)h(complex)f(v)-5 b(ariables)523 4858 y Fu(z)562 4870 y Fs(j)620 4858 y Fw(=)745 4825 y Ft(1)p 718 4839 88 4 v 718 4847 a Fn(p)p 772 4847 34 3 v 772 4896 a Ft(2)815 4858 y Fw(\()p Fu(q)884 4870 y Fs(j)938 4858 y Fr(\000)18 b Fu({p)1090 4870 y Fs(j)1125 4858 y Fw(\))p Fu(;)37 b(j)29 b Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g(:)g(:)g(:)f(;)h(n)p Fw(.)37 b(In)28 b(this)g(complex)f(v)-5 b(ariables)27 b(the)h(Hamiltonian)f Fu(H)35 b Fw(is)28 b(giv)n(en)e(b)n(y)523 4915 y Fo(P)611 4936 y Fs(n)611 5003 y(j)s Ft(=1)744 4978 y Fu(z)783 4990 y Fs(j)p 817 4932 43 4 v 817 4978 a Fu(z)859 4990 y Fs(j)894 4978 y Fw(.)37 b(This)28 b(is)f(the)h(classical)f(analog)e (of)j(equation\(2\).)648 5090 y(A)h(remark)-5 b(able)27 b(and)i(fundamen)n(tal)f(idea)h(for)f(this)h(pap)r(er)f(is)h(the)g (follo)n(wing:)38 b(as)28 b(in)h(reference)3649 5060 y Ft(5)3686 5090 y Fw(,)g(let)523 5189 y(us)f(consider)e(the)i(argumen) n(t)f(of)g(the)h(k)n(ernel)f(of)g(the)h(Bargmann)e(transform)h(in)h (equation)f(\(1\))1363 5409 y(\010)1423 5421 y Fs(n)1468 5409 y Fw(\()p Fv(q)p Fu(;)14 b Fv(z)p Fw(\))24 b Fr(\021)f(\000)p Fu({)1879 5292 y Fo(\022)1939 5409 y Fr(\000)2014 5353 y Fw(1)p 2014 5390 42 4 v 2014 5466 a(2)2065 5409 y(\()p Fv(z)2139 5375 y Ft(2)2195 5409 y Fw(+)c Fv(q)2329 5375 y Ft(2)2366 5409 y Fw(\))g(+)2500 5336 y Fr(p)p 2569 5336 V 73 x Fw(2)p Fv(z)f Fr(\001)h Fv(q)2763 5292 y Fo(\023)3724 5409 y Fw(\(5\))523 5629 y Fl(It)24 b(turns)g(out)g(that)h (the)g(function)g Fw(\010)1639 5641 y Fs(n)1684 5629 y Fw(\()p Fv(q)p Fu(;)14 b Fv(z)p Fw(\))26 b Fl(is)f(a)h(gener)l(ating) f(function)f(of)i(a)f(c)l(anonic)l(al)h(tr)l(ansformation)523 5728 y(\(symple)l(ctomorphism\))35 b(fr)l(om)e(the)f(phase)i(sp)l(ac)l (e)f Fw(\()p Fv(R)2223 5698 y Ft(2)p Fs(n)2301 5728 y Fu(;)14 b(!)2390 5740 y Fs(n)2435 5728 y Fw(\))32 b Fl(onto)h(the)f (phase)i(sp)l(ac)l(e)f Fw(\()p Fv(C)3383 5698 y Fs(n)3429 5728 y Fu(;)14 b(\026)3516 5740 y Fs(n)3561 5728 y Fw(\))32 b Fl(wher)l(e)p eop %%Page: 4 4 4 3 bop 3789 324 a Fw(4)523 614 y Fu(\026)573 626 y Fs(n)655 614 y Fl(is)37 b(the)g(symple)l(ctic)h(form)f Fu(\026)1559 626 y Fs(n)1640 614 y Fw(=)e Fr(\000)p Fu({)1846 552 y Fo(P)1933 573 y Fs(n)1933 639 y(j)s Ft(=1)2066 614 y Fu(dz)2148 626 y Fs(j)2183 614 y Fr(^)p Fu(d)p 2281 569 43 4 v(z)2324 626 y Fs(j)2359 614 y Fl(.)60 b Fw(\(see)35 b(section)g(I)r(I)r(I)g(for)g(details\).)60 b(Th)n(us)35 b(w)n(e)523 714 y(shall)g(sa)n(y)g(that)h Fr(C)1109 726 y Fs(n)1190 714 y Fw(is)g(the)g(classical)e(analog)g(of)i(the)g (Bargmann)f(transform)f(\(or)i(that)g Fv(B)3487 726 y Fq(R)3543 710 y Fp(n)3623 714 y Fw(is)g(the)523 814 y(Quan)n(tization) 27 b(of)g Fr(C)1162 826 y Fs(n)1207 814 y Fw(\))h(in)g(the)g(sense)f (that)h Fv(B)1968 826 y Fq(R)2024 809 y Fp(n)2096 814 y Fw(is)g(a)f(unitary)g(transformation)f(b)r(et)n(w)n(een)i(the)g(Hilb) r(ert)523 913 y(spaces)c Fu(L)832 883 y Ft(2)869 913 y Fw(\()p Fv(R)973 883 y Fs(n)1018 913 y Fw(\))h(and)g Fr(B)1289 925 y Fs(n)1334 913 y Fw(,)g(and)g Fr(C)1585 925 y Fs(n)1655 913 y Fw(is)g(a)g(di\013eomorphism)f(b)r(et)n(w)n(een)h (the)h(phase)e(spaces)g(\()p Fv(R)3427 883 y Ft(2)p Fs(n)3505 913 y Fu(;)14 b(!)3594 925 y Fs(n)3639 913 y Fw(\))25 b(and)523 1013 y(\()p Fv(C)624 983 y Fs(n)669 1013 y Fu(;)14 b(\026)756 1025 y Fs(n)802 1013 y Fw(\).)648 1112 y(This)27 b(pap)r(er)g(concerns)f(ab)r(out)h(the)h(same)f (relationship)f(b)r(et)n(w)n(een)i(the)f(Bargmann)f(transform)g(and)523 1212 y(a)31 b(corresp)r(onding)f(canonical)h(transformation)f(when)i (the)g(con\014guration)e(space)h Fv(R)3246 1182 y Fs(n)3323 1212 y Fw(is)h(replaced)e(b)n(y)523 1312 y(the)23 b(2-sphere)e Fu(S)1040 1282 y Ft(2)1099 1312 y Fw(or)h(the)g(three)g(sphere)g Fu(S)1850 1282 y Ft(3)1910 1312 y Fw(\(here)g(w)n(e)g(mean)g(b)n(y)g Fr(S)2616 1282 y Fs(n)2684 1312 y Fw(the)h(set)f(of)g(unit)h(v)n (ectors)e(in)i Fv(R)3646 1282 y Fs(n)p Ft(+1)3775 1312 y Fw(\).)523 1411 y(F)-7 b(or)26 b(the)i Fu(S)870 1381 y Ft(2)934 1411 y Fw(case,)e(L.Thomas)g(and)h(S.)g(W)-7 b(assell)2051 1381 y Ft(6)2115 1411 y Fw(in)n(tro)r(duced)27 b(a)f(Bargmann)f(transform)h(from)h Fu(L)3636 1381 y Ft(2)3673 1411 y Fw(\()p Fu(S)3761 1381 y Ft(2)3798 1411 y Fw(\))523 1511 y(on)n(to)38 b(a)g(prop)r(er)g(subspace)f(of)i Fr(B)1596 1523 y Ft(2)1633 1511 y Fw(.)70 b(F)-7 b(or)37 b(the)i Fu(S)2095 1481 y Ft(3)2171 1511 y Fw(case,)h(a)f(Bargmann)e (transform)g(w)n(as)g(in)n(tro)r(duced)523 1611 y(b)n(y)32 b(C.)g(Villegas-Blas)1228 1580 y Ft(7)1264 1611 y Fw(.)51 b(This)32 b(last)g(transform)f(go)r(es)g(from)h Fu(L)2525 1580 y Ft(2)2562 1611 y Fw(\()p Fu(S)2650 1580 y Ft(3)2687 1611 y Fw(\))h(on)n(to)e(a)h(prop)r(er)g(subspace)f(of)h Fr(B)3793 1623 y Ft(4)523 1710 y Fw(determined)19 b(b)n(y)f(the)h(k)n (ernel)f(of)h(an)f(op)r(erator)f(whic)n(h)h(carries)f(the)i (restriction)f(in)h(quan)n(tum)f(mec)n(hanics)g(of)523 1810 y(the)k(classical)e(fact)i(that)g(the)g(Angular)f(Momen)n(tum)h (and)f(Runge-Lenz)g(v)n(ectors)f(should)h(b)r(e)h(orthogonal)523 1910 y(for)i(the)i(Kepler)e(problem.)35 b(W)-7 b(e)25 b(sho)n(w)f(in)h(b)r(oth)h(cases)d Fu(S)2299 1879 y Ft(2)2361 1910 y Fw(and)i Fu(S)2576 1879 y Ft(3)2613 1910 y Fw(,)g(that)h(the)f (k)n(ernel)f(of)h(the)g(Bargmann)523 2009 y(transform)j(is)i(a)f(p)r(o) n(w)n(er)f(series)g(in)i(a)f(function)h(whic)n(h)f(pla)n(ys)g(the)g (role)g(of)g(a)g(generating)f(function)i(of)g(a)523 2109 y(canonical)c(transformation)g(whic)n(h)i(in)g(turn)f(is)h(the)g (classical)e(analog)g(of)i(the)g(Bargmann)e(transform.)648 2208 y(In)31 b(section)g(I)r(I,)h(w)n(e)f(describ)r(e)g(the)h(Bargmann) e(transform)g(for)h(the)h(three)g(cases)e Fv(R)3324 2178 y Fs(n)3369 2208 y Fw(,)i Fu(S)3480 2178 y Ft(2)3549 2208 y Fw(and)f Fu(S)3770 2178 y Ft(3)3807 2208 y Fw(.)523 2308 y(The)d(men)n(tioned)f(relationship)f(b)r(et)n(w)n(een)i(the)g (Bargmann)d(transform)i(and)g(corresp)r(onding)e(canonical)523 2408 y(transformations)30 b(is)i(established)g(in)h(section)e(I)r(I)r (I.)i(F)-7 b(or)32 b(the)g Fu(S)2534 2378 y Ft(2)2603 2408 y Fw(and)g Fu(S)2825 2378 y Ft(3)2895 2408 y Fw(cases)f(w)n(e)g (do)h(this)h(b)n(y)f(using)523 2507 y(spherical)j(co)r(ordinates)f(\(a) i(lo)r(cal)f(c)n(hart\).)61 b(In)36 b(order)f(to)h(de\014ne)g(and)f (extend)i(our)e(canonical)f(trans-)523 2607 y(formations)d(to)g(the)i (whole)e(spaces)g Fu(T)1753 2577 y Fn(\003)1790 2607 y Fu(S)1846 2577 y Ft(2)1904 2607 y Fr(\000)21 b(f)p Fw(0)p Fr(g)31 b Fw(and)g Fu(T)2373 2577 y Fn(\003)2410 2607 y Fu(S)2466 2577 y Ft(3)2525 2607 y Fr(\000)20 b(f)p Fw(0)p Fr(g)31 b Fw(\(w)n(e)h(need)f(to)h(tak)n(e)f(o\013)h(the)g(zero) 523 2707 y(section)20 b(in)h(b)r(oth)g(cases)e(to)h(a)n(v)n(oid)f (singularities\))g(w)n(e)i(use)f(a)g(k)n(ey)f(prop)r(ert)n(y)h(of)g (the)h(generating)e(functions.)523 2806 y(Namely)-7 b(,)32 b(that)g(these)f(functions)h(can)f(b)r(e)h(written)f(as)g(the)h(inner)f (pro)r(duct)g(of)h(the)f(p)r(osition)h(v)n(ector)e Fv(x)523 2906 y Fw(\(in)25 b Fu(S)705 2876 y Ft(2)766 2906 y Fw(or)e Fu(S)920 2876 y Ft(3)957 2906 y Fw(\))i(and)f(a)g(complex)f(v)n(ector)g Fu(\013)p Fw(\()p Fv(z)p Fw(\))j(in)e(the)h(quadric)e Fv(Q)2589 2918 y Ft(3)2650 2906 y Fw(\(for)h(the)h Fu(S)3002 2876 y Ft(2)3063 2906 y Fw(case\))f(and)g Fv(Q)3498 2918 y Ft(4)3559 2906 y Fw(\(for)g(the)523 3005 y Fu(S)579 2975 y Ft(3)641 3005 y Fw(case\).)35 b(Here,)26 b(for)e Fu(n)h Fw(a)f(p)r(ositiv)n(e)h(in)n(teger,)f(the)i(quadric)e Fv(Q)2471 3017 y Fs(n)2541 3005 y Fw(is)g(the)i(set)f(of)g(v)n(ectors)e (in)i Fv(C)3423 2975 y Fs(n)3493 3005 y Fw(suc)n(h)g(that)523 3105 y(the)33 b(addition)f(of)h(the)g(square)e(of)h(their)h(comp)r (onen)n(ts)f(is)g(zero.)51 b(Note)32 b(that)h(a)f(v)n(ector)f(in)i Fv(C)3482 3075 y Fs(n)3560 3105 y Fw(b)r(elongs)523 3205 y(to)h Fv(Q)703 3217 y Fs(n)783 3205 y Fw(if)h(and)f(only)g(if)h(the)g (norm)f(of)h(its)f(real)g(and)g(imaginary)f(parts)h(is)g(the)h(same)f (and)g(these)h(t)n(w)n(o)523 3304 y(v)n(ectors)22 b(are)h(orthogonal)f (to)h(eac)n(h)g(other.)35 b(Th)n(us)24 b Fv(Q)2155 3316 y Ft(3)2203 3304 y Fr(\000)11 b(f)p Fw(0)p Fr(g)22 b Fw(and)h Fv(Q)2656 3316 y Ft(4)2704 3304 y Fr(\000)11 b(f)p Fw(0)p Fr(g)22 b Fw(can)h(b)r(e)i(iden)n(ti\014ed)f(through)523 3404 y(symplectomorphisms)k Fu(\032)1351 3416 y Ft(2)1418 3404 y Fw(and)i Fu(\032)1625 3416 y Ft(3)1691 3404 y Fw(with)h Fu(T)1944 3374 y Fn(\003)1981 3404 y Fu(S)2037 3374 y Ft(2)2093 3404 y Fr(\000)20 b(f)p Fw(0)p Fr(g)28 b Fw(and)h Fu(T)2556 3374 y Fn(\003)2594 3404 y Fu(S)2650 3374 y Ft(3)2706 3404 y Fr(\000)20 b(f)p Fw(0)p Fr(g)28 b Fw(resp)r(ectiv)n(ely)-7 b(.)42 b(W)-7 b(e)30 b(remark)523 3504 y(here)e(that)i Fu(\032)929 3516 y Ft(2)995 3504 y Fw(\()p Fu(\032)1070 3516 y Ft(3)1107 3504 y Fw(\))f(is)g(not)g(the)g (restriction)f(to)h Fv(Q)2120 3516 y Ft(3)2176 3504 y Fr(\000)19 b(f)p Fw(0)p Fr(g)27 b Fw(\()p Fv(Q)2517 3516 y Ft(4)2574 3504 y Fr(\000)19 b(f)p Fw(0)p Fr(g)p Fw(\))27 b(of)i(the)h(canonical)d(symplectic)523 3603 y(form)g(for)g(the)h(am)n (bien)n(t)g Fv(C)1377 3573 y Ft(3)1441 3603 y Fw(\()p Fv(C)1542 3573 y Ft(4)1580 3603 y Fw(\).)648 3703 y(Th)n(us)d(w)n(e)f (are)h(able)f(to)i(extend)f(our)f(canonical)g(transformation)g(as)g (the)i(in)n(v)n(erse)e(of)h(a)g(comp)r(osition)523 3802 y(of)30 b(t)n(w)n(o)e(maps.)43 b(F)-7 b(or)29 b(the)g Fu(S)1385 3772 y Ft(2)1452 3802 y Fw(case,)g(one)g(map)h(\()p Fu(\013)p Fw(\()p Fv(z)p Fw(\)\))h(go)r(es)e(from)g Fv(C)2700 3772 y Ft(2)2757 3802 y Fr(\000)19 b(f)p Fw(0)p Fr(g)29 b Fw(on)n(to)f Fv(Q)3256 3814 y Ft(3)3313 3802 y Fr(\000)19 b(f)p Fw(0)p Fr(g)28 b Fw(and)i(the)523 3902 y(other)36 b(one,)p Fu(\032)940 3914 y Ft(2)977 3902 y Fw(,)i(go)r(es)d(from)h Fv(Q)1506 3914 y Ft(3)1568 3902 y Fr(\000)23 b(f)p Fw(0)p Fr(g)35 b Fw(on)n(to)h Fu(T)2074 3872 y Fn(\003)2111 3902 y Fu(S)2167 3872 y Ft(2)2228 3902 y Fr(\000)24 b(f)p Fw(0)p Fr(g)p Fw(.)62 b(F)-7 b(or)35 b(the)i Fu(S)2893 3872 y Ft(3)2966 3902 y Fw(case,)h(w)n(e)e(ha)n(v)n(e)f(a)h(similar)523 4002 y(extension,)c(but)f(instead)g(of)h Fv(C)1531 3972 y Ft(2)1589 4002 y Fr(\000)20 b(f)p Fw(0)p Fr(g)p Fw(,)31 b(w)n(e)g(ha)n(v)n(e)e(a)i(six)g(real-dimensional)e(phase)i(space)f (whic)n(h)i(is)f(a)523 4101 y(submanifold)d(of)f Fv(C)1155 4071 y Ft(4)1220 4101 y Fw(and)h(determined)f(b)n(y)h(reduction.)648 4201 y(Once)d(w)n(e)g(ha)n(v)n(e,)g(for)g(eac)n(h)g(case)f Fu(S)1730 4171 y Ft(2)1793 4201 y Fw(or)h Fu(S)1949 4171 y Ft(3)1985 4201 y Fw(,)i(the)f(symplectomorphisms)e(b)r(et)n(w)n(een)h (the)h(three)g(phase)523 4301 y(spaces,)k(a)h(natural)e(question)i(to)f (ask)g(is)g(ab)r(out)h(the)g(description)f(of)g(their)h(corresp)r (onding)d(quan)n(tiza-)523 4400 y(tions.)37 b(This)27 b(is)h(done)f(in)h(section)f(IV.)648 4500 y(The)35 b(fact)g(that)g(our) g(generating)e(functions)j(for)f(the)g Fu(S)2460 4470 y Ft(2)2532 4500 y Fw(and)g Fu(S)2757 4470 y Ft(3)2829 4500 y Fw(cases)f(are)g(inner)h(pro)r(ducts)g(of)523 4599 y(p)r(osition)27 b(v)n(ectors)e(and)h(a)h(v)n(ectors)e(in)i(the)g (quadric)f(has)g(an)g(imp)r(ortan)n(t)h(implication.)36 b(The)27 b(Bargmann)523 4699 y(transforms)20 b(describ)r(ed)h(in)g (section)g(I)r(I)h(for)e Fu(S)1913 4669 y Ft(2)1971 4699 y Fw(and)i Fu(S)2183 4669 y Ft(3)2241 4699 y Fw(are)e Fl(c)l(oher)l(ent)k(states)f(tr)l(ansforms)p Fw(.)35 b(That)22 b(is,)g(they)523 4799 y(can)k(b)r(e)h(written)g(as)f(an)g (in\014nite)i(sum)e(of)h(inner)f(pro)r(ducts)g(with)i(coheren)n(t)d (states.)36 b(This)27 b(is)f(in)h(analogy)523 4898 y(with)k(the)f (Bargmann)e(transform)h Fv(B)1719 4910 y Fq(R)1775 4894 y Fp(n)1850 4898 y Fw(whic)n(h)h(acting)f(on)h(a)f(function)i(is)f(the) g(inner)g(pro)r(duct)g(of)g(the)523 4998 y(function)c(and)g(a)f (coheren)n(t)g(state)g(of)h(the)g(n-dimensional)f(harmonic)g (oscillator.)34 b(The)26 b(coheren)n(t)f(states)523 5098 y(app)r(earing)32 b(here)h(are)g(w)n(ell)g(kno)n(wn)1655 5067 y Ft(8)1692 5098 y Fw(.)55 b(They)33 b(concen)n(trate)g(on)g(geo)r (desics)f(of)i Fu(S)3084 5067 y Ft(2)3154 5098 y Fw(or)f Fu(S)3318 5067 y Ft(3)3389 5098 y Fw(in)h(the)g(semi-)523 5197 y(classical)21 b(regime)h(and)h(they)g(giv)n(e)f(a)g(resolution)g (of)h(the)g(iden)n(tit)n(y)g(for)f(eac)n(h)g(eigenspace)f(of)i(the)g (spherical)523 5297 y(Laplacian)h(on)h Fu(S)1069 5267 y Ft(2)1131 5297 y Fw(or)f Fu(S)1286 5267 y Ft(3)1348 5297 y Fw(resp)r(ectiv)n(ely)-7 b(.)36 b(\(these)25 b(eigenspaces)f (are)g(irreducible)h(represen)n(tations)e(of)i(the)523 5396 y(groups)34 b(SO\(3\))h(or)g(SO\(4\))g(\).)60 b(Section)36 b(V)f(is)h(dev)n(oted)e(to)i(describ)r(e)f(the)g(Bargmann)f(transforms) g(as)523 5496 y(coheren)n(t)23 b(states)g(transforms.)34 b(These)24 b(coheren)n(t)f(states)g(also)g(help)h(us)g(to)f(obtain)h (repro)r(ducing)e(k)n(ernels)523 5596 y(for)35 b(the)i(range)d(of)i (our)g(Bargmann)e(transforms)g(for)i(b)r(oth)g(the)h Fr(S)2687 5566 y Ft(2)2760 5596 y Fw(and)f Fr(S)2986 5566 y Ft(3)3060 5596 y Fw(cases.)61 b(W)-7 b(e)36 b(also)f(sho)n(w)523 5695 y(repro)r(ducing)29 b(k)n(ernels)f(for)h(spaces)g(of)h (holomorphic)e(functions)i(on)g(the)g(quadric)f Fv(Q)3219 5707 y Ft(3)3286 5695 y Fw(and)g(Q)3461 5768 y(\026)3514 5732 y Ft(4)3581 5695 y Fw(related)p eop %%Page: 5 5 5 4 bop 3789 324 a Fw(5)523 614 y(with)20 b(the)f(range)f(and)h(domain) f(of)i(the)f(quan)n(tizations)f(describ)r(ed)h(in)g(section)g(IV.)g(F) -7 b(or)19 b(the)g(quan)n(tizations)523 714 y(going)25 b(from)g(these)h(spaces)f(of)g(holomorphic)g(functions)h(on)f(the)h (quadric)f(on)n(to)g Fu(L)3115 684 y Ft(2)3152 714 y Fw(\()p Fu(S)3240 684 y Ft(2)3277 714 y Fw(\))h(or)f Fu(L)3492 684 y Ft(2)3529 714 y Fw(\()p Fu(S)3617 684 y Ft(3)3654 714 y Fw(\),)i(w)n(e)523 814 y(relate)g(our)g(descriptions) f(with)j(w)n(ork)d(of)h(V.)h(Bargmann)2338 783 y Ft(14)2435 814 y Fw(and)f(V.)h(Guillemin)3064 783 y Ft(15)3135 814 y Fw(.)648 913 y(W)-7 b(e)25 b(also)e(include)i(t)n(w)n(o)f(app)r (endices)h(to)f(pro)n(v)n(e)f(that)i(the)g(k)n(ernel)f(of)g(the)h (Bargmann)e(transforms)h(for)523 1013 y(b)r(oth)k(spheres)f Fr(S)1067 983 y Ft(2)1132 1013 y Fw(and)h Fr(S)1350 983 y Ft(3)1415 1013 y Fw(is)g(a)f(p)r(o)n(w)n(er)f(series)h(in)h(a)f (certain)g(function.)648 1112 y(W)-7 b(e)37 b(remark)e(that)j(the)f (case)f(of)h(the)g(3-sphere)f(is)h(ph)n(ysically)f(relev)-5 b(an)n(t)36 b(in)h(b)r(oth)g(quan)n(tum)g(and)523 1212 y(classical)24 b(mec)n(hanics.)35 b(The)26 b(symmetries)f(of)g(the)h(b) r(ound)g(states)f(of)g(the)h(h)n(ydrogen)e(atom)g(are)h(giv)n(en)f(b)n (y)523 1312 y(the)g(lie)g(algebra)e(of)i(the)g(group)e Fu(S)5 b(O)r Fw(\(4\))25 b(\(see)1908 1282 y Ft(7)p Fs(;)p Ft(9)1997 1312 y Fw(\).)36 b(In)24 b(this)g(w)n(a)n(y)f(w)n(e)g(can)h (explain)f(the)h(degeneracy)e(of)i(the)523 1411 y(negativ)n(e)e (energies)h(for)g(the)g(h)n(ydrogen)f(atom.)35 b(In)24 b(classical)e(Mec)n(hanics,)i(J.)f(Moser)3164 1381 y Ft(10)3257 1411 y Fw(sho)n(w)n(ed)f(that)i(the)523 1511 y(Hamiltonian)29 b(\015o)n(w)g(for)f(the)i(Kepler)e(problem)h(\(with)h (negativ)n(e)e(energy\))g(is)i(equiv)-5 b(alen)n(t,)29 b(after)g(a)g(time)523 1611 y(reparametrization,)23 b(to)h(the)h(geo)r (desic)e(\015o)n(w)h(on)g Fu(S)2116 1580 y Ft(3)2153 1611 y Fw(.)36 b(In)25 b(this)g(w)n(a)n(y)-7 b(,)24 b(J.)g(Moser)f(is)i (able)f(to)g(regularize)f(the)523 1710 y(motion)k(of)h(the)g(Kepler)f (problem.)648 1810 y(There)37 b(are)g(other)h(Bargmann)f(T)-7 b(ransforms)36 b(in)n(tro)r(duced)i(b)n(y)g(B.)g(Hall)2958 1780 y Ft(2)p Fs(;)p Ft(11)p Fs(;)p Ft(12)3206 1810 y Fw(\(for)g(compact)f(Lie)523 1910 y(groups\),)30 b(M.)g(Stenzel)1237 1879 y Ft(13)1338 1910 y Fw(\(for)g(symmetric)g(spaces)f(of)h(compact)f (t)n(yp)r(e\),)j(K.)d(Ii)3011 1879 y Ft(14)3112 1910 y Fw(and)h(R.)g(W)-7 b(ada)3598 1879 y Ft(15)3698 1910 y Fw(\(for)523 2009 y(n-spheres\),)39 b(but)f(they)f(are)g(di\013eren)n (t)g(from)g(the)h(transformations)d(in)n(tro)r(duced)i(in)g(this)h(pap) r(er.)65 b(J.)523 2109 y(Ra)n(wnsley)867 2079 y Ft(16)970 2109 y Fw(also)33 b(studies)h(a)f(transformation)g(for)g(the)h (n-spheres,)h(but)f(his)g(transformation)e(is)i(not)523 2208 y(unitary)-7 b(.)p eop %%Page: 6 6 6 5 bop 3789 324 a Fw(6)648 614 y(,)523 714 y Fv(I)s(I.)31 b(BAR)m(GMANN)h(TRANSF)m(ORM)f(F)m(OR)g Fu(L)2235 684 y Ft(2)2272 714 y Fw(\()p Fu(S)2360 684 y Ft(2)2397 714 y Fw(\))h Fv(AND)g Fu(L)2770 684 y Ft(2)2807 714 y Fw(\()p Fu(S)2895 684 y Ft(3)2932 714 y Fw(\))648 861 y(The)39 b(purp)r(ose)g(of)h(this)g(section)f(is)h(to)f(describ)r(e)g(b)r(oth)h (the)g(Bargmann)e(transform)h(for)g Fu(L)3636 831 y Ft(2)3673 861 y Fw(\()p Fu(S)3761 831 y Ft(2)3798 861 y Fw(\))523 960 y(\(in)n(tro)r(duced)32 b(b)n(y)g(L.Thomas)g(and)g(S.)h(W)-7 b(assell)2031 930 y Ft(6)2067 960 y Fw(\))33 b(and)f Fu(L)2355 930 y Ft(2)2392 960 y Fw(\()p Fu(S)2480 930 y Ft(3)2517 960 y Fw(\))h(\(in)n(tro)r(duced)f(b)n(y)g(C.)h (Villegas-Blas)3740 930 y Ft(7)3775 960 y Fw(\).)523 1060 y(In)21 b(b)r(oth)h(cases)e(w)n(e)h(need)g(the)h(usual)e(Bargmann) g(spaces)g(in)n(tro)r(duced)h(b)n(y)g(Bargmann)3206 1030 y Ft(1)3263 1060 y Fw(in)g(his)g(transform)523 1160 y(for)32 b Fu(L)712 1130 y Ft(2)749 1160 y Fw(\()p Fv(R)853 1130 y Fs(n)898 1160 y Fw(\).)52 b(T)-7 b(o)32 b(that)h(end,)h(and)e(as)g(a) g(reference)f(for)h(next)h(sections,)g(w)n(e)f(\014rst)g(remind)h(the)g (reader)523 1259 y(what)28 b(these)f(spaces)g(and)g(the)h(Bargmann)e (transform)h(are.)523 1430 y Fv(I)s(I.1)k(Bargmann)g(T)-8 b(ransform)32 b(for)g Fu(L)1837 1400 y Ft(2)1874 1430 y Fw(\()p Fv(R)1978 1400 y Fs(n)2023 1430 y Fw(\))648 1577 y(In)h(this)h(section)f(w)n(e)g(mak)n(e)g(a)g(review)f(of)i(imp)r (ortan)n(t)f(prop)r(erties)g(of)g(the)h(Bargmann)e(T)-7 b(ransform)523 1676 y(that)33 b(w)n(e)g(need)g(for)f(further)h (sections)f(in)h(the)h(pap)r(er.)52 b(F)-7 b(or)32 b(a)h(detailed)g (explanation)f(and)h(pro)r(ofs)f(see)523 1776 y(references)878 1746 y Ft(1)p Fs(;)p Ft(2)967 1776 y Fw(.)648 1876 y(The)g(Bargmann)e (space)h Fr(B)1513 1888 y Fs(n)1590 1876 y Fw(is)h(the)g(space)g(of)g (analytic)f(functions)h(of)g Fu(n)g Fw(complex)g(v)-5 b(ariables)31 b Fv(z)g Fw(=)523 1975 y(\()p Fu(z)594 1987 y Ft(1)631 1975 y Fu(;)14 b(z)707 1987 y Ft(2)744 1975 y Fu(;)g(:)g(:)g(:)g(;)g(z)968 1987 y Fs(n)1012 1975 y Fw(\))28 b(whic)n(h)g(are)e(square)h(in)n(tegrable)f(with)i (resp)r(ect)f(to)h(the)g(Gaussian)f(measure)1436 2224 y Fu(d\027)1520 2236 y Fs(n)1566 2224 y Fw(\()p Fv(z)p Fw(\))d Fr(\021)1820 2168 y Fw(1)p 1793 2205 96 4 v 1793 2281 a Fu(\031)1843 2257 y Fs(n)1913 2224 y Fw(exp\()p Fr(\000j)p Fv(z)p Fr(j)2225 2190 y Ft(2)2262 2224 y Fw(\))2345 2121 y Fs(n)2313 2146 y Fo(Y)2308 2322 y Fs(j)s Ft(=1)2437 2224 y Fu(dx)2527 2236 y Fs(j)2563 2224 y Fu(dy)2647 2236 y Fs(j)2728 2224 y Fu(;)908 b Fw(\(2.1\))523 2501 y(where,)27 b Fr(j)p Fv(z)p Fr(j)874 2471 y Ft(2)935 2501 y Fr(\021)c(j)p Fu(z)1085 2513 y Ft(1)1122 2501 y Fr(j)1145 2471 y Ft(2)1200 2501 y Fw(+)18 b Fr(j)p Fu(z)1345 2513 y Ft(2)1382 2501 y Fr(j)1405 2471 y Ft(2)1461 2501 y Fw(+)g Fu(:)c(:)g(:)k Fw(+)g Fr(j)p Fu(z)1804 2513 y Fs(n)1849 2501 y Fr(j)1872 2471 y Ft(2)1937 2501 y Fw(and)28 b Fu(z)2138 2513 y Fs(j)2195 2501 y Fw(=)23 b Fu(x)2330 2513 y Fs(j)2384 2501 y Fw(+)18 b Fu({y)2535 2513 y Fs(j)2597 2501 y Fw(,)28 b Fu(x)2695 2513 y Fs(j)2730 2501 y Fu(;)14 b(y)2808 2513 y Fs(j)2866 2501 y Fr(2)23 b Fv(R)p Fw(,)28 b(with)g Fu(j)g Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g(:)g(:)g(:)f(;)h(n)p Fw(.)648 2601 y(The)27 b(space)g Fr(B)1095 2613 y Fs(n)1167 2601 y Fw(is)h(a)f(Hilb)r(ert)h (space)f(with)h(inner)g(pro)r(duct)1501 2821 y Fr(h)p Fu(F)r(;)14 b(G)p Fr(i)1722 2833 y Fn(B)1766 2841 y Fp(n)1835 2821 y Fw(=)1923 2708 y Fo(Z)1969 2897 y Fq(C)2023 2880 y Fp(n)2082 2821 y Fu(F)e Fw(\()p Fv(z)p Fw(\))p 2253 2754 66 4 v Fu(G)q Fw(\()p Fv(z)p Fw(\))p Fu(d\027)2509 2833 y Fs(n)2556 2821 y Fw(\()p Fv(z)p Fw(\))p Fu(:)974 b Fw(\(2.2\))648 3084 y(The)29 b(set)h(of)f(all)g(monomials)1596 3038 y Fs(z)1630 3006 y Fp(a)1662 3018 y Fm(1)1628 3058 y(1)1699 3038 y Fs(z)1733 3006 y Fp(a)1765 3018 y Fm(2)1731 3058 y(2)1802 3038 y Fs(:::)o(z)1895 3013 y Fp(a)1927 3021 y(n)1893 3055 y(n)p 1593 3065 384 4 v 1593 3074 a Fn(p)p 1647 3074 330 3 v 1647 3120 a Fs(a)1683 3128 y Fm(1)1715 3120 y Ft(!)p Fs(a)1770 3128 y Fm(2)1803 3120 y Ft(!)p Fs(:::)o(a)1917 3128 y Fp(n)1957 3120 y Ft(!)1986 3084 y Fw(,)h(with)g Fu(a)2274 3096 y Ft(1)2311 3084 y Fu(;)14 b(a)2392 3096 y Ft(2)2429 3084 y Fu(;)g(:)g(:)g(:)g(;)g (a)2658 3096 y Fs(n)2732 3084 y Fw(non-negativ)n(e)28 b(in)n(teger)h(n)n(um)n(b)r(ers,)523 3195 y(is)f(an)f(orthonormal)e (basis)i(of)h Fr(B)1551 3207 y Fs(n)1596 3195 y Fw(.)648 3295 y(The)f(coheren)n(t)g(states)g(in)h Fr(B)1541 3307 y Fs(n)1509 3477 y Fw(\011)1574 3489 y Fq(a)1615 3477 y Fw(\()p Fv(z)p Fw(\))c Fr(\021)e Fw(exp\()p 1991 3431 47 4 v Fv(a)q(z)p Fw(\))95 b(with)48 b Fv(a)23 b Fr(2)h Fv(C)2633 3443 y Fs(n)3659 3477 y Fw(\(2.3\))523 3660 y(pro)n(vide)i(a)i(repro)r(ducing)e(k)n(ernel)h(for)g(the)h(space)f Fr(B)2139 3672 y Fs(n)2211 3660 y Fw(.)37 b(That)28 b(is,)f(for)g(\010) c Fr(2)h(B)2929 3672 y Fs(n)1239 3881 y Fw(\010\()p Fv(z)p Fw(\))g(=)f Fr(h)p Fw(\010)p Fu(;)14 b Fw(\011)1711 3893 y Fq(z)1748 3881 y Fr(i)1780 3893 y Fn(B)1824 3901 y Fp(n)1892 3881 y Fw(=)1980 3768 y Fo(Z)2026 3956 y Fq(C)2080 3940 y Fp(n)2139 3881 y Fw(exp\()p Fv(z)p 2340 3834 71 4 v(w)r Fw(\)\010\()p Fv(w)q Fw(\))p Fu(d\027)2721 3893 y Fs(n)2767 3881 y Fw(\()p Fv(w)q Fw(\))24 b Fu(:)711 b Fw(\(2.4\))648 4115 y(The)32 b(Bargmann)f(T)-7 b(ransform)31 b(for)i Fu(L)1830 4085 y Ft(2)1866 4115 y Fw(\()p Fv(R)1970 4085 y Fs(n)2015 4115 y Fw(\))g(is)g(de\014ned)g(as)f(the)h(op)r (erator)e Fv(B)3123 4127 y Fq(R)3179 4111 y Fp(n)3255 4115 y Fw(:)g Fu(L)3366 4085 y Ft(2)3403 4115 y Fw(\()p Fv(R)3507 4085 y Fs(n)3552 4115 y Fw(\))h Fr(7!)f(B)3785 4127 y Fs(n)523 4215 y Fw(giv)n(en)c(b)n(y)1139 4435 y Fv(B)1207 4447 y Fq(R)1263 4431 y Fp(n)1307 4435 y Fw(\011\()p Fv(z)p Fw(\))d Fr(\021)1660 4379 y Fw(1)p 1599 4416 163 4 v 1599 4494 a Fu(\031)1649 4470 y Fs(n=)p Ft(4)1786 4322 y Fo(Z)1832 4511 y Fq(R)1888 4494 y Fp(n)1946 4435 y Fu(A)p Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\)\011\()p Fv(x)p Fw(\))p Fu(d)p Fv(x)p Fu(;)111 b Fw(\011)23 b Fr(2)g Fu(L)2830 4401 y Ft(2)2867 4435 y Fw(\()p Fv(R)2971 4401 y Fs(n)3016 4435 y Fw(\))611 b(\(2.5\))523 4660 y(where)27 b Fu(d)p Fv(x)h Fw(is)g(the)g(usual)f(Leb)r(esgue)g(measure) g(on)g Fv(R)2198 4630 y Fs(n)2270 4660 y Fw(and)1370 4877 y Fu(A)p Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b Fr(\021)f Fw(exp\()p Fr(\000)1971 4821 y Fw(1)p 1971 4858 42 4 v 1971 4934 a(2)2022 4877 y(\()p Fv(z)2096 4843 y Ft(2)2153 4877 y Fw(+)18 b Fv(x)2286 4843 y Ft(2)2323 4877 y Fw(\))h(+)2457 4804 y Fr(p)p 2526 4804 V 73 x Fw(2)p Fv(z)g Fr(\017)f Fv(x)p Fw(\))23 b Fu(;)842 b Fw(\(2.6\))523 5093 y(with)34 b Fv(z)760 5063 y Ft(2)831 5093 y Fw(=)f Fu(z)972 5063 y Ft(2)968 5113 y(1)1031 5093 y Fw(+)22 b Fu(z)1161 5063 y Ft(2)1157 5113 y(2)1220 5093 y Fw(+)h Fu(:)14 b(:)g(:)22 b Fw(+)g Fu(z)1557 5063 y Ft(2)1553 5113 y Fs(n)1598 5093 y Fw(,)35 b Fv(x)f Fw(=)f(\()p Fu(x)1917 5105 y Ft(1)1955 5093 y Fu(;)14 b(x)2039 5105 y Ft(2)2077 5093 y Fu(;)g(:)g(:)g(:)f(;)h(x)2308 5105 y Fs(n)2354 5093 y Fw(\),)35 b Fv(x)2494 5063 y Ft(2)2565 5093 y Fw(=)e Fu(x)2710 5063 y Ft(2)2710 5113 y(1)2771 5093 y Fw(+)22 b Fu(x)2905 5063 y Ft(2)2905 5113 y(2)2965 5093 y Fw(+)g Fu(:)14 b(:)g(:)23 b Fw(+)f Fu(x)3306 5063 y Ft(2)3306 5113 y Fs(n)3385 5093 y Fw(and)34 b Fv(z)23 b Fr(\017)f Fv(x)34 b Fw(=)523 5192 y Fu(z)562 5204 y Ft(1)599 5192 y Fu(x)646 5204 y Ft(1)694 5192 y Fw(+)11 b Fu(z)809 5204 y Ft(2)846 5192 y Fu(x)893 5204 y Ft(2)941 5192 y Fw(+)g Fu(:)j(:)g(:)c Fw(+)h Fu(z)1239 5204 y Fs(n)1283 5192 y Fu(x)1330 5204 y Fs(n)1376 5192 y Fw(.)35 b(The)24 b(Bargmann)e(transform)h Fv(B)2448 5204 y Fq(R)2504 5188 y Fp(n)2572 5192 y Fw(is)h(a)f(unitary)g(op)r (erator)f(\(with)j Fu(L)3612 5162 y Ft(2)3649 5192 y Fw(\()p Fv(R)3753 5162 y Fs(n)3798 5192 y Fw(\))523 5292 y(endo)n(w)n(ed)i(with)h(the)g(usual)f(inner)g(pro)r(duct\).)648 5392 y(The)g(in)n(v)n(erse)f Fv(B)1161 5356 y Fn(\000)p Ft(1)1161 5416 y Fq(R)1217 5400 y Fp(n)1290 5392 y Fw(is)h(giv)n(en)g (b)n(y)1310 5621 y Fv(B)1378 5585 y Fn(\000)p Ft(1)1378 5645 y Fq(R)1434 5629 y Fp(n)1478 5621 y Fu(f)9 b Fw(\()p Fv(x)p Fw(\))24 b(=)66 b(lim)1754 5674 y Fs(M)6 b Fn(7!1)1969 5508 y Fo(Z)2015 5696 y Fn(j)p Fq(z)p Fn(j\024)p Fs(M)p 2228 5548 257 4 v 2228 5621 a Fu(A)p Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))q Fu(f)9 b Fw(\()p Fv(z)p Fw(\))p Fu(d\027)2724 5633 y Fs(n)2770 5621 y Fw(\()p Fv(z)p Fw(\))783 b(\(2.7\))p eop %%Page: 7 7 7 6 bop 3789 324 a Fw(7)523 614 y(Here)31 b(w)n(e)g(need)h(to)f(tak)n (e)g(the)h(limit)g(b)r(ecause)f(the)h(in)n(tegral)e(o)n(v)n(er)g(all)h (the)h(space)f Fv(C)3192 584 y Fs(n)3268 614 y Fw(migh)n(t)h(not)f (exist)523 714 y(\(note)d(that,)g(for)f(\014xed)g Fv(x)p Fw(,)i Fu(A)p Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b Fr(62)g(B)1785 726 y Fs(n)1829 714 y Fw(\).)523 884 y Fv(I)s(I.2)31 b(Bargmann)g(T)-8 b(ransform)32 b(for)g Fu(L)1837 854 y Ft(2)1874 884 y Fw(\()p Fu(S)1962 854 y Ft(2)1999 884 y Fw(\))648 1031 y(Let)c(us)g(\014rst)f(de\014ne)h(the) h(space)e Fu(L)1738 1001 y Ft(2)1775 1031 y Fw(\()p Fu(S)1863 1001 y Fs(n)1908 1031 y Fw(\))h(for)g(a)f(giv)n(en)g(in)n(teger)g(n)n (um)n(b)r(er)h Fu(n)p Fw(.)38 b(Let)28 b Fu(d\033)3310 1043 y Fs(n)3384 1031 y Fw(b)r(e)g(the)h(usual)523 1131 y(normalized)24 b(Leb)r(esgue)h(surface)f(measure)h(on)g Fu(S)2070 1101 y Fs(n)2115 1131 y Fw(.)36 b(Let)25 b(us)h(de\014ne)f Fu(L)2719 1101 y Ft(2)2756 1131 y Fw(\()p Fu(S)2844 1101 y Fs(n)2889 1131 y Fw(\))h(as)e(the)i(space)e(of)i(Leb)r(esgue)523 1231 y(measurable)g(functions)i(on)g Fu(S)1487 1200 y Fs(n)1559 1231 y Fw(whic)n(h)g(are)e(square)h(in)n(tegrable)f(with)i (resp)r(ect)f(to)h(the)g(measure)e Fu(d\033)3713 1243 y Fs(n)3759 1231 y Fw(.)648 1330 y(The)h(space)g Fu(L)1097 1300 y Ft(2)1134 1330 y Fw(\()p Fu(S)1222 1300 y Fs(n)1267 1330 y Fw(\))h(is)f(a)h(Hilb)r(ert)g(space)f(with)h(inner)f(pro)r(duct) 1480 1555 y Fr(h)p Fu(F)r(;)14 b(G)p Fr(i)1701 1567 y Fs(S)1745 1551 y Fp(n)1814 1555 y Fw(=)1902 1442 y Fo(Z)1948 1631 y Fs(S)1992 1614 y Fp(n)2050 1555 y Fu(F)e Fw(\()p Fv(x)p Fw(\))p 2229 1489 66 4 v Fu(G)r Fw(\()p Fv(x)p Fw(\))p Fu(d\033)2500 1567 y Fs(n)2546 1555 y Fw(\()p Fv(x)p Fw(\))24 b Fu(;)952 b Fw(\(2.8\))648 1790 y(In)27 b(order)e(to)i(de\014ne)g(the)g(Bargmann)e(transform)g(for)i Fu(L)2417 1760 y Ft(2)2453 1790 y Fw(\()p Fu(S)2541 1760 y Ft(2)2579 1790 y Fw(\),)g(W)-7 b(e)27 b(\014rst)g(remind)f(the)h (reader)f(some)523 1889 y(w)n(ell)h(kno)n(wn)g(facts)g(ab)r(out)h(t)n (w)n(o)e(di\013eren)n(t)i(represen)n(tations)d(of)j(the)g Fu(S)5 b(O)r Fw(\(3\))28 b(group)e(of)h(rotations)f(of)i(the)523 1989 y(space)f Fv(R)817 1959 y Ft(3)881 1989 y Fw(.)648 2089 y(Let)35 b(us)h(consider)f(the)h(Laplacian)e(op)r(erator)g(on)h (the)h(2-sphere)e Fu(S)2798 2059 y Ft(2)2871 2089 y Fw(denoted)i(b)n(y) f(\001)3385 2104 y Fs(S)3429 2087 y Fm(2)3466 2089 y Fw(.)61 b(W)-7 b(e)36 b(will)523 2188 y(actually)g(consider)g(the)h (normalized)e(Laplacian)h(op)r(erator)e(suc)n(h)j(that)g(its)f(sp)r (ectrum)h(is)g(the)g(set)f(of)523 2288 y(n)n(um)n(b)r(ers)858 2221 y Fo(\010)907 2288 y Fw(\()p Fu(`)18 b Fw(+)g(1)p Fu(=)p Fw(2\))1233 2258 y Ft(2)1269 2288 y Fr(j)p Fu(`)23 b Fw(=)f(0)p Fu(;)14 b Fw(1)p Fu(;)g Fw(2)g Fu(:)g(:)g(:)1746 2221 y Fo(\011)648 2388 y Fw(If)28 b(w)n(e)f(in)n(tro)r(duce)g (spherical)g(co)r(ordinates)f(on)h(the)h(2-sphere)654 2570 y Fu(x)701 2582 y Ft(1)761 2570 y Fw(=)23 b(sin\()p Fu(\022)r Fw(\))14 b(cos\()p Fu(\036)p Fw(\))p Fu(;)84 b(x)1449 2582 y Ft(2)1510 2570 y Fw(=)22 b(sin\()p Fu(\022)r Fw(\))14 b(sin)q(\()p Fu(\036)p Fw(\))p Fu(;)84 b(x)2188 2582 y Ft(3)2249 2570 y Fw(=)22 b(cos\()p Fu(\022)r Fw(\))93 b(0)22 b Fu(<)h(\022)i(<)e(\031)s(;)60 b Fw(0)23 b Fr(\024)f Fu(\036)i Fr(\024)f Fw(2)p Fu(\031)i(;)523 2753 y Fw(the)j(expression)e (for)h(the)h(Laplacian)f(on)g Fu(S)1887 2723 y Ft(2)1951 2753 y Fw(in)h(spherical)f(co)r(ordinates)f(is)1243 3068 y(\001)1312 3083 y Fs(S)1356 3066 y Fm(2)1415 3068 y Fw(=)d Fr(\000)1661 3011 y Fw(1)p 1578 3049 208 4 v 1578 3125 a(sin\()p Fu(\022)r Fw(\))1826 3011 y Fu(@)p 1805 3049 90 4 v 1805 3125 a(@)5 b(\022)1919 3068 y Fw(sin\()p Fu(\022)r Fw(\))2157 3011 y Fu(@)p 2136 3049 V 2136 3125 a(@)g(\022)2255 3068 y Fr(\000)2450 3011 y Fw(1)p 2348 3049 246 4 v 2348 3131 a(sin)2450 3096 y Ft(2)2487 3131 y Fw(\()p Fu(\022)r Fw(\))2638 3011 y Fu(@)2687 2981 y Ft(2)p 2613 3049 136 4 v 2613 3125 a Fu(@)g(\036)2711 3101 y Ft(2)2800 3068 y Fw(+)2893 3011 y(1)p 2893 3049 42 4 v 2893 3125 a(4)3659 3068 y(\(2.9\))648 3301 y(The)28 b(Laplacian)f(\001)1268 3316 y Fs(S)1312 3299 y Fm(2)1377 3301 y Fw(is)h(in)h(turn)f(related)g(to)g(the)h(addition)f(of)g(the)h (square)e(of)h(the)h(generators)d Fu(L)3796 3313 y Fs(j)523 3400 y Fw(of)i(rotations)e(in)i Fv(R)1138 3370 y Ft(3)1202 3400 y Fw(around)f(the)h(plane)f(p)r(erp)r(endicular)g(to)h(the)g(axis) f Fu(x)2835 3412 y Fs(j)2898 3400 y Fw(for)g Fu(j)h Fw(=)23 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3:)1608 3583 y(\001)1677 3598 y Fs(S)1721 3581 y Fm(2)1781 3583 y Fw(=)23 b Fu(L)1926 3549 y Ft(2)1926 3603 y(1)1981 3583 y Fw(+)18 b Fu(L)2121 3549 y Ft(2)2121 3603 y(2)2176 3583 y Fw(+)g Fu(L)2316 3549 y Ft(2)2316 3603 y(3)2371 3583 y Fw(+)g(1)p Fu(=)p Fw(4)1038 b(\(2.10\))523 3766 y(where)1411 3971 y Fu(L)1468 3983 y Ft(1)1588 3971 y Fw(=)82 b Fu({)14 b Fw(sin\()p Fu(\036)p Fw(\))2023 3914 y Fu(@)p 2001 3952 90 4 v 2001 4028 a(@)5 b(\022)2120 3971 y Fw(+)18 b Fu({)c Fw(cot\()p Fu(\022)r Fw(\))g(cos)o(\()p Fu(\036)p Fw(\))2734 3914 y Fu(@)p 2708 3952 99 4 v 2708 4028 a(@)5 b(\036)2841 3971 y(;)1411 4191 y(L)1468 4203 y Ft(2)1588 4191 y Fw(=)82 b Fr(\000)p Fu({)14 b Fw(cos)o(\()p Fu(\036)p Fw(\))2096 4135 y Fu(@)p 2075 4172 90 4 v 2075 4248 a(@)5 b(\022)2194 4191 y Fw(+)18 b Fu({)c Fw(cot)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))2799 4135 y Fu(@)p 2773 4172 99 4 v 2773 4248 a(@)5 b(\036)2905 4191 y(;)1411 4411 y(L)1468 4423 y Ft(3)1588 4411 y Fw(=)82 b Fr(\000)p Fu({)1861 4355 y(@)p 1837 4392 V 1837 4468 a(@)5 b(\036)1945 4411 y(:)1650 b Fw(\(2.11\))648 4643 y(The)24 b(op)r(erators)d Fu(L)1235 4655 y Fs(j)1294 4643 y Fw(act)i(on)h(a)g(dense)f(subspace)g(of)h Fu(L)2319 4613 y Ft(2)2356 4643 y Fw(\()p Fu(S)2444 4613 y Ft(2)2481 4643 y Fw(\))g(and)g(pro)n(vide)f(a)g(represen)n(tation)f (for)i(the)523 4743 y(Lie)k(algebra)d Fu(so)p Fw(\(3\))j(of)g Fu(S)5 b(O)r Fw(\(3\).)37 b(They)28 b(also)e(satisfy)h(the)h(comm)n (utation)f(relations)1421 4925 y([)p Fu(L)1501 4937 y Fs(i)1528 4925 y Fu(;)14 b(L)1622 4937 y Fs(j)1657 4925 y Fw(])23 b(=)g Fu({\017)1852 4937 y Fs(ij)s(k)1946 4925 y Fu(L)2003 4937 y Fs(k)2228 4925 y Fu(i;)14 b(j;)g(k)26 b Fw(=)c(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)22 b Fu(;)852 b Fw(\(2.12\))523 5108 y(where)28 b Fr(f)p Fu(i;)14 b(j;)g(k)s Fr(g)27 b Fw(is)i(a)f(p)r(erm)n(utation)g(of)h Fr(f)p Fw(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)p Fr(g)p Fw(.)37 b(If)29 b(this)g(p)r(erm)n(utation)f(is)g(ev)n(en)g(\(o)r(dd\))i(then)f Fu(\017)3580 5120 y Fs(ij)s(k)3699 5108 y Fw(=)c(1)523 5207 y(\()p Fu(\017)589 5219 y Fs(ij)s(k)707 5207 y Fw(=)e Fr(\000)p Fw(1\).)648 5307 y(It)36 b(is)g(kno)n(wn)f(that)h(the)g (eigenfunctions)g(of)g(the)g(Laplacian)f(\001)2710 5322 y Fs(S)2754 5305 y Fm(2)2827 5307 y Fw(pro)n(vide)g(a)g(basis)g(for)h Fu(L)3613 5277 y Ft(2)3650 5307 y Fw(\()p Fu(S)3738 5277 y Ft(2)3775 5307 y Fw(\).)523 5407 y(Moreo)n(v)n(er,)k Fu(L)983 5377 y Ft(2)1020 5407 y Fw(\()p Fu(S)1108 5377 y Ft(2)1145 5407 y Fw(\))g(is)g(the)g(direct)g(sum)g(of)g(the)g (eigenspaces)e(of)i(the)g(Laplacian.)72 b(F)-7 b(or)40 b(eac)n(h)f Fu(`)k Fw(=)523 5506 y(0)p Fu(;)14 b Fw(1)p Fu(;)g Fw(2)g Fu(:)g(:)g(:)n Fw(,)44 b(w)n(e)c(shall)g(denote)h(b)n(y)f Fu(V)1698 5518 y Fs(`)1771 5506 y Fw(the)h(eigenspace)e(of)i(the)g (Laplacian)e(\001)3071 5521 y Fs(S)3115 5504 y Fm(2)3192 5506 y Fw(asso)r(ciated)g(to)i(the)523 5606 y(eigen)n(v)-5 b(alue)27 b(\()p Fu(`)18 b Fw(+)g(1)p Fu(=)p Fw(2\))1246 5576 y Ft(2)1282 5606 y Fw(.)p eop %%Page: 8 8 8 7 bop 3789 324 a Fw(8)648 614 y(The)25 b(group)f Fu(S)5 b(O)r Fw(\(3\))26 b(acts)f(on)f Fu(L)1640 584 y Ft(2)1677 614 y Fw(\()p Fu(S)1765 584 y Ft(2)1802 614 y Fw(\))i(as)f(follo)n(ws.) 35 b(Giv)n(en)25 b Fu(R)f Fr(2)f Fu(S)5 b(O)r Fw(\(3\),)26 b(w)n(e)f(de\014ne)h Fu(T)3350 626 y Fs(R)3427 614 y Fw(:)d Fu(L)3530 584 y Ft(2)3567 614 y Fw(\()p Fu(S)3655 584 y Ft(2)3692 614 y Fw(\))g Fr(7!)523 714 y Fu(L)580 684 y Ft(2)617 714 y Fw(\()p Fu(S)705 684 y Ft(2)742 714 y Fw(\))28 b(b)n(y)1496 877 y Fu(T)1545 889 y Fs(R)1599 877 y Fw(\011\()p Fv(x)p Fw(\))c(=)e(\011\()p Fu(R)2050 843 y Fn(\000)p Ft(1)2139 877 y Fv(x)p Fw(\))p Fu(;)156 b Fv(x)24 b Fr(2)f Fu(S)2608 843 y Ft(2)2668 877 y Fu(:)927 b Fw(\(2.13\))648 1040 y(A)32 b(particular)f(and)h(imp)r(ortan)n(t)f (basis)h(of)g Fu(L)2057 1010 y Ft(2)2093 1040 y Fw(\()p Fu(S)2181 1010 y Ft(2)2218 1040 y Fw(\))h(is)f(obtained)g(b)n(y)f (taking)h(common)f(normalized)523 1140 y(eigenfunctions)f(of)g(b)r(oth) g(op)r(erators)e(\001)1802 1155 y Fs(S)1846 1138 y Fm(2)1913 1140 y Fw(and)h Fu(L)2133 1152 y Ft(3)2200 1140 y Fw(\(these)h(t)n(w)n (o)f(op)r(erators)f(comm)n(ute\).)44 b(These)30 b(func-)523 1239 y(tions)d(are)g(kno)n(wn)g(as)g(\\spherical)f(harmonics")g(and)h (w)n(e)h(shall)f(denote)g(them)h(b)n(y)1137 1402 y Fu(Y)1185 1414 y Fs(`m)1322 1402 y Fw(with)46 b Fu(`)23 b Fw(=)g(0)p Fu(;)14 b Fw(1)p Fu(;)g Fw(2)g Fu(:)g(:)g(:)58 b Fw(and)45 b Fu(m)23 b Fw(=)g Fr(\000)p Fu(`;)14 b Fr(\000)p Fu(`)j Fw(+)h(1)p Fu(;)c(:)g(:)g(:)f(;)h(`)22 b(;)568 b Fw(\(2.14\))523 1565 y(Moreo)n(v)n(er,)25 b(if,)j(for)f(eac)n(h)g Fu(`)c Fw(=)f(0)p Fu(;)14 b Fw(1)p Fu(;)g Fw(2)p Fu(;)g(:)g(:)g(:)n Fw(,)28 b(w)n(e)f(let)h(the)g(group)f Fr(f)p Fu(T)2564 1577 y Fs(R)2617 1565 y Fr(j)p Fu(R)d Fr(2)f Fu(S)5 b(O)r Fw(\(3\))q Fr(g)27 b Fw(to)g(act)h(on)f(the)h(space)1361 1728 y Fu(V)1409 1740 y Fs(`)1464 1728 y Fw(=)23 b Fu(span)14 b Fr(f)o Fu(Y)1830 1740 y Fs(`m)1921 1728 y Fr(j)p Fu(m)23 b Fw(=)f Fr(\000)p Fu(`;)14 b Fr(\000)p Fu(`)j Fw(+)h(1)p Fu(;)c(:)g(:)g(:)f(;)h(`)p Fr(g)36 b Fu(;)792 b Fw(\(2.15\))523 1891 y(then)28 b(w)n(e)f(obtain)h(all)f(the)h(irreducible)f(represen)n (tations)f(of)h Fu(S)5 b(O)r Fw(\(3\).)648 1991 y(In)26 b(the)h(other)f(hand,)h(the)g(group)e Fu(S)5 b(O)r Fw(\(3\))27 b(can)f(b)r(e)h(regarded)e(as)h(a)g(subgroup)g(of)g Fu(S)5 b(U)k Fw(\(2\))26 b(\(the)h(group)523 2091 y(of)h(t)n(w)n(o)f(b)n(y)h (t)n(w)n(o)f(unitary)g(matrices)g(with)i(complex)e(en)n(tries)g(and)h (determinan)n(t)g(one\).)38 b(W)-7 b(e)28 b(can)g(obtain)523 2190 y(irreducible)33 b(represen)n(tations)f(for)h Fu(S)5 b(O)r Fw(\(3\))35 b(on)e(base)h(of)f(the)i(irreducible)e(represen)n (tations)f(of)i Fu(S)5 b(U)k Fw(\(2\).)523 2290 y(An)35 b(imp)r(ortan)n(t)f(represen)n(tation)f(of)h(the)h Fu(S)5 b(U)k Fw(\(2\))34 b(group)f(is)h(obtained)g(b)n(y)h(letting)f(this)h (group)e(to)h(act)523 2390 y(on)c(the)g(Bargmann)e(space)h Fr(B)1472 2402 y Ft(2)1539 2390 y Fw(of)h(analytic)f(functions)h(in)g (t)n(w)n(o)f(complex)g(v)-5 b(ariables)29 b Fv(z)e Fw(=)f(\()p Fu(z)3477 2402 y Ft(1)3515 2390 y Fu(;)14 b(z)3591 2402 y Ft(2)3627 2390 y Fw(\))31 b(in)f(a)523 2489 y(similar)d(w)n(a)n(y)f (as)h(w)n(e)g(made)h(the)g Fu(S)5 b(O)r Fw(\(3\))28 b(group)e(to)i(act) f(on)g Fu(L)2455 2459 y Ft(2)2492 2489 y Fw(\()p Fu(S)2580 2459 y Ft(2)2617 2489 y Fw(\).)648 2589 y(Let)g(us)g(denote)h(b)n(y)f Fv(S)g Fw(the)h(set)g(of)f(all)g(n)n(um)n(b)r(ers)g(of)g(the)h(form)f Fu(p=)p Fw(2)f(with)i Fu(p)g Fw(a)f(non)g(negativ)n(e)f(in)n(teger)523 2688 y(n)n(um)n(b)r(er.)36 b(F)-7 b(or)24 b Fu(s)p Fr(2)q Fv(S)p Fw(,)h(let)h(us)f(also)f(denote)h(b)n(y)f Fr(W)2044 2700 y Fs(s)2105 2688 y Fw(the)i(subspace)e(of)h Fr(B)2737 2700 y Ft(2)2799 2688 y Fw(of)g(homogeneous)e(p)r(olynomials)523 2788 y(in)k(t)n(w)n(o)g(complex)f(v)-5 b(ariables)26 b(\()p Fu(z)1515 2800 y Ft(1)1552 2788 y Fu(;)14 b(z)1628 2800 y Ft(2)1665 2788 y Fw(\))27 b(of)g(degree)f(2)p Fu(s)p Fw(.)36 b(An)28 b(orthonormal)d(basis)h(for)g Fr(W)3238 2800 y Fs(s)3301 2788 y Fw(is)h(giv)n(en)f(b)n(y)h(the)523 2888 y(monomials:)1595 3087 y Fu(y)1636 3099 y Fs(sm)1753 3087 y Fw(=)2005 3031 y Fu(z)2048 2995 y Fs(s)p Ft(+)p Fs(m)2044 3053 y Ft(1)2193 3031 y Fu(z)2236 2995 y Fs(s)p Fn(\000)p Fs(m)2232 3053 y Ft(2)p 1851 3068 685 4 v 1851 3085 a Fo(p)p 1934 3085 602 4 v 71 x Fw(\()p Fu(s)19 b Fw(+)f Fu(m)p Fw(\)!\()p Fu(s)h Fr(\000)f Fu(m)p Fw(\)!)2569 3087 y Fu(;)1026 b Fw(\(2.16\))523 3310 y(with)28 b Fu(m)23 b Fw(=)g Fr(\000)p Fu(s;)14 b Fr(\000)p Fu(s)j Fw(+)h(1)p Fu(;)c(:)g(:)g(:)f(;)h(s)k Fr(\000)g Fw(1)p Fu(;)c(s)p Fw(.)648 3410 y(When)26 b(w)n(e)g(restrict)g(the)h(action)e(of)i Fu(S)5 b(U)k Fw(\(2\))26 b(to)g(the)h(spaces)e Fr(W)2609 3422 y Fs(s)2644 3410 y Fw(,)i(w)n(e)f(get)g(all)g(the)h(irreducible)e (repre-)523 3510 y(sen)n(tations)i(of)i Fu(S)5 b(U)k Fw(\(2\).)38 b(In)28 b(particular,)f(when)i(w)n(e)f(tak)n(e)56 b Fu(s)24 b Fw(=)f Fu(`)28 b Fw(a)g(non)g(negativ)n(e)f(in)n(teger)g(n) n(um)n(b)r(er,)i(w)n(e)523 3609 y(obtain)e(all)h(the)g(irreducible)f (represen)n(tations)e(of)j Fu(S)5 b(O)r Fw(\(3\).)648 3709 y(In)27 b(order)g(to)g(de\014ne)h(a)f(Bargmann)g(T)-7 b(ransform)26 b(for)h Fu(L)2371 3679 y Ft(2)2408 3709 y Fw(\()p Fu(S)2496 3679 y Ft(2)2533 3709 y Fw(\),)h(the)g(description) f(ab)r(o)n(v)n(e)f(of)i(t)n(w)n(o)f(t)n(yp)r(e)523 3808 y(of)h(irreducible)f(represen)n(tations)e(of)j Fu(S)5 b(O)r Fw(\(3\))28 b(suggests)e(to)i(de\014ne)f(the)h(follo)n(wing)f (assignmen)n(t:)922 3971 y Fu(Y)970 3983 y Fs(`m)1084 3971 y Fr(7\000)-14 b(!)23 b Fu(y)1282 3983 y Fs(`m)1442 3971 y Fw(for)f(all)46 b Fu(`)23 b Fw(=)f(0)p Fu(;)14 b Fw(1)p Fu(;)g Fw(2)p Fu(;)g(:)g(:)g(:)81 b Fw(and)46 b Fu(m)23 b Fw(=)g Fr(\000)p Fu(`;)14 b Fr(\000)p Fu(`)i Fw(+)i(1)p Fu(;)c(:)g(:)g(:)g(;)g(`)22 b(:)353 b Fw(\(2.17\))523 4134 y(Th)n(us)37 b(w)n(e)f(de\014ne)h(the)h(Bargmann)d(T)-7 b(ransform)36 b Fv(B)2170 4149 y Fs(S)2214 4133 y Fm(2)2287 4134 y Fw(for)g Fu(L)2480 4104 y Ft(2)2517 4134 y Fw(\()p Fu(S)2605 4104 y Ft(2)2642 4134 y Fw(\))i(as)e(the)h(linear)g (extension)f(of)h(this)523 4234 y(assignmen)n(t.)52 b(Notice)33 b(that)g Fv(B)1518 4249 y Fs(S)1562 4232 y Fm(2)1632 4234 y Fw(is)g(a)f(unitary)h(op)r(erator)e(from)i Fu(L)2693 4204 y Ft(2)2729 4234 y Fw(\()p Fu(S)2817 4204 y Ft(2)2855 4234 y Fw(\))g(on)n(to)f(the)i(closed)e(subspace)523 4334 y Fr(B)578 4349 y Fs(S)622 4332 y Fm(2)692 4334 y Fw(of)i Fr(B)848 4346 y Ft(2)919 4334 y Fw(generated)f(b)n(y)h(the)h (monomials)e(with)i(ev)n(en)e(degree,)i(that)g(is)f Fr(B)3011 4349 y Fs(S)3055 4332 y Fm(2)3125 4334 y Fw(=)f Fr(\010W)3370 4346 y Fs(`)3436 4334 y Fw(\(where)h(the)523 4433 y(direct)27 b(sum)h(is)g(tak)n(en)f(o)n(v)n(er)f(all)h(non)g(negativ)n(e)g(in)n (teger)f(n)n(um)n(b)r(ers\).)648 4533 y(The)h(Bargmann)f(transform)h Fv(B)1673 4548 y Fs(S)1717 4531 y Fm(2)1781 4533 y Fw(can)g(b)r(e)h (written)g(as)f(an)g(in)n(tegral)f(op)r(erator)1359 4734 y Fv(B)1427 4749 y Fs(S)1471 4732 y Fm(2)1507 4734 y Fw(\011\()p Fv(z)p Fw(\))e(=)1790 4621 y Fo(Z)1836 4809 y Fq(x)p Fn(2)p Fs(S)1965 4793 y Fm(2)2014 4734 y Fu(K)2085 4749 y Fs(S)2129 4732 y Fm(2)2165 4734 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\)\011\()p Fv(x)p Fw(\))p Fu(d\033)2627 4746 y Ft(2)2667 4734 y Fw(\()p Fv(x)p Fw(\))24 b Fu(;)790 b Fw(\(2.18\))523 4943 y(with)28 b(k)n(ernel)1418 5167 y Fu(K)1489 5182 y Fs(S)1533 5166 y Fm(2)1569 5167 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))25 b(=)1901 5064 y Fn(1)1874 5089 y Fo(X)1878 5267 y Fs(`)p Ft(=0)2034 5064 y Fs(m)p Ft(=)p Fs(`)2043 5089 y Fo(X)2008 5267 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 2212 5101 67 4 v 2212 5167 a Fu(Y)2278 5179 y Fs(`m)2369 5167 y Fw(\()p Fv(x)p Fw(\))p Fu(y)2524 5179 y Fs(`m)2616 5167 y Fw(\()p Fv(z)p Fw(\))f Fu(:)849 b Fw(\(2.19\))523 5412 y(This)28 b(k)n(ernel)e(has)h (the)h(prop)r(ert)n(y)f(of)g(b)r(eing)h(a)f(p)r(o)n(w)n(er)g(series)f (of)i(the)g(function)1506 5649 y(\010)1566 5664 y Fs(S)1610 5647 y Fm(2)1646 5649 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)1977 5545 y Fs(m)p Ft(=1)1989 5570 y Fo(X)1951 5746 y Fs(m)p Ft(=)p Fn(\000)p Ft(1)p 2160 5583 V 2160 5649 a Fu(Y)2226 5661 y Ft(1)p Fs(m)2323 5649 y Fw(\()p Fv(x)p Fw(\))p Fu(y)2478 5661 y Ft(1)p Fs(m)2574 5649 y Fw(\()p Fv(z)p Fw(\))938 b(\(2.20\))p eop %%Page: 9 9 9 8 bop 3789 324 a Fw(9)523 614 y(in)28 b(the)g(follo)n(wing)e(w)n(a)n (y)1243 859 y Fu(K)1314 874 y Fs(S)1358 857 y Fm(2)1394 859 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)1726 755 y Fn(1)1699 780 y Fo(X)1703 959 y Fs(`)p Ft(=0)1842 735 y Fr(p)p 1912 735 220 4 v 1912 803 a Fw(2)p Fu(`)17 b Fw(+)h(1)p 1842 840 289 4 v 1958 916 a Fu(`)p Fw(!)2155 742 y Fo(\022)2260 803 y Fw(1)p 2226 840 111 4 v 2226 856 a Fr(p)p 2295 856 42 4 v 69 x Fw(3)2346 742 y Fo(\023)2407 759 y Fs(`)2453 859 y Fw(\(\010)2545 874 y Fs(S)2589 857 y Fm(2)2626 859 y Fw(\()p Fv(x)p Fu(;)c Fv(z)p Fw(\)\))2852 817 y Fs(`)2921 859 y Fu(;)674 b Fw(\(2.21\))523 1117 y(with)28 b(the)g(follo)n(wing)f(explicit)h(expression)e(for)h(\010) 2086 1132 y Fs(S)2130 1115 y Fm(2)2166 1117 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\):)754 1358 y(\010)814 1373 y Fs(S)858 1356 y Fm(2)894 1358 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))25 b(=)1209 1233 y Fr(p)p 1278 1233 V 68 x Fw(3)p 1209 1338 111 4 v 1244 1415 a(2)1344 1290 y Fo(\000)1382 1358 y Fr(\000)14 b Fw(sin)o(\()p Fu(\022)r Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))p Fu(z)2057 1323 y Ft(2)2053 1378 y(1)2112 1358 y Fw(+)k(2)c(cos)n(\()p Fu(\022)r Fw(\))p Fu(z)2505 1370 y Ft(1)2543 1358 y Fu(z)2582 1370 y Ft(2)2637 1358 y Fw(+)k(sin\()p Fu(\022)r Fw(\))c(exp)q(\()p Fu({\036)p Fw(\))p Fu(z)3252 1323 y Ft(2)3248 1378 y(2)3289 1290 y Fo(\001)3364 1358 y Fu(:)231 b Fw(\(2.22\))523 1560 y(W)-7 b(e)28 b(giv)n(e)f(a)g(pro)r(of)g(of)g(this)h(fact)g(in)g (app)r(endix)g(A.)648 1660 y(The)f(Bargmann)f(transform)h Fv(B)1673 1675 y Fs(S)1717 1658 y Fm(2)1781 1660 y Fw(in)n(tert)n (wines)g Fu(L)2263 1672 y Ft(1)2299 1660 y Fu(;)14 b(L)2393 1672 y Ft(2)2430 1660 y Fu(;)g(L)2524 1672 y Ft(3)2561 1660 y Fu(;)g Fw(\001)2667 1630 y Ft(2)2667 1683 y Fs(S)2743 1660 y Fw(with)28 b(the)g(follo)n(wing)e(op)r(erators)1511 1891 y Fu(L)1568 1857 y Fn(0)1568 1912 y Ft(1)1688 1891 y Fw(=)1846 1835 y(1)p 1846 1872 42 4 v 1846 1948 a(2)1911 1774 y Fo(\022)1972 1891 y Fu(z)2011 1903 y Ft(1)2096 1835 y Fu(@)p 2058 1872 125 4 v 2058 1948 a(@)5 b(z)2146 1960 y Ft(2)2211 1891 y Fw(+)18 b Fu(z)2333 1903 y Ft(2)2418 1835 y Fu(@)p 2380 1872 V 2380 1948 a(@)5 b(z)2468 1960 y Ft(1)2514 1774 y Fo(\023)1511 2124 y Fu(L)1568 2089 y Fn(0)1568 2144 y Ft(2)1688 2124 y Fw(=)1853 2067 y Fu({)p 1846 2105 42 4 v 1846 2181 a Fw(2)1911 2007 y Fo(\022)1972 2124 y Fu(z)2011 2136 y Ft(1)2096 2067 y Fu(@)p 2058 2105 125 4 v 2058 2181 a(@)g(z)2146 2193 y Ft(2)2211 2124 y Fr(\000)18 b Fu(z)2333 2136 y Ft(2)2418 2067 y Fu(@)p 2380 2105 V 2380 2181 a(@)5 b(z)2468 2193 y Ft(1)2514 2007 y Fo(\023)1511 2356 y Fu(L)1568 2322 y Fn(0)1568 2377 y Ft(3)1688 2356 y Fw(=)1846 2300 y(1)p 1846 2337 42 4 v 1846 2413 a(2)1911 2239 y Fo(\022)1972 2356 y Fu(z)2011 2368 y Ft(1)2096 2300 y Fu(@)p 2058 2337 125 4 v 2058 2413 a(@)g(z)2146 2425 y Ft(1)2211 2356 y Fr(\000)18 b Fu(z)2333 2368 y Ft(2)2418 2300 y Fu(@)p 2380 2337 V 2380 2413 a(@)5 b(z)2468 2425 y Ft(2)2514 2239 y Fo(\023)1455 2606 y Fw(\001)1524 2571 y Fn(0)1524 2628 y Fs(S)1568 2611 y Fm(2)1688 2606 y Fw(=)1836 2488 y Fo(\022)1907 2549 y Fw(1)p 1907 2586 42 4 v 1907 2662 a(2)1972 2488 y Fo(\022)2033 2606 y Fu(z)2072 2618 y Ft(1)2157 2549 y Fu(@)p 2119 2586 125 4 v 2119 2662 a(@)g(z)2207 2674 y Ft(1)2272 2606 y Fw(+)18 b Fu(z)2394 2618 y Ft(2)2479 2549 y Fu(@)p 2441 2586 V 2441 2662 a(@)5 b(z)2529 2674 y Ft(2)2575 2488 y Fo(\023)2655 2606 y Fw(+)2748 2549 y(1)p 2748 2586 42 4 v 2748 2662 a(2)2799 2488 y Fo(\023)2860 2506 y Ft(2)3618 2606 y Fw(\(2.23\))523 2830 y(resp)r(ectiv)n(ely)-7 b(.)648 2930 y(Notice)37 b(that)g(the)h(op)r(erator)d Fu(L)1661 2942 y Ft(0)1737 2930 y Fw(=)k Fu(z)1880 2942 y Ft(1)1959 2897 y Fs(@)p 1927 2911 104 4 v 1927 2958 a(@)t(z)1998 2966 y Fm(1)2065 2930 y Fw(+)24 b Fu(z)2193 2942 y Ft(2)2272 2897 y Fs(@)p 2240 2911 V 2240 2958 a(@)t(z)2311 2966 y Fm(2)2390 2930 y Fw(is)37 b(the)h(addition)f(of)g (Hamiltonians)g(of)g(t)n(w)n(o)523 3048 y(indep)r(enden)n(t)28 b(harmonic)f(oscillators)f Fu(z)1786 3060 y Fs(j)1862 3015 y(@)p 1830 3029 102 4 v 1830 3077 a(@)t(z)1901 3085 y Fp(j)1942 3048 y Fu(;)37 b(j)28 b Fw(=)23 b(1)p Fu(;)14 b Fw(2)27 b(with)h(the)g(same)f(frequency)-7 b(.)523 3240 y Fv(I)s(I.3)31 b(Bargmann)g(T)-8 b(ransform)32 b(for)g Fu(L)1837 3210 y Ft(2)1874 3240 y Fw(\()p Fu(S)1962 3210 y Ft(3)1999 3240 y Fw(\))648 3387 y(In)e(analogy)e(with)i(the)h (case)e Fu(L)1635 3356 y Ft(2)1671 3387 y Fw(\()p Fu(S)1759 3356 y Ft(2)1797 3387 y Fw(\),)i(let)f(us)g(consider)f(the)h(spherical) f(normalized)g(Laplacian)f(on)523 3486 y Fu(S)579 3456 y Ft(3)616 3486 y Fw(.)37 b(In)28 b(spherical)e(co)r(ordinates)1144 3666 y Fu(x)1191 3678 y Ft(1)1312 3666 y Fw(=)82 b(sin\()p Fu(\025)p Fw(\))14 b(sin)q(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))p Fu(;)153 b(x)2357 3678 y Ft(2)2418 3666 y Fw(=)23 b(sin\()p Fu(\025)p Fw(\))14 b(sin\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))p Fu(;)1144 3790 y(x)1191 3802 y Ft(3)1312 3790 y Fw(=)82 b(sin\()p Fu(\025)p Fw(\))14 b(cos\()p Fu(\022)r Fw(\))p Fu(;)379 b(x)2353 3802 y Ft(4)2413 3790 y Fw(=)23 b(cos)o(\()p Fu(\025)p Fw(\))h Fu(;)523 3970 y Fw(where)j(0)c Fu(<)f(\022)r(;)14 b(\025)24 b(<)f(\031)31 b Fw(and)c(0)c Fr(\024)f Fu(\036)i Fr(\024)e Fw(2)p Fu(\031)s Fw(.)648 4069 y(The)27 b(expression)f(for)h (\001)1414 4084 y Fs(S)1458 4067 y Fm(3)1523 4069 y Fw(is)1324 4297 y(\001)1393 4312 y Fs(S)1437 4295 y Fm(3)1497 4297 y Fw(=)22 b Fr(\000)1683 4241 y Fu(@)1732 4211 y Ft(2)p 1659 4278 135 4 v 1659 4354 a Fu(@)5 b(\025)1756 4330 y Ft(2)1822 4297 y Fr(\000)18 b Fw(2)c(cot)o(\()p Fu(\025)p Fw(\))2218 4241 y Fu(@)p 2193 4278 98 4 v 2193 4354 a(@)5 b(\025)2319 4297 y Fw(+)2485 4241 y(\001)2554 4253 y Ft(2)p 2412 4278 253 4 v 2412 4360 a Fw(sin)2514 4325 y Ft(2)2551 4360 y Fw(\()p Fu(\025)p Fw(\))2693 4297 y(+)18 b(1)k Fu(;)755 b Fw(\(2.24\))523 4527 y(where)1366 4739 y(\001)1435 4751 y Ft(2)1496 4739 y Fw(=)1644 4682 y Fr(\000)p Fw(1)p 1594 4720 208 4 v 1594 4796 a(sin\()p Fu(\022)r Fw(\))1842 4682 y Fu(@)p 1821 4720 90 4 v 1821 4796 a(@)5 b(\022)1935 4739 y Fw(sin\()p Fu(\022)r Fw(\))2173 4682 y Fu(@)p 2152 4720 V 2152 4796 a(@)g(\022)2271 4739 y Fw(+)2434 4682 y Fr(\000)p Fw(1)p 2364 4720 246 4 v 2364 4802 a(sin)2466 4767 y Ft(2)2503 4802 y Fw(\()p Fu(\022)r Fw(\))2654 4682 y Fu(@)2703 4652 y Ft(2)p 2629 4720 136 4 v 2629 4796 a Fu(@)g(\036)2727 4772 y Ft(2)2798 4739 y Fu(:)797 b Fw(\(2.25\))523 4978 y(Note)27 b(that)g(w)n(e)f(are)g (taking)g(\001)1485 4993 y Fs(S)1529 4976 y Fm(3)1592 4978 y Fw(suc)n(h)g(that)h(its)g(sp)r(ectrum)g(is)g(the)g(set)f Fr(f)p Fu(n)2879 4948 y Ft(2)2939 4978 y Fr(j)d Fu(n)k Fw(is)f(a)g(p)r(ositiv)n(e)h(in)n(teger)p Fr(g)p Fw(.)648 5078 y(Let)35 b(us)g(also)g(consider)f(generators)f(for)i(rotations)f (on)h(the)g(\()p Fu(\026;)14 b(\027)5 b Fw(\))37 b(plane)e(with)h Fu(\026;)14 b(\027)41 b Fw(=)35 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)p Fu(;)g Fw(4,)523 5178 y(giv)n(en)27 b(b)n(y)g (restrictions)f(to)i Fu(S)1443 5147 y Ft(3)1508 5178 y Fw(of)f(the)h(follo)n(wing)f(op)r(erators)1243 5402 y Fu(E)1304 5414 y Fs(\026\027)1409 5402 y Fw(=)1507 5346 y(1)p 1507 5383 42 4 v 1514 5459 a Fu({)1572 5285 y Fo(\022)1633 5402 y Fu(x)1680 5414 y Fs(\026)1779 5346 y Fu(@)p 1735 5383 138 4 v 1735 5459 a(@)5 b(x)1831 5471 y Fs(\027)1901 5402 y Fr(\000)18 b Fu(x)2031 5414 y Fs(\027)2129 5346 y Fu(@)p 2083 5383 141 4 v 2083 5459 a(@)5 b(x)2179 5471 y Fs(\026)2233 5285 y Fo(\023)2377 5402 y Fu(\026;)14 b(\027)29 b Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)p Fu(;)g Fw(4)21 b Fu(:)674 b Fw(\(2.26\))648 5629 y(There)25 b(are)f(six)i(linearly)e(indep)r(enden)n(t)j(op)r(erators)c(among)i (the)h(op)r(erators)e Fu(E)3111 5641 y Fs(\026\027)3193 5629 y Fw(.)36 b(These)25 b(op)r(erators)523 5728 y(pro)n(vide)35 b(a)g(represen)n(tation)f(of)h(the)h(Lie)g(algebra)e Fu(so)p Fw(\(4\).)61 b(It)36 b(is)g(w)n(ell)f(kno)n(wn)g(that)h(this)g (Lie)g(algebra)p eop %%Page: 10 10 10 9 bop 3747 324 a Fw(10)523 614 y(describ)r(es)18 b(the)h(symmetries) f(of)h(the)g(h)n(ydrogen)e(atom)h(problem)2500 584 y Ft(7)p Fs(;)p Ft(9)2589 614 y Fw(.)34 b(This)19 b(can)f(b)r(e)h(sho)n (wn)f(b)n(y)h(considering)523 714 y(restrictions)28 b(of)h(the)g (Angular)f(Momen)n(tum)h Fv(L)d Fw(=)f(\()p Fu(L)2223 726 y Ft(1)2260 714 y Fu(;)14 b(L)2354 726 y Ft(2)2391 714 y Fu(;)g(L)2485 726 y Ft(3)2521 714 y Fw(\))30 b(and)e(Runge-Lenz)h Fv(A)c Fw(=)g(\()p Fu(A)3487 726 y Ft(1)3525 714 y Fu(;)14 b(A)3624 726 y Ft(2)3661 714 y Fu(;)g(A)3760 726 y Ft(3)3798 714 y Fw(\))523 814 y(op)r(erators)34 b(to)i(eigenspaces)e(of)i(the)g (Hamiltonian)g(of)g(the)g(h)n(ydrogen)e(atom)i(problem.)61 b(In)36 b(order)e(to)523 913 y(k)n(eep)29 b(trac)n(k)f(of)h(this)g (fact,)h(w)n(e)f(rename)f(the)i(op)r(erators)d Fu(L)2363 925 y Ft(1)2425 913 y Fw(=)e Fu(E)2576 925 y Ft(23)2647 913 y Fw(,)30 b Fu(L)2757 925 y Ft(2)2819 913 y Fw(=)25 b Fu(E)2970 925 y Ft(31)3041 913 y Fw(,)30 b Fu(L)3151 925 y Ft(3)3213 913 y Fw(=)25 b Fu(E)3364 925 y Ft(12)3435 913 y Fw(,)30 b Fu(A)3550 925 y Fs(j)3611 913 y Fw(=)25 b Fu(E)3762 925 y Ft(4)p Fs(j)523 1013 y Fu(j)j Fw(=)23 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3.)648 1112 y(The)27 b(expressions)f(for)h(the)h(op)r(erators)e Fv(L)i Fw(and)f Fv(A)h Fw(in)g(spherical)f(co)r(ordinates)f(are:)903 1331 y Fu(L)960 1343 y Ft(1)1079 1331 y Fw(=)83 b Fu({)1268 1214 y Fo(\022)1329 1331 y Fw(sin\()p Fu(\036)p Fw(\))1575 1275 y Fu(@)p 1554 1312 90 4 v 1554 1388 a(@)5 b(\022)1673 1331 y Fw(+)18 b(cot\()p Fu(\022)r Fw(\))c(cos\()p Fu(\036)p Fw(\))2246 1275 y Fu(@)p 2221 1312 99 4 v 2221 1388 a(@)5 b(\036)2330 1214 y Fo(\023)2405 1331 y Fu(;)903 1564 y(L)960 1576 y Ft(2)1079 1564 y Fw(=)83 b Fr(\000)p Fu({)1333 1447 y Fo(\022)1393 1564 y Fw(cos\()p Fu(\036)p Fw(\))1649 1508 y Fu(@)p 1628 1545 90 4 v 1628 1621 a(@)5 b(\022)1747 1564 y Fr(\000)18 b Fw(cot\()p Fu(\022)r Fw(\))c(sin\()p Fu(\036)p Fw(\))2311 1508 y Fu(@)p 2285 1545 99 4 v 2285 1621 a(@)5 b(\036)2394 1447 y Fo(\023)2469 1564 y Fu(;)903 1790 y(L)960 1802 y Ft(3)1079 1790 y Fw(=)83 b Fr(\000)p Fu({)1353 1733 y(@)p 1329 1771 V 1329 1847 a(@)5 b(\036)1436 1790 y(;)897 2016 y(A)959 2028 y Ft(1)1079 2016 y Fw(=)83 b Fr(\000)p Fu({)14 b Fw(cot)o(\()p Fu(\025)p Fw(\))1569 1899 y Fo(\022)1631 2016 y Fw(cos)o(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))2118 1960 y Fu(@)p 2096 1997 90 4 v 2096 2073 a(@)5 b(\022)2215 2016 y Fr(\000)2308 1960 y Fw(sin\()p Fu(\036)p Fw(\))p 2308 1997 216 4 v 2312 2073 a(sin\()p Fu(\022)r Fw(\))2569 1960 y Fu(@)p 2544 1997 99 4 v 2544 2073 a(@)g(\036)2652 1899 y Fo(\023)2732 2016 y Fr(\000)18 b Fu({)c Fw(sin)o(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))3336 1960 y Fu(@)p 3311 1997 98 4 v 3311 2073 a(@)5 b(\025)3419 2016 y(;)897 2249 y(A)959 2261 y Ft(2)1079 2249 y Fw(=)83 b Fr(\000)p Fu({)14 b Fw(cot)o(\()p Fu(\025)p Fw(\))1569 2132 y Fo(\022)1631 2249 y Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))2108 2193 y Fu(@)p 2087 2230 90 4 v 2087 2306 a(@)5 b(\022)2206 2249 y Fw(+)2299 2193 y(cos)o(\()p Fu(\036)p Fw(\))p 2299 2230 226 4 v 2308 2306 a(sin\()p Fu(\022)r Fw(\))2569 2193 y Fu(@)p 2544 2230 99 4 v 2544 2306 a(@)g(\036)2652 2132 y Fo(\023)2732 2249 y Fr(\000)18 b Fu({)c Fw(sin)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))3327 2193 y Fu(@)p 3302 2230 98 4 v 3302 2306 a(@)5 b(\025)3410 2249 y(;)897 2481 y(A)959 2493 y Ft(3)1079 2481 y Fw(=)83 b Fr(\000)p Fu({)1333 2364 y Fo(\022)1393 2481 y Fw(cos\()p Fu(\022)r Fw(\))1645 2425 y Fu(@)p 1620 2462 V 1620 2538 a(@)5 b(\025)1746 2481 y Fr(\000)18 b Fw(sin\()p Fu(\022)r Fw(\))c(cot\()p Fu(\025)p Fw(\))2305 2425 y Fu(@)p 2283 2462 90 4 v 2283 2538 a(@)5 b(\022)2384 2364 y Fo(\023)2482 2481 y Fu(:)1113 b Fw(\(2.27\))523 2710 y(Note)28 b(that)g(if)g(w)n(e)f(denote)g(b)n(y)h Fu(L)1542 2680 y Ft(2)1602 2710 y Fw(=)22 b Fu(L)1746 2680 y Ft(2)1746 2731 y(1)1801 2710 y Fw(+)c Fu(L)1941 2680 y Ft(2)1941 2731 y(2)1997 2710 y Fw(+)g Fu(L)2137 2680 y Ft(2)2137 2731 y(3)2201 2710 y Fw(and)27 b Fu(A)2424 2680 y Ft(2)2485 2710 y Fw(=)c Fu(A)2635 2680 y Ft(2)2635 2731 y(1)2691 2710 y Fw(+)18 b Fu(A)2836 2680 y Ft(2)2836 2731 y(2)2892 2710 y Fw(+)g Fu(A)3037 2680 y Ft(2)3037 2731 y(3)3102 2710 y Fw(then)28 b(w)n(e)f(ha)n(v)n(e)1721 2884 y(\001)1790 2899 y Fs(S)1834 2882 y Fm(3)1894 2884 y Fw(=)c Fu(L)2039 2849 y Ft(2)2094 2884 y Fw(+)18 b Fu(A)2239 2849 y Ft(2)2295 2884 y Fw(+)g(1)23 b Fu(:)1152 b Fw(\(2.28\))648 3057 y(Since)24 b(\001)930 3072 y Fs(S)974 3055 y Fm(3)1011 3057 y Fw(,)h Fu(L)1116 3027 y Ft(2)1178 3057 y Fw(and)f Fu(L)1393 3069 y Ft(3)1454 3057 y Fw(comm)n(ute,)h(w)n (e)g(can)f(\014nd)h(common)f(normalized)f(eigenfunctions)i(of)f(these) 523 3157 y(op)r(erators.)51 b(W)-7 b(e)34 b(shall)e(call)h(these)g (functions)h(h)n(yp)r(er-spherical)d(harmonics)2983 3127 y Ft(17)3085 3157 y Fw(and)i(denote)g(them)g(b)n(y)523 3256 y Fu(Y)571 3268 y Fs(n`m)738 3256 y Fw(with)i Fu(n)f Fw(=)h(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)g Fu(:)g(:)g(:)n Fw(,)36 b Fu(`)e Fw(=)h(0)p Fu(;)14 b Fw(1)p Fu(;)g(:)g(:)g(:)e(;)i(n) 23 b Fr(\000)g Fw(1,)36 b Fu(m)e Fw(=)h Fr(\000)p Fu(`;)14 b Fr(\000)p Fu(`)21 b Fw(+)i(1)p Fu(;)14 b(:)g(:)g(:)f(`)p Fw(.)57 b(F)-7 b(or)34 b(eac)n(h)g Fu(n)p Fw(,)i(w)n(e)e(ha)n(v)n(e)523 3356 y Fu(n)573 3326 y Ft(2)639 3356 y Fw(linearly)29 b(indep)r(enden)n(t)h(h)n(yp)r(er-spherical)e(harmonics,)g(whic)n(h)h (is)g(exactly)g(the)h(m)n(ultiplicit)n(y)g(of)f(the)523 3456 y(corresp)r(onding)i(eigen)n(v)-5 b(alues)32 b(of)g(\001)1665 3471 y Fs(S)1709 3454 y Fm(3)1779 3456 y Fw(and)g(the)i(Hamiltonian)e (of)h(the)g(h)n(ydrogen)e(atom)3374 3426 y Ft(7)p Fs(;)p Ft(9)3464 3456 y Fw(.)53 b(The)32 b Fu(n)3765 3426 y Ft(2)3802 3456 y Fw(-)523 3555 y(dimensional)18 b(v)n(ector)g(spaces)g (generated)f(b)n(y)i Fu(Y)1982 3567 y Fs(n`m)2133 3555 y Fw(\(one)g(space)f(for)g(eac)n(h)g Fu(n)p Fw(\))h(pro)n(vide)f(all)h (the)g(irreducible)523 3655 y(represen)n(tations)26 b(of)h(the)h(group) f(SO\(4\).)648 3755 y(In)k(order)e(to)i(\014nd)g(a)f(Hilb)r(ert)i (space)e(of)h(analytic)f(functions)h(related)f(to)h Fu(L)3056 3724 y Ft(2)3093 3755 y Fw(\()p Fu(S)3181 3724 y Ft(3)3218 3755 y Fw(\),)h(w)n(e)e(notice)h(that)523 3854 y(the)d(comm)n(utation)f (relations)f(satis\014ed)i(b)n(y)f Fv(L)h Fw(and)f Fv(A)h Fw(are:)1770 4028 y([)p Fu(L)1850 4040 y Fs(i)1877 4028 y Fu(;)14 b(L)1971 4040 y Fs(j)2005 4028 y Fw(])23 b(=)g Fu({\017)2200 4040 y Fs(ij)s(k)2295 4028 y Fu(L)2352 4040 y Fs(k)2392 4028 y Fu(;)1758 4152 y Fw([)p Fu(L)1838 4164 y Fs(i)1865 4152 y Fu(;)14 b(A)1964 4164 y Fs(j)2000 4152 y Fw(])23 b(=)f Fu({\017)2194 4164 y Fs(ij)s(k)2289 4152 y Fu(A)2351 4164 y Fs(k)2392 4152 y Fu(;)1772 4277 y Fw([)p Fu(A)1857 4289 y Fs(i)1885 4277 y Fu(;)14 b(A)1984 4289 y Fs(j)2019 4277 y Fw(])23 b(=)g Fu({\017)2214 4289 y Fs(ij)s(k)2309 4277 y Fu(L)2366 4289 y Fs(k)2406 4277 y Fu(:)1189 b Fw(\(2.29\))648 4450 y(No)n(w)27 b(let)h(us)f(de\014ne)h (the)g(op)r(erators)1550 4662 y Fv(C)23 b Fw(=)1740 4606 y Fv(L)c Fw(+)f Fv(A)p 1740 4643 232 4 v 1834 4719 a Fw(2)1981 4662 y Fu(;)152 b Fv(D)23 b Fw(=)2350 4606 y Fv(L)c Fr(\000)f Fv(A)p 2350 4643 V 2445 4719 a Fw(2)2614 4662 y Fu(:)981 b Fw(\(2.30\))523 4859 y(The)28 b(comm)n(utation)f (relation)f(for)h(these)h(op)r(erators)e(are:)1787 5032 y([)p Fu(C)1869 5044 y Fs(i)1897 5032 y Fu(;)14 b(C)1993 5044 y Fs(j)2029 5032 y Fw(])83 b(=)f Fu({\017)2343 5044 y Fs(ij)s(k)2438 5032 y Fu(C)2497 5044 y Fs(k)2538 5032 y Fu(;)1769 5157 y Fw([)p Fu(D)1861 5169 y Fs(i)1888 5157 y Fu(;)14 b(D)1994 5169 y Fs(j)2029 5157 y Fw(])83 b(=)f Fu({\017)2343 5169 y Fs(ij)s(k)2438 5157 y Fu(D)2507 5169 y Fs(k)2548 5157 y Fu(;)1778 5281 y Fw([)p Fu(C)1860 5293 y Fs(i)1888 5281 y Fu(;)14 b(D)1994 5293 y Fs(j)2029 5281 y Fw(])83 b(=)f(0)p Fu(:)1271 b Fw(\(2.31\))523 5455 y(Th)n(us)27 b(w)n(e)h(obtain)f Fu(so)p Fw(\(4\))c(=)g Fu(su)p Fw(\(2\))18 b Fr(\002)g Fu(su)o Fw(\(2\).)648 5555 y(Notice)27 b(that)h(the)g(op)r(erators)e Fv(L)i Fw(and)f Fv(A)h Fw(satisfy)f(the)h(relation)1430 5728 y Fv(L)19 b Fr(\017)f Fv(A)p Fw(\011)k(=)h(\()p Fu(L)1902 5740 y Ft(1)1939 5728 y Fu(A)2001 5740 y Ft(1)2057 5728 y Fw(+)18 b Fu(L)2197 5740 y Ft(2)2234 5728 y Fu(A)2296 5740 y Ft(2)2352 5728 y Fw(+)g Fu(L)2492 5740 y Ft(3)2529 5728 y Fu(A)2591 5740 y Ft(3)2628 5728 y Fw(\)\011)23 b(=)g(0)f Fu(:)695 b Fw(\(2.32\))p eop %%Page: 11 11 11 10 bop 3747 324 a Fw(11)648 614 y(This)27 b(relation)g(implies:)1532 811 y Fu(C)1597 776 y Ft(2)1635 811 y Fw(\011)22 b(=)h Fu(D)1881 776 y Ft(2)1918 811 y Fw(\011)g(=)2094 694 y Fo(\022)2165 754 y Fu(L)2222 724 y Ft(2)2277 754 y Fw(+)18 b Fu(A)2422 724 y Ft(2)p 2165 792 295 4 v 2291 868 a Fw(4)2469 694 y Fo(\023)2544 811 y Fw(\011)23 b Fu(:)963 b Fw(\(2.33\))523 1013 y(Since)24 b(the)g(eigen)n(v)-5 b(alues)22 b(of)i(the)g(op)r(erators)e Fu(C)1959 983 y Ft(2)2020 1013 y Fw(and)i Fu(D)2249 983 y Ft(2)2310 1013 y Fw(are)e(of)i(the)g(form)f Fu(u)p Fw(\()p Fu(u)11 b Fw(+)g(1\))22 b(and)h Fu(v)s Fw(\()p Fu(v)14 b Fw(+)d(1\))23 b(\(with)523 1113 y Fu(u;)14 b(v)26 b Fr(2)d Fv(S)p Fw(\))k(resp)r (ectiv)n(ely)-7 b(,)26 b(then)i(w)n(e)e(m)n(ust)h(ha)n(v)n(e)f Fu(u)p Fw(\()p Fu(u)16 b Fw(+)g(1\))23 b(=)f Fu(v)s Fw(\()p Fu(v)f Fw(+)16 b(1\).)37 b(This)26 b(equation)h(implies)g Fu(u)22 b Fw(=)h Fu(v)s Fw(.)648 1213 y(Since)18 b(\001)924 1228 y Fs(S)968 1211 y Fm(3)1028 1213 y Fw(=)23 b Fu(L)1173 1183 y Ft(2)1210 1213 y Fw(+)p Fu(A)1337 1183 y Ft(2)1374 1213 y Fw(+1)18 b(and)h(the)f(eigen)n(v)-5 b(alues)18 b(of)g(\001)2360 1228 y Fs(S)2404 1211 y Fm(3)2459 1213 y Fw(are)g Fu(n)2639 1183 y Ft(2)2676 1213 y Fw(,)i(then)f(w)n(e)g(ha)n (v)n(e)e Fu(n)3245 1183 y Ft(2)3305 1213 y Fw(=)23 b(4)p Fu(u)p Fw(\()p Fu(u)p Fw(+1\)+1,)523 1312 y(whic)n(h)29 b(in)h(turn)g(implies)g Fu(u)25 b Fw(=)h Fu(v)k Fw(=)1666 1280 y Fs(n)p Fn(\000)p Ft(1)p 1666 1294 127 4 v 1713 1341 a(2)1802 1312 y Fw(.)43 b(Since)30 b Fu(n)f Fw(is)g(a)g(p)r (ositiv)n(e)h(in)n(teger,)f Fv(S)g Fw(is)h(the)f(set)h(of)f(p)r (ermitted)523 1412 y(v)-5 b(alues)27 b(of)h Fu(u)f Fw(and)g Fu(v)k Fw(\(including)1574 1379 y Ft(1)p 1574 1393 34 4 v 1574 1441 a(2)1617 1412 y Fu(;)1664 1379 y Ft(3)p 1664 1393 V 1664 1441 a(2)1707 1412 y Fu(;)14 b(:)g(:)g(:)p Fw(\).)648 1512 y(F)-7 b(rom)24 b(the)h(discussion)f(ab)r(o)n(v)n(e,)g (w)n(e)h(w)n(an)n(t)f(to)h(consider)f(t)n(w)n(o)g(copies)g(of)h(the)g (represen)n(tation)e(of)i(su\(2\))523 1611 y(on)f(the)g(Bargmann)e (space)h Fr(B)1448 1623 y Ft(2)1509 1611 y Fw(of)g(analytical)g (functions)h(of)g(t)n(w)n(o)f(complex)g(v)-5 b(ariables)23 b(describ)r(ed)h(ab)r(o)n(v)n(e.)648 1711 y(Let)e(us)h(consider)f(the)h (Bargmann)e(space)g Fr(B)2021 1723 y Ft(4)2081 1711 y Fw(=)i Fr(B)2224 1723 y Ft(2)2269 1711 y Fr(\012)9 b(B)2398 1723 y Ft(2)2457 1711 y Fw(of)22 b(analytical)g(functions)h(of)f(four)h (complex)523 1810 y(v)-5 b(ariables)27 b(\()p Fu(z)939 1822 y Ft(1)976 1810 y Fu(;)14 b(z)1052 1822 y Ft(2)1088 1810 y Fu(;)g(z)1164 1822 y Ft(3)1201 1810 y Fu(;)g(z)1277 1822 y Ft(4)1314 1810 y Fw(\).)38 b(F)-7 b(or)27 b(eac)n(h)g(p)r (ositiv)n(e)g(in)n(teger)g Fu(n)p Fw(,)h(let)g(us)f(also)g(consider)g (the)h Fu(n)3336 1780 y Ft(2)3373 1810 y Fw(-dimensional)523 1910 y(subspaces)f Fr(M)1003 1922 y Fs(n)1075 1910 y Fw(of)h Fr(B)1225 1922 y Ft(4)1289 1910 y Fw(giv)n(en)f(b)n(y)1695 2058 y Fr(M)1795 2070 y Fs(n)1863 2058 y Fw(=)22 b Fr(W)2042 2057 y Fp(n)p Fk(\000)p Fm(1)p 2042 2071 110 4 v 2083 2103 a(2)2166 2058 y Fr(\012W)2323 2057 y Fp(n)p Fk(\000)p Fm(1)p 2323 2071 V 2363 2103 a(2)2470 2058 y Fu(;)1125 b Fw(\(2.34\))523 2217 y(with)33 b Fr(W)809 2216 y Fp(n)p Fk(\000)p Fm(1)p 809 2230 V 850 2263 a(2)966 2217 y Fw(the)g(v)n(ector) e(space)h(of)g(homogeneous)f(p)r(olynomials)h(of)g(degree)g Fu(n)21 b Fr(\000)h Fw(1)32 b(in)h(t)n(w)n(o)f(complex)523 2329 y(v)-5 b(ariables.)648 2428 y(The)27 b(functions)939 2644 y Fu(y)980 2656 y Fs(nm)1080 2664 y Fm(1)1112 2656 y Fs(m)1171 2664 y Fm(2)1231 2644 y Fw(=)1622 2587 y Fu(z)1675 2512 y Fp(n)p Fk(\000)p Fm(1)p 1675 2526 V 1715 2559 a(2)1794 2539 y Ft(+)p Fs(m)1904 2547 y Fm(1)1661 2610 y Ft(1)1941 2587 y Fu(z)1994 2512 y Fp(n)p Fk(\000)p Fm(1)p 1993 2526 V 2034 2559 a(2)2113 2539 y Fn(\000)p Fs(m)2224 2547 y Fm(1)1980 2610 y Ft(2)2260 2587 y Fu(z)2313 2512 y Fp(n)p Fk(\000)p Fm(1)p 2312 2526 V 2353 2559 a(2)2432 2539 y Ft(+)p Fs(m)2542 2547 y Fm(2)2299 2610 y Ft(3)2579 2587 y Fu(z)2632 2512 y Fp(n)p Fk(\000)p Fm(1)p 2631 2526 V 2672 2559 a(2)2750 2539 y Fn(\000)p Fs(m)2861 2547 y Fm(2)2618 2610 y Ft(4)p 1328 2625 1864 4 v 1328 2641 a Fo(q)p 1411 2641 1781 4 v 96 x Fw(\()1453 2704 y Fs(n)p Fn(\000)p Ft(1)p 1453 2718 127 4 v 1500 2766 a(2)1608 2737 y Fw(+)18 b Fu(m)1764 2749 y Ft(1)1801 2737 y Fw(\)!\()1898 2704 y Fs(n)p Fn(\000)p Ft(1)p 1899 2718 V 1945 2766 a(2)2053 2737 y Fr(\000)g Fu(m)2209 2749 y Ft(1)2246 2737 y Fw(\)!\()2343 2704 y Fs(n)p Fn(\000)p Ft(1)p 2344 2718 V 2390 2766 a(2)2498 2737 y Fw(+)g Fu(m)2654 2749 y Ft(2)2691 2737 y Fw(\)!\()2788 2704 y Fs(n)p Fn(\000)p Ft(1)p 2789 2718 V 2836 2766 a(2)2943 2737 y Fr(\000)h Fu(m)3100 2749 y Ft(2)3137 2737 y Fw(\)!)3225 2644 y Fu(;)370 b Fw(\(2.35\))523 2914 y(with)28 b Fr(\000)787 2881 y Fs(n)p Fn(\000)p Ft(1)p 787 2895 V 833 2942 a(2)946 2914 y Fr(\024)22 b Fu(m)1106 2926 y Ft(1)1143 2914 y Fu(;)14 b(m)1253 2926 y Ft(2)1314 2914 y Fr(\024)1411 2881 y Fs(n)p Fn(\000)p Ft(1)p 1411 2895 V 1458 2942 a(2)1575 2914 y Fw(pro)n(vide)26 b(a)h(basis)g(for)g(the)h(space)f Fr(M)2735 2926 y Fs(n)2780 2914 y Fw(.)648 3013 y(Since)k(w)n(e)h(w)n (an)n(t)f(to)h(assign)e(an)i(analytic)f(function)h(\(of)g(four)f (complex)h(v)-5 b(ariables\))30 b(to)i(the)g(h)n(yp)r(er-)523 3113 y(spherical)g(harmonics)f Fu(Y)1325 3125 y Fs(n`m)1490 3113 y Fw(and)i(these)f(functions)h(are)f(in)h(turn)g(common)f (eigenfunctions)h(of)f(\001)3726 3128 y Fs(S)3770 3111 y Fm(3)3807 3113 y Fw(,)523 3213 y Fu(L)580 3183 y Ft(2)644 3213 y Fw(and)c Fu(L)863 3225 y Ft(3)900 3213 y Fw(,)f(then)h(w)n(e)g (w)n(an)n(t)f(actually)g(to)g(consider)g(the)h(follo)n(wing)e (functions)843 3395 y Fu(y)884 3407 y Fs(n`m)1015 3395 y Fw(\()p Fu(z)1086 3407 y Ft(1)1124 3395 y Fu(;)14 b(z)1200 3407 y Ft(2)1236 3395 y Fu(;)g(z)1312 3407 y Ft(3)1349 3395 y Fu(;)g(z)1425 3407 y Ft(4)1462 3395 y Fw(\))23 b(=)1717 3316 y Fo(X)1605 3490 y Fs(m)1664 3498 y Fm(1)1696 3490 y Ft(+)p Fs(m)1806 3498 y Fm(2)1838 3490 y Ft(=)p Fs(m)1962 3395 y Fu(C)6 b Fw(\()2069 3339 y Fu(n)19 b Fr(\000)f Fw(1)p 2069 3376 193 4 v 2145 3452 a(2)2272 3395 y Fu(;)c(m)2382 3407 y Ft(1)2419 3395 y Fu(;)2466 3339 y(n)19 b Fr(\000)f Fw(1)p 2466 3376 V 2542 3452 a(2)2669 3395 y Fu(;)c(m)2779 3407 y Ft(2)2816 3395 y Fw(;)g Fu(`;)g(m)p Fw(\))p Fu(y)3071 3407 y Fs(nm)3171 3415 y Fm(1)3203 3407 y Fs(m)3262 3415 y Fm(2)3321 3395 y Fu(;)274 b Fw(\(2.36\))523 3633 y(where)27 b Fu(C)6 b Fw(\()870 3601 y Fs(n)p Fn(\000)p Ft(1)p 871 3615 127 4 v 917 3662 a(2)1007 3633 y Fu(;)14 b(m)1117 3645 y Ft(1)1154 3633 y Fu(;)1201 3601 y Fs(n)p Fn(\000)p Ft(1)p 1201 3615 V 1247 3662 a(2)1337 3633 y Fu(;)g(m)1447 3645 y Ft(2)1484 3633 y Fw(;)g Fu(`;)g(m)p Fw(\))27 b(are)f(Clebsh-Gordon)h (co)r(e\016cien)n(ts)2830 3603 y Ft(18)2899 3633 y Fw(.)648 3733 y(Let)e(us)g(de\014ne)h(the)g(Bargmann)e(transform)g(for)h Fu(L)2241 3703 y Ft(2)2277 3733 y Fw(\()p Fu(S)2365 3703 y Ft(3)2403 3733 y Fw(\))g(as)g(the)h(linear)e(extension)h(of)h(the)f (previous)523 3833 y(assignmen)n(t.)36 b(Th)n(us)1746 3980 y Fv(B)1814 3995 y Fs(S)1858 3979 y Fm(3)1918 3980 y Fw(:)23 b Fu(L)2021 3946 y Ft(2)2057 3980 y Fw(\()p Fu(S)2145 3946 y Ft(3)2183 3980 y Fw(\))g Fr(7!)g(F)2404 3992 y Ft(4)3618 3980 y Fw(\(2.37\))523 4128 y(where)j Fr(F)822 4140 y Ft(4)886 4128 y Fw(is)h(the)h(closed)e(subspace)g(of)h Fr(B)1851 4140 y Ft(4)1915 4128 y Fw(generated)f(b)n(y)g(the)i(spaces)e Fr(M)2904 4140 y Fs(n)2948 4128 y Fw(,)i(that)f(is)g Fr(F)3321 4140 y Ft(4)3381 4128 y Fw(=)22 b Fr(\010)3533 4098 y Fn(1)3533 4149 y Fs(n)p Ft(=1)3662 4128 y Fr(M)3762 4140 y Fs(n)3807 4128 y Fw(.)648 4228 y(The)32 b(Bargmann)e(transform)h Fv(B)1686 4243 y Fs(S)1730 4226 y Fm(3)1799 4228 y Fw(is)h(a)g(unitary) g(op)r(erator)e(and)i(in)n(tert)n(wines)g(the)g(op)r(erators)e Fu(L)3770 4240 y Ft(1)3807 4228 y Fw(,)523 4327 y Fu(L)580 4339 y Ft(2)617 4327 y Fw(,)e Fu(L)725 4339 y Ft(3)761 4327 y Fw(,)g Fu(A)874 4339 y Ft(1)912 4327 y Fw(,)p Fu(A)997 4339 y Ft(2)1034 4327 y Fw(,)p Fu(A)1119 4339 y Ft(3)1184 4327 y Fw(with)g(the)g(follo)n(wing)f(op)r(erators:)1287 4520 y Fu(L)1344 4486 y Fn(0)1344 4541 y Ft(1)1464 4520 y Fw(=)1621 4464 y(1)p 1621 4501 42 4 v 1621 4577 a(2)1687 4403 y Fo(\022)1748 4520 y Fu(z)1787 4532 y Ft(1)1871 4464 y Fu(@)p 1833 4501 125 4 v 1833 4577 a(@)5 b(z)1921 4589 y Ft(2)1986 4520 y Fw(+)18 b Fu(z)2108 4532 y Ft(3)2193 4464 y Fu(@)p 2155 4501 V 2155 4577 a(@)5 b(z)2243 4589 y Ft(4)2308 4520 y Fw(+)18 b Fu(z)2430 4532 y Ft(2)2515 4464 y Fu(@)p 2477 4501 V 2477 4577 a(@)5 b(z)2565 4589 y Ft(1)2630 4520 y Fw(+)18 b Fu(z)2752 4532 y Ft(4)2837 4464 y Fu(@)p 2799 4501 V 2799 4577 a(@)5 b(z)2887 4589 y Ft(3)2933 4403 y Fo(\023)3008 4520 y Fu(;)1287 4753 y(L)1344 4718 y Fn(0)1344 4773 y Ft(2)1464 4753 y Fw(=)1635 4697 y(1)p 1621 4734 69 4 v 1621 4810 a(2)p Fu({)1713 4636 y Fo(\022)1774 4753 y Fu(z)1813 4765 y Ft(1)1898 4697 y Fu(@)p 1860 4734 125 4 v 1860 4810 a(@)g(z)1948 4822 y Ft(2)2013 4753 y Fw(+)18 b Fu(z)2135 4765 y Ft(3)2220 4697 y Fu(@)p 2182 4734 V 2182 4810 a(@)5 b(z)2270 4822 y Ft(4)2335 4753 y Fr(\000)18 b Fu(z)2457 4765 y Ft(2)2542 4697 y Fu(@)p 2504 4734 V 2504 4810 a(@)5 b(z)2592 4822 y Ft(1)2657 4753 y Fr(\000)18 b Fu(z)2779 4765 y Ft(4)2864 4697 y Fu(@)p 2826 4734 V 2826 4810 a(@)5 b(z)2914 4822 y Ft(3)2960 4636 y Fo(\023)3035 4753 y Fu(;)1287 4985 y(L)1344 4951 y Fn(0)1344 5006 y Ft(3)1464 4985 y Fw(=)1621 4929 y(1)p 1621 4966 42 4 v 1621 5042 a(2)1687 4868 y Fo(\022)1748 4985 y Fu(z)1787 4997 y Ft(1)1871 4929 y Fu(@)p 1833 4966 125 4 v 1833 5042 a(@)g(z)1921 5054 y Ft(1)1986 4985 y Fw(+)18 b Fu(z)2108 4997 y Ft(3)2193 4929 y Fu(@)p 2155 4966 V 2155 5042 a(@)5 b(z)2243 5054 y Ft(3)2308 4985 y Fr(\000)18 b Fu(z)2430 4997 y Ft(2)2515 4929 y Fu(@)p 2477 4966 V 2477 5042 a(@)5 b(z)2565 5054 y Ft(2)2630 4985 y Fr(\000)18 b Fu(z)2752 4997 y Ft(4)2837 4929 y Fu(@)p 2799 4966 V 2799 5042 a(@)5 b(z)2887 5054 y Ft(4)2933 4868 y Fo(\023)3008 4985 y Fu(;)1281 5218 y(A)1343 5183 y Fn(0)1343 5238 y Ft(1)1464 5218 y Fw(=)1621 5161 y(1)p 1621 5199 42 4 v 1621 5275 a(2)1687 5101 y Fo(\022)1748 5218 y Fu(z)1787 5230 y Ft(1)1871 5161 y Fu(@)p 1833 5199 125 4 v 1833 5275 a(@)g(z)1921 5287 y Ft(2)1986 5218 y Fw(+)18 b Fu(z)2108 5230 y Ft(2)2193 5161 y Fu(@)p 2155 5199 V 2155 5275 a(@)5 b(z)2243 5287 y Ft(1)2308 5218 y Fr(\000)18 b Fu(z)2430 5230 y Ft(3)2515 5161 y Fu(@)p 2477 5199 V 2477 5275 a(@)5 b(z)2565 5287 y Ft(4)2630 5218 y Fr(\000)18 b Fu(z)2752 5230 y Ft(4)2837 5161 y Fu(@)p 2799 5199 V 2799 5275 a(@)5 b(z)2887 5287 y Ft(3)2933 5101 y Fo(\023)3008 5218 y Fu(;)1281 5450 y(A)1343 5416 y Fn(0)1343 5471 y Ft(2)1464 5450 y Fw(=)1635 5394 y(1)p 1621 5431 69 4 v 1621 5507 a(2)p Fu({)1713 5333 y Fo(\022)1774 5450 y Fu(z)1813 5462 y Ft(1)1898 5394 y Fu(@)p 1860 5431 125 4 v 1860 5507 a(@)g(z)1948 5519 y Ft(2)2013 5450 y Fw(+)18 b Fu(z)2135 5462 y Ft(4)2220 5394 y Fu(@)p 2182 5431 V 2182 5507 a(@)5 b(z)2270 5519 y Ft(3)2335 5450 y Fr(\000)18 b Fu(z)2457 5462 y Ft(2)2542 5394 y Fu(@)p 2504 5431 V 2504 5507 a(@)5 b(z)2592 5519 y Ft(1)2657 5450 y Fr(\000)18 b Fu(z)2779 5462 y Ft(3)2864 5394 y Fu(@)p 2826 5431 V 2826 5507 a(@)5 b(z)2914 5519 y Ft(4)2960 5333 y Fo(\023)3035 5450 y Fu(;)1281 5683 y(A)1343 5648 y Fn(0)1343 5703 y Ft(3)1464 5683 y Fw(=)1621 5626 y(1)p 1621 5663 42 4 v 1621 5739 a(2)1687 5565 y Fo(\022)1748 5683 y Fu(z)1787 5695 y Ft(1)1871 5626 y Fu(@)p 1833 5663 125 4 v 1833 5739 a(@)g(z)1921 5751 y Ft(1)1986 5683 y Fw(+)18 b Fu(z)2108 5695 y Ft(4)2193 5626 y Fu(@)p 2155 5663 V 2155 5739 a(@)5 b(z)2243 5751 y Ft(4)2308 5683 y Fr(\000)18 b Fu(z)2430 5695 y Ft(2)2515 5626 y Fu(@)p 2477 5663 V 2477 5739 a(@)5 b(z)2565 5751 y Ft(2)2630 5683 y Fr(\000)18 b Fu(z)2752 5695 y Ft(3)2837 5626 y Fu(@)p 2799 5663 V 2799 5739 a(@)5 b(z)2887 5751 y Ft(3)2933 5565 y Fo(\023)3618 5683 y Fw(\(2.38\))p eop %%Page: 12 12 12 11 bop 3747 324 a Fw(12)523 614 y(resp)r(ectiv)n(ely)-7 b(.)648 714 y(The)27 b(Laplacian)g(on)g Fu(S)1369 684 y Ft(3)1434 714 y Fw(is)g(in)n(tert)n(wined)g(with)i(the)e(op)r(erator) 1121 948 y(\001)1190 914 y Fn(0)1190 970 y Fs(S)1234 954 y Fm(3)1293 948 y Fw(=)1381 831 y Fo(\022)1452 892 y Fw(1)p 1452 929 42 4 v 1452 1005 a(2)1517 831 y Fo(\022)1578 948 y Fu(z)1617 960 y Ft(1)1702 892 y Fu(@)p 1664 929 125 4 v 1664 1005 a(@)5 b(z)1752 1017 y Ft(1)1817 948 y Fw(+)18 b Fu(z)1939 960 y Ft(2)2024 892 y Fu(@)p 1986 929 V 1986 1005 a(@)5 b(z)2074 1017 y Ft(2)2139 948 y Fw(+)18 b Fu(z)2261 960 y Ft(3)2346 892 y Fu(@)p 2308 929 V 2308 1005 a(@)5 b(z)2396 1017 y Ft(3)2461 948 y Fw(+)18 b Fu(z)2583 960 y Ft(4)2668 892 y Fu(@)p 2630 929 V 2630 1005 a(@)5 b(z)2718 1017 y Ft(4)2764 831 y Fo(\023)2844 948 y Fw(+)18 b(1)2969 831 y Fo(\023)3029 848 y Ft(2)3618 948 y Fw(\(2.39\))523 1170 y(Th)n(us)28 b(the)h(Laplacian)e(\001)1327 1140 y Fn(0)1327 1196 y Fs(S)1371 1180 y Fm(3)1436 1170 y Fw(is)i(basically)e(the)i(square)e (of)h(the)h(addition)f(of)h(four)f(harmonic)f(oscillators)523 1269 y(with)h(the)g(same)f(frequency)-7 b(.)648 1369 y(The)27 b(expression)f(for)h Fv(L)19 b Fr(\017)f Fv(A)28 b Fw(in)g(the)g(space)f Fr(F)2103 1381 y Ft(4)2167 1369 y Fw(is)h(giv)n(en)f(b)n(y)g(the)h(follo)n(wing)e(op)r(erator)1417 1579 y Fr(L)d(\021)g Fu(z)1624 1591 y Ft(1)1708 1523 y Fu(@)p 1671 1560 V 1671 1636 a(@)5 b(z)1759 1648 y Ft(1)1823 1579 y Fw(+)18 b Fu(z)1945 1591 y Ft(2)2030 1523 y Fu(@)p 1992 1560 V 1992 1636 a(@)5 b(z)2080 1648 y Ft(2)2145 1579 y Fr(\000)18 b Fu(z)2267 1591 y Ft(3)2352 1523 y Fu(@)p 2314 1560 V 2314 1636 a(@)5 b(z)2402 1648 y Ft(3)2467 1579 y Fr(\000)18 b Fu(z)2589 1591 y Ft(4)2674 1523 y Fu(@)p 2636 1560 V 2636 1636 a(@)5 b(z)2724 1648 y Ft(4)3618 1579 y Fw(\(2.40\))523 1791 y(The)19 b(restriction)e(\()q Fv(L)i Fr(\017)f Fv(A)p Fw(\))c(\011)22 b(=)h(0)18 b(is)h(already)e (included)i(in)g(the)g(space)f Fr(F)2772 1803 y Ft(4)2809 1791 y Fw(,)j(as)d(the)h(follo)n(wing)e(prop)r(osition)523 1891 y(establishes)27 b(it:)523 2048 y Fv(Prop)s(osition)j(2.1)41 b Fl(The)32 b(r)l(ange)f(of)h(the)f(Bar)l(gmann)h(T)-6 b(r)l(ansform)32 b Fv(B)2721 2063 y Fs(S)2765 2047 y Fm(3)2832 2048 y Fl(is)g(e)l(qual)f(to)g(the)g(kernel)h(of)g(the)523 2148 y(op)l(er)l(ator)f Fr(L)p Fl(,)f(that)g(is)1535 2320 y Fr(F)1595 2332 y Ft(4)1655 2320 y Fw(=)22 b Fr(f)p Fu(f)32 b Fr(2)23 b(B)1990 2332 y Ft(4)2027 2320 y Fr(jL)p Fu(f)32 b Fw(=)22 b(0)p Fr(g)g(\021)h Fw(Ker)o Fr(L)966 b Fw(\(2.41\))523 2492 y Fl(Pr)l(o)l(of)p Fw(:)38 b(Let)28 b(f)g(in)g Fr(F)1135 2504 y Ft(4)1172 2492 y Fw(.)37 b(Then)1557 2725 y Fu(f)31 b Fw(=)1746 2621 y Fn(1)1719 2646 y Fo(X)1717 2822 y Fs(n)p Ft(=0)1897 2646 y Fo(X)1856 2821 y Fs(m)1915 2829 y Fm(1)1947 2821 y Fs(;m)2026 2829 y Fm(2)2072 2725 y Fu(A)2134 2737 y Fs(nm)2234 2745 y Fm(1)2267 2737 y Fs(m)2326 2745 y Fm(2)2362 2725 y Fu(y)2403 2737 y Fs(nm)2503 2745 y Fm(1)2535 2737 y Fs(m)2594 2745 y Fm(2)3618 2725 y Fw(\(2.42\))523 2978 y(where)21 b(the)h(con)n(v)n (ergence)d(is)i(in)h(the)g(norm)e(of)i Fr(B)2006 2990 y Ft(4)2043 2978 y Fw(.)34 b(Since)22 b(norm-con)n(v)n(ergence)c(in)k Fr(B)3127 2990 y Ft(4)3185 2978 y Fw(implies)g(p)r(oin)n(t-wise)523 3078 y(con)n(v)n(ergence)i(\(this)j(is)g(b)r(ecause)f Fr(B)1618 3090 y Ft(4)1681 3078 y Fw(has)g(a)g(repro)r(ducing)f(k)n (ernel)h(and)g(the)h(Sc)n(h)n(w)n(artz)e(inequalit)n(y\))i(then)523 3178 y(f)h(is)g(an)f(analytic)g(function)h(with)g(the)g(follo)n(wing)f (p)r(o)n(w)n(er)f(series)749 3433 y Fu(f)32 b Fw(=)939 3329 y Fn(1)912 3354 y Fo(X)909 3530 y Fs(n)p Ft(=0)1089 3354 y Fo(X)1048 3529 y Fs(m)1107 3537 y Fm(1)1139 3529 y Fs(;m)1218 3537 y Fm(2)1264 3433 y Fu(A)1326 3445 y Fs(nm)1426 3453 y Fm(1)1459 3445 y Fs(m)1518 3453 y Fm(2)1859 3377 y Fu(z)1912 3302 y Fp(n)p Fk(\000)p Fm(1)p 1911 3316 110 4 v 1952 3348 a(2)2031 3328 y Ft(+)p Fs(m)2141 3336 y Fm(1)1898 3399 y Ft(1)2177 3377 y Fu(z)2230 3302 y Fp(n)p Fk(\000)p Fm(1)p 2229 3316 V 2270 3348 a(2)2349 3328 y Fn(\000)p Fs(m)2460 3336 y Fm(1)2216 3399 y Ft(2)2496 3377 y Fu(z)2549 3302 y Fp(n)p Fk(\000)p Fm(1)p 2549 3316 V 2589 3348 a(2)2668 3328 y Ft(+)p Fs(m)2778 3336 y Fm(2)2535 3399 y Ft(3)2815 3377 y Fu(z)2868 3302 y Fp(n)p Fk(\000)p Fm(1)p 2867 3316 V 2908 3348 a(2)2987 3328 y Fn(\000)p Fs(m)3098 3336 y Fm(2)2854 3399 y Ft(4)p 1564 3414 1864 4 v 1564 3431 a Fo(q)p 1647 3431 1781 4 v 96 x Fw(\()1689 3494 y Fs(n)p Fn(\000)p Ft(1)p 1690 3508 127 4 v 1736 3555 a(2)1844 3527 y Fw(+)18 b Fu(m)2000 3539 y Ft(1)2037 3527 y Fw(\)!\()2134 3494 y Fs(n)p Fn(\000)p Ft(1)p 2135 3508 V 2181 3555 a(2)2289 3527 y Fr(\000)g Fu(m)2445 3539 y Ft(1)2482 3527 y Fw(\)!\()2579 3494 y Fs(n)p Fn(\000)p Ft(1)p 2580 3508 V 2627 3555 a(2)2734 3527 y Fw(+)h Fu(m)2891 3539 y Ft(2)2928 3527 y Fw(\)!\()3025 3494 y Fs(n)p Fn(\000)p Ft(1)p 3025 3508 V 3072 3555 a(2)3180 3527 y Fr(\000)f Fu(m)3336 3539 y Ft(2)3373 3527 y Fw(\)!)3618 3433 y(\(2.43\))523 3715 y(Since)28 b(f)g(is)f(analytic,)g(w)n(e)h(can)f(di\013eren)n(tiate)g(its)h(p)r(o)n (w)n(er)e(series)h(term)n(wise)g(and)g(obtain)865 4001 y Fr(L)p Fu(f)32 b Fw(=)1112 3897 y Fn(1)1085 3922 y Fo(X)1083 4098 y Fs(n)p Ft(=0)1263 3922 y Fo(X)1222 4096 y Fs(m)1281 4104 y Fm(1)1313 4096 y Fs(;m)1392 4104 y Fm(2)1512 3914 y Fu(A)1574 3926 y Fs(nm)1674 3934 y Fm(1)1707 3926 y Fs(m)1766 3934 y Fm(2)1802 3914 y Fr(L)1873 3822 y Fo(\020)1923 3914 y Fu(z)1976 3839 y Fp(n)p Fk(\000)p Fm(1)p 1975 3853 110 4 v 2016 3886 a(2)2095 3866 y Ft(+)p Fs(m)2205 3874 y Fm(1)1962 3936 y Ft(1)2241 3914 y Fu(z)2294 3839 y Fp(n)p Fk(\000)p Fm(1)p 2293 3853 V 2334 3886 a(2)2413 3866 y Fn(\000)p Fs(m)2524 3874 y Fm(1)2280 3936 y Ft(2)2560 3914 y Fu(z)2613 3839 y Fp(n)p Fk(\000)p Fm(1)p 2613 3853 V 2653 3886 a(2)2732 3866 y Ft(+)p Fs(m)2842 3874 y Fm(2)2599 3936 y Ft(3)2879 3914 y Fu(z)2932 3839 y Fp(n)p Fk(\000)p Fm(1)p 2931 3853 V 2972 3886 a(2)3051 3866 y Fn(\000)p Fs(m)3162 3874 y Fm(2)2918 3936 y Ft(4)3198 3822 y Fo(\021)p 1448 3981 1864 4 v 1448 3998 a(q)p 1531 3998 1781 4 v 96 x Fw(\()1573 4061 y Fs(n)p Fn(\000)p Ft(1)p 1573 4075 127 4 v 1620 4123 a(2)1728 4094 y Fw(+)18 b Fu(m)1884 4106 y Ft(1)1921 4094 y Fw(\)!\()2018 4061 y Fs(n)p Fn(\000)p Ft(1)p 2018 4075 V 2065 4123 a(2)2173 4094 y Fr(\000)g Fu(m)2329 4106 y Ft(1)2366 4094 y Fw(\)!\()2463 4061 y Fs(n)p Fn(\000)p Ft(1)p 2464 4075 V 2510 4123 a(2)2618 4094 y Fw(+)g Fu(m)2774 4106 y Ft(2)2811 4094 y Fw(\)!\()2908 4061 y Fs(n)p Fn(\000)p Ft(1)p 2909 4075 V 2955 4123 a(2)3063 4094 y Fr(\000)g Fu(m)3219 4106 y Ft(2)3256 4094 y Fw(\)!)3618 4001 y(\(2.44\))523 4327 y(Since)28 b Fr(L)811 4235 y Fo(\020)861 4327 y Fu(z)914 4252 y Fp(n)p Fk(\000)p Fm(1)p 913 4266 110 4 v 954 4299 a(2)1033 4279 y Ft(+)p Fs(m)1143 4287 y Fm(1)900 4349 y Ft(1)1179 4327 y Fu(z)1232 4252 y Fp(n)p Fk(\000)p Fm(1)p 1232 4266 V 1272 4299 a(2)1351 4279 y Fn(\000)p Fs(m)1462 4287 y Fm(1)1218 4349 y Ft(2)1499 4327 y Fu(z)1552 4252 y Fp(n)p Fk(\000)p Fm(1)p 1551 4266 V 1592 4299 a(2)1671 4279 y Ft(+)p Fs(m)1781 4287 y Fm(2)1538 4349 y Ft(3)1817 4327 y Fu(z)1870 4252 y Fp(n)p Fk(\000)p Fm(1)p 1869 4266 V 1910 4299 a(2)1989 4279 y Fn(\000)p Fs(m)2100 4287 y Fm(2)1856 4349 y Ft(4)2136 4235 y Fo(\021)2210 4327 y Fw(=)23 b(0)28 b(for)g(all)g(p)r(ermitted)g(v)-5 b(alues)28 b(of)g Fu(n;)14 b(m)3500 4339 y Ft(1)3537 4327 y Fu(;)g(m)3647 4339 y Ft(2)3684 4327 y Fw(,)29 b(w)n(e)523 4447 y(conclude)e Fu(f)32 b Fr(2)23 b Fw(Ker)o Fr(L)p Fw(.)648 4547 y(Con)n(v)n(ersely)-7 b(,)25 b(supp)r(ose)i Fu(f)32 b Fr(2)23 b Fw(Ker)p Fr(L)p Fw(.)37 b(Since)850 4647 y Fo(\032)935 4708 y Fu(z)978 4671 y Fs(a)1014 4679 y Fm(1)974 4730 y Ft(1)1050 4708 y Fu(z)1093 4671 y Fs(a)1129 4679 y Fm(2)1089 4730 y Ft(2)1165 4708 y Fu(z)1208 4671 y Fs(a)1244 4679 y Fm(3)1204 4730 y Ft(3)1280 4708 y Fu(z)1323 4671 y Fs(a)1359 4679 y Fm(4)1319 4730 y Ft(4)p 922 4745 486 4 v 922 4761 a Fr(p)p 991 4761 417 4 v 65 x Fu(a)1035 4838 y Ft(1)1072 4826 y Fw(!)p Fu(a)1139 4838 y Ft(2)1176 4826 y Fw(!)p Fu(a)1243 4838 y Ft(3)1280 4826 y Fw(!)p Fu(a)1347 4838 y Ft(4)1385 4826 y Fw(!)1441 4764 y Fr(j)23 b Fu(a)1531 4776 y Fs(j)1589 4764 y Fw(is)g(a)45 b(nonnegativ)n(e)22 b(in)n(teger)g(n)n(um)n(b)r(er)p Fu(;)46 b(j)28 b Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)p Fu(;)g Fw(4)p Fu(:)3277 4647 y Fo(\033)523 4981 y Fw(is)28 b(an)f(orthonormal)e(basis)i(of)h Fr(B)1551 4993 y Ft(4)1615 4981 y Fw(then)g(f)g(can)g(b)r(e)g(written)f(as)1642 5192 y Fu(f)32 b Fw(=)1802 5113 y Fo(X)1844 5287 y Fq(a)1936 5192 y Fu(A)1998 5204 y Fq(a)2062 5136 y Fu(z)2105 5099 y Fs(a)2141 5107 y Fm(1)2101 5158 y Ft(1)2177 5136 y Fu(z)2220 5099 y Fs(a)2256 5107 y Fm(2)2216 5158 y Ft(2)2292 5136 y Fu(z)2335 5099 y Fs(a)2371 5107 y Fm(3)2331 5158 y Ft(3)2407 5136 y Fu(z)2450 5099 y Fs(a)2486 5107 y Fm(4)2446 5158 y Ft(4)p 2049 5173 486 4 v 2049 5189 a Fr(p)p 2118 5189 417 4 v 65 x Fu(a)2162 5266 y Ft(1)2199 5254 y Fw(!)p Fu(a)2266 5266 y Ft(2)2304 5254 y Fw(!)p Fu(a)2371 5266 y Ft(3)2408 5254 y Fw(!)p Fu(a)2475 5266 y Ft(4)2512 5254 y Fw(!)3618 5192 y(\(2.45\))523 5438 y(with)c Fv(a)c Fw(=)e(\()p Fu(a)945 5450 y Ft(1)983 5438 y Fu(;)14 b(a)1064 5450 y Ft(2)1101 5438 y Fu(;)g(a)1182 5450 y Ft(3)1219 5438 y Fu(;)g(a)1300 5450 y Ft(4)1337 5438 y Fw(\))28 b(Th)n(us)f(w)n(e)g(ha)n(v)n(e)1191 5658 y(0)22 b(=)h Fr(L)p Fu(f)32 b Fw(=)1560 5579 y Fo(X)1602 5753 y Fq(a)1694 5658 y Fu(A)1756 5670 y Fq(a)1797 5658 y Fw(\()p Fu(a)1873 5670 y Ft(1)1929 5658 y Fw(+)18 b Fu(a)2056 5670 y Ft(2)2112 5658 y Fr(\000)g Fu(a)2239 5670 y Ft(3)2294 5658 y Fr(\000)g Fu(a)2421 5670 y Ft(4)2458 5658 y Fw(\))2514 5601 y Fu(z)2557 5565 y Fs(a)2593 5573 y Fm(1)2553 5624 y Ft(1)2629 5601 y Fu(z)2672 5565 y Fs(a)2708 5573 y Fm(2)2668 5624 y Ft(2)2744 5601 y Fu(z)2787 5565 y Fs(a)2823 5573 y Fm(3)2783 5624 y Ft(3)2858 5601 y Fu(z)2901 5565 y Fs(a)2937 5573 y Fm(4)2897 5624 y Ft(4)p 2501 5639 486 4 v 2501 5655 a Fr(p)p 2570 5655 417 4 v 65 x Fu(a)2614 5732 y Ft(1)2651 5720 y Fw(!)p Fu(a)2718 5732 y Ft(2)2755 5720 y Fw(!)p Fu(a)2822 5732 y Ft(3)2859 5720 y Fw(!)p Fu(a)2926 5732 y Ft(4)2963 5720 y Fw(!)3618 5658 y(\(2.46\))p eop %%Page: 13 13 13 12 bop 3747 324 a Fw(13)523 614 y(No)n(w)26 b(consider)g(the)h(case) f(when)h Fv(b)d Fw(=)e(\()p Fu(b)1799 626 y Ft(1)1836 614 y Fu(;)14 b(b)1909 626 y Ft(2)1946 614 y Fu(;)g(b)2019 626 y Ft(3)2056 614 y Fu(;)g(b)2129 626 y Ft(4)2166 614 y Fw(\))27 b(is)g(suc)n(h)f(that)h Fu(b)2709 626 y Ft(1)2763 614 y Fw(+)17 b Fu(b)2881 626 y Ft(2)2935 614 y Fr(\000)f Fu(b)3052 626 y Ft(3)3106 614 y Fr(\000)g Fu(b)3223 626 y Ft(4)3283 614 y Fr(6)p Fw(=)23 b(0.)36 b(By)27 b(taking)523 714 y(the)38 b(inner)g(pro)r(duct)g(of)f Fr(L)p Fu(f)47 b Fw(with)38 b Fu(z)1712 677 y Fs(b)1741 685 y Fm(1)1708 736 y Ft(1)1777 714 y Fu(z)1820 677 y Fs(b)1849 685 y Fm(2)1816 736 y Ft(2)1885 714 y Fu(z)1928 677 y Fs(b)1957 685 y Fm(3)1924 736 y Ft(3)1993 714 y Fu(z)2036 677 y Fs(b)2065 685 y Fm(4)2032 736 y Ft(4)2139 714 y Fw(w)n(e)g(obtain)f Fu(A)2602 726 y Fq(b)2688 714 y Fw(=)j(0.)67 b(Th)n(us)37 b(f)i(m)n(ust)e(b)r(e)i(a)e(linear)523 814 y(com)n(bination)29 b(of)g(monomials)f Fu(z)1554 777 y Fs(b)1583 785 y Fm(1)1550 836 y Ft(1)1619 814 y Fu(z)1662 777 y Fs(b)1691 785 y Fm(2)1658 836 y Ft(2)1727 814 y Fu(z)1770 777 y Fs(b)1799 785 y Fm(3)1766 836 y Ft(3)1835 814 y Fu(z)1878 777 y Fs(b)1907 785 y Fm(4)1874 836 y Ft(4)1972 814 y Fw(where)h Fu(b)2250 826 y Ft(1)2307 814 y Fw(+)19 b Fu(b)2427 826 y Ft(2)2483 814 y Fr(\000)g Fu(b)2603 826 y Ft(3)2660 814 y Fr(\000)g Fu(b)2780 826 y Ft(4)2843 814 y Fw(=)25 b(0.)42 b(But)30 b(eac)n(h)e(one)h(of)g(these)523 913 y(monomials)e(b)r(elongs)g(to)g(some)g(space)g Fr(M)1868 925 y Fs(n)1913 913 y Fw(,)h(therefore)e Fu(f)32 b Fr(2)23 b(F)2523 925 y Ft(4)2560 913 y Fw(.)1185 b Fj(2)648 1013 y Fw(The)27 b(Bargmann)f(transform)h Fv(B)1673 1028 y Fs(S)1717 1011 y Fm(3)1781 1013 y Fw(is)g(an)h(in)n(tegral)e(op)r (erator)1383 1233 y Fv(B)1451 1248 y Fs(S)1495 1231 y Fm(3)1531 1233 y Fw(\011\()p Fv(z)p Fw(\))e(=)1814 1120 y Fo(Z)1860 1309 y Fs(x)p Fn(2)p Fs(S)1987 1292 y Fm(3)2036 1233 y Fu(K)2107 1248 y Fs(S)2151 1231 y Fm(3)2187 1233 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\)\011\()p Fv(x)p Fw(\))p Fu(d\033)2649 1245 y Ft(3)2689 1233 y Fw(\()p Fv(x)p Fw(\))815 b(\(2.47\))523 1462 y(with)28 b(k)n(ernel)1328 1706 y Fu(K)1399 1721 y Fs(S)1443 1704 y Fm(3)1479 1706 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)1813 1602 y Fn(1)1786 1627 y Fo(X)1784 1803 y Fs(n)p Ft(=1)1923 1602 y Fs(n)p Fn(\000)p Ft(1)1926 1627 y Fo(X)1930 1806 y Fs(`)p Ft(=0)2144 1602 y Fs(`)2098 1627 y Fo(X)2063 1806 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 2266 1639 67 4 v 2266 1706 a Fu(Y)2333 1718 y Fs(n`m)2465 1706 y Fw(\()p Fv(x)p Fw(\))p Fu(y)2620 1718 y Fs(n`m)2752 1706 y Fw(\()p Fv(z)p Fw(\))760 b(\(2.48\))648 1970 y(As)27 b(in)h(the)g(case)f(of)g (the)h(2-sphere,)f Fu(K)1846 1985 y Fs(S)1890 1968 y Fm(3)1953 1970 y Fw(can)g(b)r(e)h(written)g(as)f(a)g(p)r(o)n(w)n(er)g (series)f(of)i(a)f(function)1511 2230 y(\010)1571 2245 y Fs(S)1615 2228 y Fm(3)1674 2230 y Fw(=)1805 2126 y Ft(1)1762 2151 y Fo(X)1766 2330 y Fs(`)p Ft(=0)1922 2126 y Fs(m)p Ft(=)p Fs(`)1931 2151 y Fo(X)1896 2330 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 2099 2163 V 2099 2230 a Fu(Y)2166 2242 y Ft(2)p Fs(`m)2290 2230 y Fw(\()p Fv(x)p Fw(\))p Fu(y)2445 2242 y Ft(2)p Fs(`m)2569 2230 y Fw(\()p Fv(z)p Fw(\))943 b(\(2.49\))523 2494 y(in)28 b(the)g(follo)n(wing)e(w)n(a)n(y) 1482 2739 y Fu(K)1553 2754 y Fs(S)1597 2737 y Fm(3)1656 2739 y Fw(=)1773 2635 y Fn(1)1746 2660 y Fo(X)1743 2836 y Fs(n)p Ft(=1)1952 2614 y Fr(p)p 2021 2614 92 4 v 69 x Fw(2)p Fu(n)p 1892 2720 281 4 v 1892 2796 a Fw(\()p Fu(n)19 b Fr(\000)f Fw(1\)!)2249 2683 y(1)p 2193 2720 154 4 v 2193 2797 a(2)2235 2773 y Fs(n=)p Ft(2)2370 2739 y Fw(\()q(\010)2463 2754 y Fs(S)2507 2737 y Fm(3)2543 2739 y Fw(\))2575 2697 y Fs(n)p Fn(\000)p Ft(1)3618 2739 y Fw(\(2.50\))523 2998 y(with)28 b(the)g(follo)n(wing)f(explicit)h (expression)e(for)h(\010)2086 3013 y Fs(S)2130 2996 y Fm(3)2166 2998 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\):)729 3189 y(\010)789 3204 y Fs(S)833 3187 y Fm(3)870 3189 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))84 b(=)1295 3116 y Fr(p)p 1364 3116 42 4 v 73 x Fw(2)13 b([)p Fu(z)1481 3201 y Ft(1)1518 3189 y Fu(z)1557 3201 y Ft(4)1594 3189 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))20 b Fr(\000)e Fu({)c Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin\()p Fu(\025)p Fw(\)\))20 b Fr(\000)e Fu(z)2611 3201 y Ft(2)2648 3189 y Fu(z)2687 3201 y Ft(3)2724 3189 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))i(+)e Fu({)c Fw(cos)n(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\)\))1295 3314 y(+)23 b Fu({z)1449 3326 y Ft(1)1486 3314 y Fu(z)1525 3326 y Ft(3)1575 3314 y Fw(sin\()p Fu(\022)r Fw(\))14 b(sin)q(\()p Fu(\025)p Fw(\))g(exp\()p Fr(\000)p Fu({\036)p Fw(\))19 b Fr(\000)f Fu({z)2525 3326 y Ft(2)2562 3314 y Fu(z)2601 3326 y Ft(4)2652 3314 y Fw(sin\()p Fu(\022)r Fw(\))c(sin\()p Fu(\025)p Fw(\))g(exp)q(\()p Fu({\036)p Fw(\)])226 b(\(2.51\))523 3496 y(The)28 b(pro)r(of)f(of)g(this)h(last)f (fact)h(is)g(giv)n(en)e(in)i(app)r(endix)g(B.)p eop %%Page: 14 14 14 13 bop 3747 324 a Fw(14)523 614 y Fv(I)s(I)s(I.)22 b(THE)g(KERNEL)g(OF)g(THE)h(BAR)m(GMANN)f(TRANSF)m(ORM)f(AND)i (CANONICAL)523 714 y(TRANSF)m(ORMA)-8 b(TIONS)648 861 y Fw(In)21 b(this)g(section)f(w)n(e)h(describ)r(e)f(a)h(b)r(eautiful)h (relationship)e(b)r(et)n(w)n(een)g(the)i(Bargmann)d(T)-7 b(ransform)19 b(and)523 960 y(a)30 b(corresp)r(onding)e(canonical)g (transformation)g(whic)n(h)i(pla)n(ys)f(the)i(role)e(of)g(its)i (classical)d(analog.)42 b(This)523 1060 y(fact)24 b(o)r(ccurs)e(in)i (the)f(three)h(cases)e(describ)r(ed)h(in)h(the)g(previous)e(section.)35 b(Namely)-7 b(,)24 b(w)n(e)f(ha)n(v)n(e)f(sho)n(wn)h(that)523 1160 y(the)28 b(k)n(ernel)e(of)h(the)g(Bargmann)f(transform)g(is)h(a)f (p)r(o)n(w)n(er)g(series)g(of)h(a)g(certain)f(function.)38 b(This)27 b(function)523 1259 y(is)k(in)g(turn)g(a)f(generating)g (function)h(of)g(the)g(men)n(tioned)g(canonical)e(transformation.)45 b(This)31 b(will)g(giv)n(e)523 1359 y(us)25 b(symplectomorphisms)e (mapping)i(the)g(phase)g(space)f(of)g(a)h(particle)f(mo)n(ving)g(in)h Fv(R)3243 1329 y Fs(n)3288 1359 y Fw(,)g Fu(S)3392 1329 y Ft(2)3454 1359 y Fw(or)f Fu(S)3609 1329 y Ft(3)3671 1359 y Fw(on)n(to)523 1459 y(a)j(phase)g(space)g(describ)r(ed)h(b)n(y)f (complex)g(co)r(ordinates.)648 1558 y(W)-7 b(e)30 b(start)f(b)n(y)h (describing)f(this)i(relationship)e(in)h(the)g(case)f(of)h(the)h(BT)f (for)f Fu(L)3127 1528 y Ft(2)3164 1558 y Fw(\()p Fv(R)3268 1528 y Fs(n)3313 1558 y Fw(\))h(and)g(then)h(for)523 1658 y(the)d(cases)e Fu(L)931 1628 y Ft(2)968 1658 y Fw(\()p Fu(S)1056 1628 y Ft(2)1093 1658 y Fw(\))i(and)g Fu(L)1372 1628 y Ft(2)1409 1658 y Fw(\()p Fu(S)1497 1628 y Ft(3)1534 1658 y Fw(\).)523 1828 y Fv(I)s(I)s(I.1)j(Canonical)h (transformation)f(for)h(a)g(particle)g(mo)m(ving)e(in)i(R)2929 1798 y Fs(n)648 1975 y Fw(In)26 b(this)g(case)f(the)h(relationship)f(b) r(et)n(w)n(een)g(the)i(k)n(ernel)d(of)i(the)g(Bargmann)e(transform)h (and)h(a)f(corre-)523 2075 y(sp)r(onding)i(canonical)g(transformation)f (is)h(w)n(ell)g(kno)n(wn)g(\(see)2488 2045 y Ft(5)p Fs(;)p Ft(19)2611 2075 y Fw(\).)648 2174 y(F)-7 b(ollo)n(wing)26 b(S.)i(Gra\016)f(and)g(A.)h(P)n(armeggiani)2081 2144 y Ft(5)2143 2174 y Fw(,)g(let)g(us)g(consider)e(the)i(function)1363 2379 y(\010)1423 2391 y Fs(n)1468 2379 y Fw(\()p Fv(q)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)f Fr(\000)p Fu({)1879 2262 y Fo(\022)1939 2379 y Fr(\000)2014 2323 y Fw(1)p 2014 2360 42 4 v 2014 2436 a(2)2065 2379 y(\()p Fv(z)2139 2345 y Ft(2)2195 2379 y Fw(+)c Fv(q)2329 2345 y Ft(2)2366 2379 y Fw(\))g(+)2500 2307 y Fr(p)p 2569 2307 V 72 x Fw(2)p Fv(z)f Fr(\001)h Fv(q)2763 2262 y Fo(\023)3659 2379 y Fw(\(3.1\))523 2584 y(whic)n(h)32 b(app)r(ears)f(in)h(the)g (argumen)n(t)f(of)h(the)g(exp)r(onen)n(tial)g(function)g(whic)n(h)g(in) g(turn)g(giv)n(es)f(the)h(k)n(ernel)523 2684 y Fu(A)p Fw(\()p Fv(q)p Fu(;)14 b Fv(z)p Fw(\))29 b(of)f Fv(B)970 2696 y Fq(R)1026 2680 y Fp(n)1070 2684 y Fw(.)648 2783 y(Let)c(us)h(consider)f(no)n(w)g(a)g(particle)g(mo)n(ving)f(in)i Fv(R)2204 2753 y Fs(n)2249 2783 y Fw(.)36 b(The)25 b(phase)f(space)g (in)h(this)g(case)e(is)i Fu(T)3490 2753 y Fn(\003)3527 2783 y Fv(R)3599 2753 y Fs(n)3669 2783 y Fw(with)523 2883 y(co)r(ordinates)h(\()p Fv(q)p Fu(;)14 b Fv(p)p Fw(\))29 b(and)e(symplectic)h(form)f Fu(!)2015 2895 y Fs(n)2083 2883 y Fw(=)2171 2821 y Fo(P)2258 2841 y Fs(n)2258 2908 y(j)s Ft(=1)2391 2883 y Fu(dq)2471 2895 y Fs(j)2506 2883 y Fr(^)q Fu(dp)2647 2895 y Fs(j)2682 2883 y Fw(.)648 2983 y(No)n(w)35 b(consider)g(the)h(function)h(\010)1722 2995 y Fs(n)1803 2983 y Fw(as)e(a)h(generating)e(function)j(of)f(a)f (canonical)g(transformation)523 3082 y(mapping)27 b(the)h(v)-5 b(ariables)27 b(\()p Fv(q)p Fu(;)14 b Fv(p)p Fw(\))28 b(to)g(the)g(v)-5 b(ariables)26 b(\()p Fv(z)p Fu(;)14 b Fv(w)q Fw(\))p Fr(2)r Fu(T)2504 3052 y Fn(\003)2541 3082 y Fw(\()p Fv(C)2642 3052 y Fs(n)2687 3082 y Fw(\))28 b(via)f(the)h(follo)n(wing)f(equations:)1583 3285 y Fu(p)1625 3297 y Fs(j)1682 3285 y Fw(=)c Fr(\000)1845 3229 y Fu(@)5 b Fw(\010)1954 3241 y Fs(n)p 1845 3266 154 4 v 1861 3342 a Fu(@)g(q)1947 3354 y Fs(j)2101 3285 y Fu(j)28 b Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g(:)g(:)g(:)f(;)h(n)1603 3513 y(w)1662 3525 y Fs(j)1720 3513 y Fw(=)23 b Fu({)1845 3457 y(@)5 b Fw(\010)1954 3469 y Fs(n)p 1845 3494 V 1860 3570 a Fu(@)g(z)1948 3582 y Fs(j)2101 3513 y Fu(j)28 b Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g(:)g(:)g(:)f(;)h(n)1054 b Fw(\(3.2\))523 3720 y(Note)28 b(that)g(b)n(y)f(considering)f(these)i (equations)f(w)n(e)g(guaran)n(tee)e(that,)1591 3837 y Fs(n)1552 3862 y Fo(X)1554 4039 y Fs(j)s Ft(=1)1685 3941 y Fu(dq)1765 3953 y Fs(j)1801 3941 y Fr(^)p Fu(dp)1941 3953 y Fs(j)1999 3941 y Fw(=)e Fr(\000)p Fu({)2231 3837 y Fs(n)2192 3862 y Fo(X)2194 4039 y Fs(j)s Ft(=1)2326 3941 y Fu(dz)2408 3953 y Fs(j)2442 3941 y Fr(^)q Fu(dw)2600 3953 y Fs(j)3659 3941 y Fw(\(3.3\))648 4189 y(The)k(expressions)f(for)h (\()p Fv(z)p Fu(;)14 b Fv(w)q Fw(\))29 b(in)f(terms)f(of)h(\()p Fv(q)p Fu(;)14 b Fv(p)p Fw(\))28 b(are)1776 4388 y Fu(z)1815 4400 y Fs(j)1872 4388 y Fw(=)2005 4332 y(1)p 1970 4369 111 4 v 1970 4385 a Fr(p)p 2039 4385 42 4 v 69 x Fw(2)2091 4388 y(\()p Fu(q)2160 4400 y Fs(j)2213 4388 y Fr(\000)18 b Fu({p)2365 4400 y Fs(j)2400 4388 y Fw(\))1755 4608 y Fu(w)1814 4620 y Fs(j)1872 4608 y Fw(=)2005 4552 y(1)p 1970 4589 111 4 v 1970 4606 a Fr(p)p 2039 4606 42 4 v 68 x Fw(2)2091 4608 y(\()p Fu(q)2160 4620 y Fs(j)2213 4608 y Fw(+)g Fu({p)2365 4620 y Fs(j)2400 4608 y Fw(\))1227 b(\(3.4\))523 4811 y(In)30 b(this)h(w)n(a)n(y)e(w)n(e)g(obtain)h(that)h Fu(w)1592 4823 y Fs(j)1657 4811 y Fw(m)n(ust)g(b)r(e)f(the)h(complex)e (conjugate)h(of)g Fu(z)2969 4823 y Fs(j)3004 4811 y Fw(.)44 b(This)30 b(fact)h(suggests)d(to)523 4911 y(de\014ne)g(the)g(follo)n (wing)e(submanifold)i(of)g Fu(T)1881 4881 y Fn(\003)1918 4911 y Fw(\()p Fv(C)2019 4881 y Fs(n)2064 4911 y Fw(\))1592 5070 y Fr(D)1656 5082 y Fs(n)1724 5070 y Fw(=)23 b Fr(f)p Fw(\()p Fv(z)p Fu(;)14 b Fv(w)q Fw(\))23 b Fr(2)h Fv(C)2238 5036 y Ft(2)p Fs(n)2316 5070 y Fr(j)p Fu(w)i Fw(=)p 2511 5024 43 4 v 22 w Fv(z)q Fr(g)1063 b Fw(\(3.5\))523 5230 y(Since)30 b Fu(T)803 5200 y Fn(\003)841 5230 y Fw(\()p Fv(C)942 5200 y Fs(n)987 5230 y Fw(\))h(is)f(a)g(symplectic)g(manifold) g(with)h(symplectic)f(form)g Fu(\024)2815 5242 y Fs(n)2887 5230 y Fw(=)d Fr(\000)p Fu({)3085 5168 y Fo(P)3172 5188 y Fs(n)3172 5255 y(j)s Ft(=1)3305 5230 y Fu(dz)3387 5242 y Fs(j)3422 5230 y Fr(^)p Fu(dw)3579 5242 y Fs(j)3615 5230 y Fw(,)k(then)523 5330 y(w)n(e)21 b(can)f(consider)g Fr(D)1165 5342 y Fs(n)1231 5330 y Fw(as)h(a)f(symplectic)h(manifold)g (with)h(symplectic)f(form)f Fu(\034)2938 5342 y Fs(n)3005 5330 y Fw(giv)n(en)g(b)n(y)h(the)g(restriction)523 5429 y(of)28 b Fu(\024)666 5441 y Fs(n)739 5429 y Fw(to)g Fr(D)905 5441 y Fs(n)950 5429 y Fw(.)38 b(Notice)28 b(that)g(w)n(e)g (can)g(regard)e(the)i(symplectic)g(manifold)g(\()p Fr(D)2979 5441 y Fs(n)3025 5429 y Fu(;)14 b(\034)3098 5441 y Fs(n)3143 5429 y Fw(\))28 b(as)g(the)g(symplectic)523 5529 y(manifold)h(\()p Fv(C)969 5541 y Fs(n)1015 5529 y Fu(;)14 b(\026)1102 5541 y Fs(n)1147 5529 y Fw(\))29 b(where)g Fu(\026)1500 5541 y Fs(n)1570 5529 y Fw(=)c Fr(\000)p Fu({)1766 5467 y Fo(P)1853 5487 y Fs(n)1853 5554 y(j)s Ft(=1)1986 5529 y Fu(dz)2068 5541 y Fs(j)2103 5529 y Fr(^)p Fu(d)p 2201 5483 V(z)2244 5541 y Fs(j)2308 5529 y Fw(and)k(\()p Fv(z)p Fu(;)p 2582 5482 V 14 w Fv(z)q Fw(\))c Fr(7!)h Fw(\()p Fv(z)p Fw(\))k(b)r(eing)f(the)h(symplectomor-)523 5629 y(phism)e(b)r(et)n(w)n(een)g(\()p Fr(D)1185 5641 y Fs(n)1230 5629 y Fu(;)14 b(\034)1303 5641 y Fs(n)1348 5629 y Fw(\))28 b(and)g(\()p Fv(C)1671 5641 y Fs(n)1716 5629 y Fu(;)14 b(\026)1803 5641 y Fs(n)1848 5629 y Fw(\).)648 5728 y(Th)n(us)27 b(w)n(e)g(obtain)g(the)h(follo)n(wing)p eop %%Page: 15 15 15 14 bop 3747 324 a Fw(15)523 614 y Fv(Theorem)30 b(3.1)41 b Fl(The)23 b(function)e Fw(\010)1632 626 y Fs(n)1699 614 y Fl(is)h(a)g(gener)l(ating)g(function)g(of)g(a)g(line)l(ar)h(c)l (anonic)l(al)f(tr)l(ansformation)523 714 y(which)39 b(is)f(a)g(symple)l (ctomorphism)i Fr(C)1728 726 y Fs(n)1810 714 y Fl(fr)l(om)e Fw(\()p Fu(T)2107 684 y Fn(\003)2144 714 y Fv(R)2216 684 y Fs(n)2261 714 y Fu(;)14 b(!)2350 726 y Fs(n)2395 714 y Fw(\))38 b Fl(onto)f Fw(\()p Fv(C)2762 726 y Fs(n)2808 714 y Fu(;)14 b(\026)2895 726 y Fs(n)2940 714 y Fw(\))37 b Fl(given)h(by)g(the)g(e)l(quations)523 814 y Fu(z)562 826 y Fs(j)640 814 y Fw(=)786 781 y Ft(1)p 759 795 88 4 v 759 803 a Fn(p)p 813 803 34 3 v 813 852 a Ft(2)856 814 y Fw(\()p Fu(q)925 826 y Fs(j)987 814 y Fr(\000)27 b Fu({p)1148 826 y Fs(j)1182 814 y Fw(\))88 b Fu(j)49 b Fw(=)43 b(1)p Fu(;)14 b Fw(2)p Fu(;)g(:)g(:)g(:)f(;)h(n)41 b Fl(The)h(tr)l(ansformation)g Fr(C)2692 826 y Fs(n)2778 814 y Fl(maps)g(the)f(Hamiltonian)h Fu(H)51 b Fw(=)533 905 y Ft(1)p 533 919 34 4 v 533 966 a(2)590 875 y Fo(P)678 895 y Fs(n)678 962 y(j)s Ft(=1)810 937 y Fu(q)850 907 y Ft(2)847 959 y Fs(j)906 937 y Fw(+)18 b Fu(p)1031 907 y Ft(2)1031 959 y Fs(j)1098 937 y Fl(to)30 b(the)g(Hamiltonian)g Fu(H)1891 907 y Fn(0)1937 937 y Fw(=)2025 875 y Fo(P)2113 895 y Fs(n)2113 962 y(j)s Ft(=1)2245 937 y Fu(z)2284 949 y Fs(j)p 2319 892 43 4 v 2319 937 a Fu(z)2361 949 y Fs(j)523 1103 y Fw(This)f(theorem)g(giv)n(es)e(the)j(classical)d (analog)h(of)h(the)g(Bargmann)e(T)-7 b(ransform)28 b Fv(B)3091 1115 y Fq(R)3147 1099 y Fp(n)3221 1103 y Fw(in)h(the)g(sense) g(that)523 1203 y Fv(B)591 1215 y Fq(R)647 1199 y Fp(n)723 1203 y Fw(is)h(a)h(unitary)f(op)r(erator)f(from)i Fu(L)1773 1173 y Ft(2)1809 1203 y Fw(\()p Fv(R)1913 1173 y Fs(n)1958 1203 y Fw(\))h(on)n(to)e(the)h(Bargmann)e(space)h Fr(B)3046 1215 y Fs(n)3122 1203 y Fw(and)h(in)n(tert)n(wines)f(the)523 1327 y(quan)n(tum)37 b(Hamiltonian)f(of)h(the)g(harmonic)f(oscillator) 2386 1306 y(^)2364 1327 y Fu(H)45 b Fw(=)2591 1295 y Ft(1)p 2591 1309 34 4 v 2591 1356 a(2)2648 1235 y Fo(\020)2697 1265 y(P)2785 1286 y Fs(n)2785 1352 y(j)s Ft(=1)2918 1327 y Fr(\000)2993 1290 y Fs(@)3032 1265 y Fm(2)p 2992 1308 106 4 v 2993 1358 a Fs(@)t(q)3064 1338 y Fm(2)3062 1379 y Fp(j)3127 1327 y Fw(+)18 b Fu(q)3250 1297 y Ft(2)3247 1349 y Fs(j)3306 1327 y Fr(\000)g Fw(1)3431 1235 y Fo(\021)3517 1327 y Fw(with)37 b(the)523 1474 y(op)r(erator)880 1453 y(^)858 1474 y Fu(H)934 1444 y Fn(0)980 1474 y Fw(=)1068 1412 y Fo(P)1155 1433 y Fs(n)1155 1499 y(j)s Ft(=1)1288 1474 y Fu(z)1327 1486 y Fs(i)1393 1442 y(@)p 1364 1456 98 4 v 1364 1503 a(@)t(z)1435 1511 y Fp(i)1471 1474 y Fw(.)523 1658 y Fv(I)s(I)s(I.2)31 b(Canonical)h(transformation)f(for)h (a)g(particle)g(mo)m(ving)e(on)i Fr(S)2934 1628 y Ft(2)648 1805 y Fw(The)38 b(description)f(ab)r(o)n(v)n(e)f(for)i(a)f(particle)g (mo)n(ving)g(in)h Fv(R)2516 1775 y Fs(n)2599 1805 y Fw(suggests)f(to)h (consider)e(the)j(function)539 1872 y Fn(\000)p Fs({)p 533 1886 88 4 v 533 1894 a Fn(p)p 588 1894 34 3 v 49 x Ft(3)631 1905 y Fw(\010)691 1920 y Fs(S)735 1903 y Fm(2)771 1905 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))34 b(\(the)f(factor)1457 1872 y Ft(1)p 1430 1886 88 4 v 1430 1894 a Fn(p)p 1485 1894 34 3 v 49 x Ft(3)1560 1905 y Fw(is)f(just)h(to)f(scale)f(v)-5 b(ariables)31 b(in)i(a)f(con)n(v)n (enien)n(t)f(w)n(a)n(y\))g(as)h(a)g(generating)523 2026 y(function)k(of)f(a)g(canonical)f(transformation)g Fr(C)2025 2041 y Fs(S)2069 2024 y Fm(2)2140 2026 y Fw(whic)n(h)h(maps)g(the)h (phase)f(space)f Fu(T)3290 1995 y Fn(\003)3327 2026 y Fu(S)3383 1995 y Ft(2)3444 2026 y Fr(\000)23 b(f)p Fw(0)p Fr(g)34 b Fw(\(i.e.)523 2125 y Fu(T)584 2095 y Fn(\003)621 2125 y Fu(S)677 2095 y Ft(2)742 2125 y Fw(min)n(us)27 b(the)h(zero)f(section\))g(on)n(to)g(a)g(submanifold)h(of)g Fu(T)2493 2095 y Fn(\003)2530 2125 y Fv(C)2599 2095 y Ft(2)2636 2125 y Fw(.)648 2225 y(Th)n(us)f(let)h(us)f(consider)g(the)h (equations)1345 2446 y Fu(p)1387 2458 y Fs(\022)1507 2446 y Fw(=)83 b Fr(\000)1750 2390 y Fu(@)p 1730 2427 90 4 v 1730 2503 a(@)5 b(\022)1849 2390 y Fr(\000)p Fu({)p 1839 2427 111 4 v 1839 2444 a Fr(p)p 1908 2444 42 4 v 68 x Fw(3)1960 2446 y(\010)2020 2461 y Fs(S)2064 2444 y Fm(2)2100 2446 y Fu(;)250 b(p)2415 2458 y Fs(\036)2482 2446 y Fw(=)23 b Fr(\000)2669 2390 y Fu(@)p 2645 2427 99 4 v 2645 2503 a(@)5 b(\036)2772 2390 y Fr(\000)p Fu({)p 2762 2427 111 4 v 2762 2444 a Fr(p)p 2832 2444 42 4 v 2832 2512 a Fw(3)2883 2446 y(\010)2943 2461 y Fs(S)2987 2444 y Fm(2)1330 2670 y Fu(w)1389 2682 y Fs(j)1507 2670 y Fw(=)83 b Fu({)1728 2614 y(@)p 1692 2651 123 4 v 1692 2727 a(@)5 b(z)1780 2739 y Fs(j)1843 2614 y Fr(\000)p Fu({)p 1834 2651 111 4 v 1834 2668 a Fr(p)p 1903 2668 42 4 v 69 x Fw(3)1954 2670 y(\010)2014 2685 y Fs(S)2058 2669 y Fm(2)2331 2670 y Fu(j)28 b Fw(=)23 b(1)p Fu(;)14 b Fw(2)1057 b(\(3.6\))523 2899 y(Since)807 3122 y(\010)867 3137 y Fs(S)911 3120 y Fm(2)947 3122 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))25 b(=)1262 2997 y Fr(p)p 1331 2997 V 69 x Fw(3)p 1262 3103 111 4 v 1297 3179 a(2)1397 3055 y Fo(\000)1435 3122 y Fr(\000)14 b Fw(sin)o(\()p Fu(\022)r Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))p Fu(z)2110 3088 y Ft(2)2106 3142 y(1)2165 3122 y Fw(+)k(2)c(cos)n(\()p Fu(\022)r Fw(\))p Fu(z)2558 3134 y Ft(1)2596 3122 y Fu(z)2635 3134 y Ft(2)2690 3122 y Fw(+)19 b(sin)o(\()p Fu(\022)r Fw(\))14 b(exp)q(\()p Fu({\036)p Fw(\))p Fu(z)3305 3088 y Ft(2)3301 3142 y(2)3342 3055 y Fo(\001)523 3328 y Fw(the)28 b(equations)f(ab)r(o)n(v)n(e)f(tak)n(e)h(the)h(follo)n(wing)e(explicit) i(expression:)996 3527 y Fu(p)1038 3539 y Fs(\022)1159 3527 y Fw(=)82 b Fr(\000)1388 3471 y Fu({)p 1381 3508 42 4 v 1381 3584 a Fw(2)1446 3460 y Fo(\000)1484 3527 y Fw(cos)o(\()p Fu(\022)r Fw(\))14 b(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))p Fu(z)2090 3493 y Ft(2)2086 3548 y(1)2145 3527 y Fw(+)k(2)c(sin)o(\()p Fu(\022)r Fw(\))p Fu(z)2529 3539 y Ft(1)2567 3527 y Fu(z)2606 3539 y Ft(2)2661 3527 y Fr(\000)k Fw(cos\()p Fu(\022)r Fw(\))c(exp\()p Fu({\036)p Fw(\))p Fu(z)3285 3493 y Ft(2)3281 3548 y(2)3322 3460 y Fo(\001)990 3727 y Fu(p)1032 3739 y Fs(\036)1159 3727 y Fw(=)1316 3671 y Fr(\000)p Fw(1)p 1316 3708 107 4 v 1348 3784 a(2)1446 3660 y Fo(\000)1484 3727 y Fw(sin\()p Fu(\022)r Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))p Fu(z)2081 3693 y Ft(2)2077 3748 y(1)2136 3727 y Fw(+)k(sin\()p Fu(\022)r Fw(\))c(exp\()p Fu({\036)p Fw(\))p Fu(z)2750 3693 y Ft(2)2746 3748 y(2)2788 3660 y Fo(\001)979 3879 y Fu(w)1038 3891 y Ft(1)1159 3879 y Fw(=)82 b Fr(\000)14 b Fw(sin)o(\()p Fu(\022)r Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))p Fu(z)1977 3891 y Ft(1)2033 3879 y Fw(+)k(cos)o(\()p Fu(\022)r Fw(\))p Fu(z)2371 3891 y Ft(2)979 4004 y Fu(w)1038 4016 y Ft(2)1159 4004 y Fw(=)82 b(cos)o(\()p Fu(\022)r Fw(\))p Fu(z)1561 4016 y Ft(1)1617 4004 y Fw(+)18 b(sin\()p Fu(\022)r Fw(\))c(exp)q(\()p Fu({\036)p Fw(\))p Fu(z)2228 4016 y Ft(2)3659 4004 y Fw(\(3.7\))648 4187 y(Note)31 b(that)h(b)n(y)g(w)n(orking)e(with)i (spherical)f(co)r(ordinates)f(for)h Fu(S)2650 4157 y Ft(2)2687 4187 y Fw(,)i(w)n(e)f(are)e(dealing)h(with)i(a)e(sp)r (eci\014c)523 4286 y(lo)r(cal)k(c)n(hart)f(of)h Fu(S)1107 4256 y Ft(2)1180 4286 y Fw(and)g(hence)g(with)h(a)f(lo)r(cal)g(c)n (hart)f(for)h Fu(T)2483 4256 y Fn(\003)2520 4286 y Fu(S)2576 4256 y Ft(2)2637 4286 y Fr(\000)23 b(f)p Fw(0)p Fr(g)p Fw(.)59 b(Later)34 b(on)h(w)n(e)g(shall)g(de\014ne)523 4386 y(the)h(canonical)e(transformation)f Fr(C)1663 4401 y Fs(S)1707 4384 y Fm(2)1779 4386 y Fw(b)n(y)i(extending)g(the)h (relationship)f(b)r(et)n(w)n(een)g(\()p Fu(\022)r(;)14 b(\036;)g(p)3468 4398 y Fs(\022)3506 4386 y Fu(;)g(p)3585 4398 y Fs(\036)3629 4386 y Fw(\))35 b(and)523 4486 y(\()p Fu(z)594 4498 y Ft(1)631 4486 y Fu(;)14 b(z)707 4498 y Ft(2)744 4486 y Fu(;)g(w)840 4498 y Ft(1)878 4486 y Fu(;)g(w)974 4498 y Ft(2)1011 4486 y Fw(\))28 b(\(determined)g(b)n(y)f (equation)g(\(3.7\))h(to)f(the)h(whole)f(space)g Fu(T)2952 4455 y Fn(\003)2990 4486 y Fu(S)3046 4455 y Ft(2)3101 4486 y Fr(\000)18 b(f)p Fw(0)p Fr(g)p Fw(.)648 4585 y(In)23 b(order)f(to)h(iden)n(tify)h(the)g(submanifold)g(of)f Fu(T)2110 4555 y Fn(\003)2147 4585 y Fv(C)2216 4555 y Ft(2)2277 4585 y Fw(where)g Fu(T)2574 4555 y Fn(\003)2611 4585 y Fu(S)2667 4555 y Ft(2)2714 4585 y Fr(\000)10 b(f)p Fw(0)p Fr(g)22 b Fw(is)h(mapp)r(ed)h(to,)g(w)n(e)f(need)g(to)523 4685 y(\014nd)j(ho)n(w)f(the)g(v)-5 b(ariables)24 b(\()p Fu(w)1431 4697 y Ft(1)1469 4685 y Fu(;)14 b(w)1565 4697 y Ft(2)1603 4685 y Fw(\))26 b(are)e(related)h(to)g(\()p Fu(z)2242 4697 y Ft(1)2279 4685 y Fu(;)14 b(z)2355 4697 y Ft(2)2392 4685 y Fw(\).)36 b(T)-7 b(o)25 b(do)g(this,)i(w)n(e)e (shall)f(use)i(the)f(condition)523 4784 y(that)i(the)g(v)-5 b(ariables)25 b Fu(\022)r(;)14 b(\036;)g(p)1393 4796 y Fs(\022)1431 4784 y Fu(;)g(p)1510 4796 y Fs(\036)1581 4784 y Fw(are)25 b(real)h(v)-5 b(ariables.)35 b(Th)n(us)27 b(from)f(the)h(\014rst)f(t)n(w)n(o)g(equations)g(\(3.7\))g(and)523 4884 y(the)i(realit)n(y)f(condition)g Fu(\022)e Fw(=)p 1441 4816 42 4 v 23 w Fu(\022)r Fw(,)j Fu(\036)23 b Fw(=)p 1693 4816 50 4 v 23 w Fu(\036)q Fw(,)k Fu(p)1835 4896 y Fs(\022)1896 4884 y Fw(=)p 1983 4838 42 4 v 22 w Fu(p)2025 4904 y Fs(\022)2063 4884 y Fw(,)g Fu(p)2155 4896 y Fs(\036)2222 4884 y Fw(=)p 2310 4838 V 23 w Fu(p)2352 4904 y Fs(\036)2396 4884 y Fw(,)g(w)n(e)h(obtain)1729 5139 y(sin\()p Fu(\022)r Fw(\))23 b(=)2109 5007 y Fo(\014)2109 5057 y(\014)p 2137 5032 43 4 v 21 x Fu(z)2179 5046 y Ft(2)2179 5098 y(1)2235 5078 y Fr(\000)18 b Fu(z)2361 5047 y Ft(2)2357 5098 y(2)2397 5007 y Fo(\014)2397 5057 y(\014)p 2057 5120 420 4 v 2057 5209 a Fr(j)p Fu(z)2119 5221 y Ft(1)2156 5209 y Fr(j)2179 5167 y Ft(2)2235 5209 y Fw(+)g Fr(j)p Fu(z)2380 5221 y Ft(2)2417 5209 y Fr(j)2440 5167 y Ft(2)2487 5139 y Fu(;)1663 5372 y Fw(cos)o(\()p Fu(\022)r Fw(\))24 b(=)1968 5336 y Ft(+)1968 5392 y Fn(\000)p 2061 5270 43 4 v 2061 5315 a Fu(z)2103 5327 y Ft(1)p 2140 5270 V 2140 5315 a Fu(z)2183 5327 y Ft(2)2238 5315 y Fw(+)18 b Fu(z)2360 5327 y Ft(1)2397 5315 y Fu(z)2436 5327 y Ft(2)p 2057 5352 420 4 v 2057 5442 a Fr(j)p Fu(z)2119 5454 y Ft(1)2156 5442 y Fr(j)2179 5400 y Ft(2)2235 5442 y Fw(+)g Fr(j)p Fu(z)2380 5454 y Ft(2)2417 5442 y Fr(j)2440 5400 y Ft(2)2487 5372 y Fu(;)1731 5621 y Fw(exp)o(\()p Fu({\036)p Fw(\))24 b(=)2086 5585 y Ft(+)2086 5642 y Fn(\000)p 2203 5519 43 4 v 2203 5565 a Fu(z)2245 5533 y Ft(2)2245 5585 y(2)2301 5565 y Fr(\000)18 b Fu(z)2427 5535 y Ft(2)2423 5585 y(1)p 2175 5602 316 4 v 2175 5612 a Fo(\014)2175 5662 y(\014)p 2203 5637 43 4 v 20 x Fu(z)2245 5651 y Ft(2)2245 5703 y(2)2301 5682 y Fr(\000)g Fu(z)2427 5654 y Ft(2)2423 5705 y(1)2463 5612 y Fo(\014)2463 5662 y(\014)2501 5621 y Fu(;)1135 b Fw(\(3.8\))p eop %%Page: 16 16 16 15 bop 3747 324 a Fw(16)648 614 y(If)28 b(w)n(e)f(substitute)h (equations)f(\(3.8\))g(in)h(the)g(last)f(t)n(w)n(o)g(equations)g(of)h (\(3.7\),)f(w)n(e)g(obtain:)1698 797 y Fu(w)1757 809 y Ft(1)1817 797 y Fw(=)1882 761 y Ft(+)1882 818 y Fn(\000)p 1961 751 43 4 v 1961 797 a Fu(z)2003 809 y Ft(1)2087 797 y Fw(and)46 b Fu(w)2326 809 y Ft(2)2386 797 y Fw(=)2451 761 y Ft(+)2451 818 y Fn(\000)p 2530 751 V 2530 797 a Fu(z)2572 809 y Ft(2)2633 797 y Fu(:)1003 b Fw(\(3.9\))523 980 y(W)-7 b(e)22 b(shall)f(restrict)g(ourselv)n(es)e(to)j(consider)e (the)i(plus)f(sign)g(in)h(equations)f(\(3.8\))g(and)g(\(3.9\))h(in)f (suc)n(h)g(a)g(w)n(a)n(y)523 1079 y(that)g(\()p Fu(w)787 1091 y Ft(1)825 1079 y Fu(;)14 b(w)921 1091 y Ft(2)959 1079 y Fw(\))23 b(=)g(\()p 1134 1034 V Fu(z)1176 1091 y Ft(1)1213 1079 y Fu(;)p 1250 1034 V 14 w(z)1293 1091 y Ft(2)1330 1079 y Fw(\).)35 b(It)21 b(can)f(b)r(e)h(sho)n(wn,)h(b)n(y) e(expressing)f(\()p Fu(z)2596 1091 y Ft(1)2634 1079 y Fu(;)14 b(z)2710 1091 y Ft(2)2746 1079 y Fw(\))21 b(in)g(terms)g(of)f (\()p Fu(\022)r(;)14 b(\036;)g(p)3439 1091 y Fs(\022)3478 1079 y Fu(;)g(p)3557 1091 y Fs(\036)3600 1079 y Fw(\),)23 b(that)523 1179 y(giv)n(en)j(\()p Fu(\022)r(;)14 b(\036;)g(p)977 1191 y Fs(\022)1016 1179 y Fu(;)g(p)1095 1191 y Fs(\036)1138 1179 y Fw(\),)28 b(there)f(are)g(four)f(solutions)h(for)g(\()p Fu(z)2289 1191 y Ft(1)2326 1179 y Fu(;)14 b(z)2402 1191 y Ft(2)2439 1179 y Fw(\))2498 1149 y Ft(1)2535 1179 y Fw(.)37 b(Tw)n(o)27 b(of)g(these)g(solutions)g(corresp)r(ond)523 1279 y(to)d(consider)f(the)h(plus)g(sign)g(in)g(equations)f(\(3.8\),)i (\(3.9\))f(and)g(the)g(other)f(t)n(w)n(o)h(corresp)r(ond)e(to)i(the)g (min)n(us)523 1378 y(sign.)36 b(The)27 b(t)n(w)n(o)f(solutions)f (corresp)r(onding)g(to)h(the)h(plus)g(sign)f(are)g(related)g(to)g(eac)n (h)g(other)g(b)n(y)g(a)g(min)n(us)523 1478 y(sign)g(\(i.e.)37 b(if)28 b(\()p Fu(z)1015 1490 y Ft(1)1052 1478 y Fu(;)14 b(z)1128 1490 y Ft(2)1165 1478 y Fw(\))27 b(is)g(a)f(solution)g(of)h (equations)f(\(3.7\),)h(with)g(\()p Fu(\022)r(;)14 b(\036;)g(p)2800 1490 y Fs(\022)2838 1478 y Fu(;)g(p)2917 1490 y Fs(\036)2961 1478 y Fw(\))27 b(giv)n(en,)f(then)i(\()p Fr(\000)p Fu(z)3584 1490 y Ft(1)3621 1478 y Fu(;)14 b Fr(\000)p Fu(z)3762 1490 y Ft(2)3798 1478 y Fw(\))523 1577 y(is)35 b(a)f(solution)h(to)r (o\).)59 b(Th)n(us)35 b(w)n(e)f(need)h(to)g(iden)n(tify)h(\()p Fu(z)2282 1589 y Ft(1)2319 1577 y Fu(;)14 b(z)2395 1589 y Ft(2)2432 1577 y Fw(\))35 b(with)g Fr(\000)p Fw(\()p Fu(z)2831 1589 y Ft(1)2868 1577 y Fu(;)14 b(z)2944 1589 y Ft(2)2981 1577 y Fw(\))35 b(in)g(order)f(to)h(mak)n(e)f(our)523 1677 y(canonical)e(transformation)f Fr(C)1508 1692 y Fs(S)1552 1675 y Fm(2)1621 1677 y Fw(a)h(function.)53 b(In)32 b(this)h(w)n(a)n(y)-7 b(,)33 b(w)n(e)g(see)f(that)h Fr(C)3030 1692 y Fs(S)3074 1675 y Fm(2)3143 1677 y Fw(tak)n(es)f(v)-5 b(alues)32 b(in)h(the)523 1777 y(follo)n(wing)27 b(submanifold)g(of)h Fu(T)1498 1747 y Fn(\003)1535 1777 y Fv(C)1604 1747 y Ft(2)1669 1777 y Fw(:)1051 1959 y Fr(D)1115 1971 y Ft(2)1175 1959 y Fw(=)1263 1892 y Fo(\010)1311 1959 y Fw(\()p Fu(z)1382 1971 y Ft(1)1419 1959 y Fu(;)14 b(z)1495 1971 y Ft(2)1532 1959 y Fu(;)g(w)1628 1971 y Ft(1)1666 1959 y Fu(;)g(w)1762 1971 y Ft(2)1799 1959 y Fw(\))24 b Fr(2)f Fv(C)2002 1925 y Ft(4)2039 1959 y Fr(j)p Fu(w)2121 1971 y Ft(1)2182 1959 y Fw(=)p 2270 1914 V 23 w Fu(z)2312 1971 y Ft(1)2349 1959 y Fu(;)60 b(w)2491 1971 y Ft(2)2552 1959 y Fw(=)p 2639 1914 V 22 w Fu(z)2682 1971 y Ft(2)2719 1892 y Fo(\011)2781 1959 y Fu(=)p Fr(f)p Fw(1)p Fu(;)14 b Fr(\000)p Fw(1)p Fr(g)20 b Fu(:)482 b Fw(\(3.10\))523 2142 y(Note)28 b(that)g(the)h (symplectic)f(manifold)g(\()p Fr(D)1897 2154 y Ft(2)1935 2142 y Fu(;)14 b(\034)2008 2154 y Ft(2)2045 2142 y Fw(\))29 b(can)e(b)r(e)i(regarded)d(as)h Fv(C)2885 2112 y Ft(2)2922 2142 y Fu(=)p Fr(f)p Fw(1)p Fu(;)14 b Fr(\000)p Fw(1)p Fr(g)26 b Fw(with)i(symplectic)523 2242 y(form)f Fu(\026)769 2254 y Ft(2)830 2242 y Fw(=)22 b Fr(\000)p Fu({)14 b Fw(\()p Fu(dz)1137 2254 y Ft(1)1192 2242 y Fr(^)19 b Fu(d)p 1309 2196 V(z)1352 2254 y Ft(1)1407 2242 y Fw(+)f Fu(dz)1572 2254 y Ft(2)1628 2242 y Fr(^)h Fu(d)p 1745 2196 V(z)1787 2254 y Ft(2)1824 2242 y Fw(\))q(.)648 2341 y(Once)33 b(w)n(e)h(ha)n(v)n(e)f(c)n(hosen)g(the)h(plus)g(sign)g(in)g (equations)f(\(3.8\))h(and)f(\(3.9\),)j(w)n(e)d(\014nd)i(the)f(follo)n (wing)523 2441 y(expressions)26 b(for)h Fu(p)1125 2453 y Fs(\022)1162 2441 y Fu(;)14 b(p)1241 2453 y Fs(\036)1313 2441 y Fw(in)28 b(terms)f(of)g(\()p Fu(z)1806 2453 y Ft(1)1844 2441 y Fu(;)14 b(z)1920 2453 y Ft(2)1956 2441 y Fu(;)p 1993 2395 V 14 w(z)2035 2453 y Ft(1)2073 2441 y Fu(;)p 2110 2395 V 14 w(z)2152 2453 y Ft(2)2189 2441 y Fw(\):)958 2680 y Fu(p)1000 2692 y Fs(\022)1061 2680 y Fw(=)1148 2563 y Fo(\022)1219 2624 y Fu(z)1258 2636 y Ft(1)1295 2624 y Fu(z)1334 2636 y Ft(2)1389 2624 y Fr(\000)p 1472 2578 V 18 w Fu(z)1515 2636 y Ft(1)p 1552 2578 V 1552 2624 a Fu(z)1594 2636 y Ft(2)p 1219 2661 413 4 v 1391 2737 a Fw(2)p Fu({)1641 2563 y Fo(\023)g(\022)1787 2624 y Fr(j)p Fu(z)1849 2636 y Ft(1)1886 2624 y Fr(j)1909 2593 y Ft(2)1965 2624 y Fw(+)k Fr(j)p Fu(z)2110 2636 y Ft(2)2147 2624 y Fr(j)2170 2593 y Ft(2)p 1787 2661 420 4 v 1867 2740 a Fu(z)1910 2711 y Ft(2)1906 2762 y(1)1965 2740 y Fr(\000)p 2048 2694 43 4 v 18 w Fu(z)2090 2708 y Ft(2)2090 2760 y(2)2217 2563 y Fo(\023)2292 2680 y Fu(;)132 b(p)2489 2692 y Fs(\036)2556 2680 y Fw(=)2654 2624 y(1)p 2654 2661 42 4 v 2654 2737 a(2)2719 2612 y Fo(\000)2757 2680 y Fr(j)p Fu(z)2819 2692 y Ft(1)2856 2680 y Fr(j)2879 2645 y Ft(2)2935 2680 y Fr(\000)18 b(j)p Fu(z)3080 2692 y Ft(2)3117 2680 y Fr(j)3140 2645 y Ft(2)3177 2612 y Fo(\001)1538 2936 y Fw(and)69 b Fu(H)30 b Fw(=)22 b Fu(p)1969 2902 y Ft(2)1969 2956 y Fs(\022)2025 2936 y Fw(+)2220 2880 y(1)p 2118 2917 246 4 v 2118 2999 a(sin)2220 2964 y Ft(2)2257 2999 y Fw(\()p Fu(\022)r Fw(\))2373 2936 y Fu(p)2415 2902 y Ft(2)2415 2956 y Fs(\036)2482 2936 y Fw(=)2570 2819 y Fo(\022)2641 2880 y Fr(j)p Fu(z)2703 2892 y Ft(1)2740 2880 y Fr(j)2763 2850 y Ft(2)2818 2880 y Fw(+)c Fr(j)p Fu(z)2963 2892 y Ft(2)3000 2880 y Fr(j)3023 2850 y Ft(2)p 2641 2917 420 4 v 2830 2993 a Fw(2)3071 2819 y Fo(\023)3132 2833 y Ft(2)3206 2936 y Fu(:)389 b Fw(\(3.11\))523 3173 y(Note)25 b(that)h(w)n(e)f(ha)n(v)n(e)f(a)g (singularit)n(y)g(in)i(equations)e(\(3.8\))h(when)g(the)h(energy)e(H)h (is)g(zero.)35 b(Th)n(us)25 b(w)n(e)g(need)523 3273 y(to)33 b(tak)n(e)f(o\013)h(the)g(zero)f(section)h(of)f Fu(T)1710 3243 y Fn(\003)1748 3273 y Fu(S)1804 3243 y Ft(2)1874 3273 y Fw(\(whic)n(h)h(w)n(e)f(shall)h(denote)g(b)n(y)f Fr(f)p Fw(0)p Fr(g)p Fw(\))g(and)h(the)g(origin)f(of)h(the)523 3373 y(manifold)c Fv(C)937 3385 y Ft(2)974 3373 y Fu(=)p Fr(f)p Fw(1)p Fu(;)14 b Fr(\000)p Fw(1)p Fr(g)p Fw(.)37 b(\(denoted)30 b(b)n(y)e Fr(f)p Fw(0)p Fr(g)f Fw(to)r(o\).)41 b(The)29 b(other)f(singularit)n(y)f(in)i(the)g(expression)f(for)g Fu(p)3793 3385 y Fs(\022)523 3472 y Fw(corresp)r(onds)k(to)i(the)h(p)r (oin)n(t)f Fu(z)1509 3442 y Ft(2)1505 3493 y(1)1579 3472 y Fw(=)p 1678 3427 43 4 v 34 w Fu(z)1720 3440 y Ft(2)1720 3493 y(2)1757 3472 y Fw(,)i(whic)n(h)e(in)h(turn)f(is)g(equiv)-5 b(alen)n(t)34 b(to)g(the)g(equation)f(sin\()p Fu(\022)r Fw(\))i(=)f(0)523 3572 y(\(north)j(and)h(south)f(p)r(oles\).)67 b(W)-7 b(e)38 b(actually)f(don't)g(ha)n(v)n(e)g(suc)n(h)g(a)g (singularit)n(y)f(since)h(w)n(e)g(are)g(taking)523 3672 y(0)23 b Fu(<)f(\022)k(<)c(\031)31 b Fw(and)d(it)g(w)n(on't)f(app)r (ear)g(in)g(our)g(extension)g(of)h(our)f(canonical)f(transformation.) 523 3842 y Fv(I)s(I)s(I.2.1)k(Canonical)i(transformation)f(for)h Fu(S)2116 3812 y Ft(2)2185 3842 y Fv(and)g(quadric.)648 3989 y Fw(In)24 b(this)g(section)f(w)n(e)h(iden)n(tify)g(the)g (equations)f(giving)g(\()p Fu(\022)r(;)14 b(\036;)g(p)2584 4001 y Fs(\022)2622 3989 y Fu(;)g(p)2701 4001 y Fs(\036)2745 3989 y Fw(\))24 b(as)f(functions)h(of)g(\()p Fu(z)3415 4001 y Ft(1)3452 3989 y Fu(;)14 b(z)3528 4001 y Ft(2)3565 3989 y Fu(;)p 3602 3943 V 14 w(z)3644 4001 y Ft(1)3681 3989 y Fu(;)p 3718 3943 V 14 w(z)3761 4001 y Ft(2)3798 3989 y Fw(\))523 4089 y(with)28 b(the)g(comp)r(osition)f(of)h(t)n(w)n (o)f(maps)g(in)n(v)n(olving)f(the)i(quadric)f(in)g Fv(C)2749 4058 y Ft(3)2814 4089 y Fw(de\014ned)h(b)n(y:)1276 4271 y Fv(Q)1348 4283 y Ft(3)1408 4271 y Fr(\021)23 b(f)p Fw(\()p Fu(\013)1623 4283 y Ft(1)1660 4271 y Fu(;)14 b(\013)1750 4283 y Ft(2)1787 4271 y Fu(;)g(\013)1877 4283 y Ft(3)1915 4271 y Fw(\))23 b Fr(2)g Fv(C)2117 4237 y Ft(3)2155 4271 y Fu(=\013)2250 4237 y Ft(2)2250 4292 y(1)2305 4271 y Fw(+)18 b Fu(\013)2441 4237 y Ft(2)2441 4292 y(2)2497 4271 y Fw(+)g Fu(\013)2633 4237 y Ft(2)2633 4292 y(3)2694 4271 y Fw(=)23 b(0)p Fr(g)f Fu(:)707 b Fw(\(3.12\))523 4454 y(One)22 b(of)h(the)g(maps)g(go)r(es)f(from)g Fv(C)1572 4424 y Ft(2)1609 4454 y Fu(=)p Fr(f)p Fw(1)p Fu(;)14 b Fr(\000)p Fw(1)p Fr(g)9 b(\000)g(f)p Fw(0)p Fr(g)18 b Fw(on)n(to)k Fv(Q)2402 4466 y Ft(3)2448 4454 y Fr(\000)9 b(f)p Fw(0)p Fr(g)21 b Fw(and)h(the)h(other)g(one)f(from)g Fv(Q)3586 4466 y Ft(3)3632 4454 y Fr(\000)9 b(f)p Fw(0)p Fr(g)523 4553 y Fw(on)n(to)28 b Fu(T)772 4523 y Fn(\003)809 4553 y Fu(S)865 4523 y Ft(2)922 4553 y Fr(\000)18 b(f)p Fw(0)p Fr(g)p Fw(.)40 b(In)29 b(this)g(manner,)g(w)n(e)f(shall)g (extend)i(our)e(canonical)f(transformation)g Fr(C)3502 4568 y Fs(S)3546 4552 y Fm(2)3612 4553 y Fw(as)h(the)523 4653 y(in)n(v)n(erse)e(of)i(suc)n(h)f(a)g(comp)r(osition.)p 523 4706 1323 4 v 615 4760 a Fi(1)650 4783 y Fh(It)k(actually)g(can)g (b)r(e)g(sho)n(wn)g(that,)i(if)c(\()p Fg(z)1780 4792 y Fi(1)1815 4783 y Fg(;)11 b(z)1879 4792 y Fi(2)1914 4783 y Fh(\))30 b(is)g(a)g(solution)h(then)g(\()p Ff(\000)p Fg(z)2670 4792 y Fi(1)2705 4783 y Fg(;)12 b Ff(\000)p Fg(z)2825 4792 y Fi(2)2859 4783 y Fh(\))31 b(,)h(\()p 2996 4745 36 4 v Fg(z)3032 4792 y Fi(2)3066 4783 y Fg(;)12 b Ff(\000)p 3153 4745 V Fg(z)3188 4792 y Fi(1)3223 4783 y Fh(\))30 b(and)h(\()p Ff(\000)p 3506 4745 V Fg(z)3542 4792 y Fi(2)3577 4783 y Fg(;)p 3608 4745 V 11 w(z)3644 4792 y Fi(1)3678 4783 y Fh(\))g(are)523 4862 y(solutions)21 b(to)r(o.)31 b(If)21 b(w)n(e)g(tak)n(e)h(real)e(and)i(imaginary)d (parts)i(of)g Fg(z)2170 4871 y Fi(1)2225 4862 y Fh(and)g Fg(z)2392 4871 y Fi(2)2426 4862 y Fh(,)g(w)n(e)g(see)h(that)g(the)f(4) 10 b Ff(\002)g Fh(4)21 b(matrices)f(taking)i(\()p Fg(z)3669 4871 y Fi(1)3704 4862 y Fg(;)11 b(z)3768 4871 y Fi(2)3803 4862 y Fh(\))523 4941 y(to)24 b(\()p Ff(\000)p Fg(z)724 4950 y Fi(1)759 4941 y Fg(;)11 b Ff(\000)p Fg(z)878 4950 y Fi(2)913 4941 y Fh(\))20 b(,)j(\()p 1030 4903 V Fg(z)1066 4950 y Fi(2)1101 4941 y Fg(;)11 b Ff(\000)p 1187 4903 V Fg(z)1222 4950 y Fi(1)1257 4941 y Fh(\))24 b(and)g(\()p Ff(\000)p 1527 4903 V Fg(z)1563 4950 y Fi(2)1598 4941 y Fg(;)p 1629 4903 V 11 w(z)1665 4950 y Fi(1)1699 4941 y Fh(\))g(are)1078 5111 y Fe(A)1139 5120 y Fd(2)1198 5111 y Fh(=)1272 5023 y Fo(\020)1363 5071 y Ff(\000)p Fe(I)59 b Fh(0)1363 5150 y(0)75 b Ff(\000)15 b Fe(I)1615 5023 y Fo(\021)1677 5111 y Fg(;)50 b Fe(A)1808 5120 y Fd(3)1867 5111 y Fh(=)1941 5023 y Fo(\020)2060 5071 y Fh(0)59 b Fe(K)2033 5150 y Ff(\000)p Fe(K)f Fh(0)2287 5023 y Fo(\021)2348 5111 y Fg(;)50 b Fe(A)2479 5120 y Fd(4)2538 5111 y Fh(=)2612 5023 y Fo(\020)2704 5071 y Fh(0)74 b Ff(\000)16 b Fe(K)2747 5150 y(K)58 b Fh(0)2989 5023 y Fo(\021)3050 5111 y Fg(;)523 5327 y Fh(resp)r(ectiv)n(ely)24 b(\(where)h Fe(I)20 b Fh(=)1263 5239 y Fo(\020)1354 5287 y Fh(1)40 b(0)1354 5366 y(0)g(1)1506 5239 y Fo(\021)1578 5327 y Fh(and)24 b Fe(K)19 b Fh(=)1873 5239 y Fo(\020)1964 5287 y Fh(1)40 b(0)1964 5366 y(0)55 b Ff(\000)16 b Fh(1)2202 5239 y Fo(\021)2274 5327 y Fh(are)24 b(2)15 b Ff(\002)g Fh(2)24 b(matrices\).)31 b(These)24 b(three)g(matrices)e(together)523 5448 y(with)j(the)g(iden)n(tit)n(y)g(matrix)f(are)g(a)h(represen)n (tation)h(of)e(the)h(cyclic)g(group)g Fg(C)2588 5457 y Fi(4)2647 5448 y Fh(of)f(four)g(elemen)n(ts\(the)i(same)e(group)g (giv)n(en)523 5526 y(b)n(y)g(the)g(four)e(complex)h(n)n(um)n(b)r(ers)f (1)p Fg(;)12 b Ff(\000)p Fh(1)p Fg(;)g({;)f Ff(\000)p Fg({)p Fh(\).)31 b(Th)n(us,)23 b(for)g(\()p Fg(\022)r(;)12 b(\036;)f(p)2356 5538 y Fc(\022)2391 5526 y Fg(;)g(p)2458 5538 y Fc(\036)2499 5526 y Fh(\))23 b(giv)n(en,)h(the)g(four)e(corresp) r(onding)i(solutions)f(to)523 5605 y(the)e(equations)h(determined)e(b)n (y)g(the)h(canonical)g(transformation)f(are)g(related)g(to)h(eac)n(h)g (other)g(b)n(y)g(elemen)n(ts)f(of)g(the)h(group)523 5684 y Fg(C)573 5693 y Fi(4)p eop %%Page: 17 17 17 16 bop 3747 324 a Fw(17)648 614 y(The)26 b(generating)e(function) 1561 582 y Fn(\000)p Fs({)p 1555 596 88 4 v 1555 604 a Fn(p)p 1610 604 34 3 v 49 x Ft(3)1653 614 y Fw(\010)1713 629 y Fs(S)1757 613 y Fm(2)1819 614 y Fw(of)i(our)g(canonical)f (transformation)f Fr(C)3030 629 y Fs(S)3074 613 y Fm(2)3137 614 y Fw(can)h(b)r(e)i(written)f(as)g(a)523 725 y(inner)h(pro)r(duct:) 1688 881 y Fr(\000)p Fu({)p 1679 918 111 4 v 1679 935 a Fr(p)p 1748 935 42 4 v 69 x Fw(3)1799 938 y(\010)1859 953 y Fs(S)1903 936 y Fm(2)1963 938 y Fw(=)22 b Fr(\000)p Fu({)p Fw(\()p Fu(\013)p Fw(\()p Fv(z)p Fw(\))e Fr(\017)e Fv(x)p Fw(\))p Fu(;)1100 b Fw(\(3.13\))523 1164 y(where)1561 1346 y Fv(x)84 b Fw(=)e(\(sin\()p Fu(\022)r Fw(\))14 b(cos\()p Fu(\036)p Fw(\))p Fu(;)g Fw(sin)q(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))p Fu(;)g Fw(cos\()p Fu(\022)r Fw(\)\))1271 1510 y(and)46 b Fu(\013)p Fw(\()p Fv(z)p Fw(\))85 b(=)1852 1454 y(1)p 1852 1491 V 1852 1567 a(2)1917 1443 y Fo(\000)1955 1510 y Fr(\000)p Fu(z)2063 1476 y Ft(2)2059 1531 y(1)2118 1510 y Fw(+)18 b Fu(z)2244 1476 y Ft(2)2240 1531 y(2)2281 1510 y Fu(;)c({z)2388 1476 y Ft(2)2384 1531 y(1)2442 1510 y Fw(+)k Fu({z)2595 1476 y Ft(2)2591 1531 y(2)2631 1510 y Fu(;)c Fw(2)p Fu(z)2749 1522 y Ft(1)2786 1510 y Fu(z)2825 1522 y Ft(2)2862 1443 y Fo(\001)2937 1510 y Fu(:)658 b Fw(\(3.14\))523 1721 y(Here)43 b(w)n(e)f(are)g(taking)g(the)h(usual)g(inner)g(pro)r(duct)f (of)h(t)n(w)n(o)f(v)n(ectors)g Fu(\013)49 b Fw(=)f(\()p Fu(\013)3122 1733 y Ft(1)3159 1721 y Fu(;)14 b(\013)3249 1733 y Ft(2)3287 1721 y Fu(;)g(\013)3377 1733 y Ft(3)3414 1721 y Fw(\))43 b(and)g Fu(\014)53 b Fw(=)523 1820 y(\()p Fu(\014)602 1832 y Ft(1)640 1820 y Fu(;)14 b(\014)724 1832 y Ft(2)761 1820 y Fu(;)g(\014)845 1832 y Ft(3)882 1820 y Fw(\))28 b(in)g Fv(C)1108 1790 y Ft(3)1145 1820 y Fw(:)1660 1920 y Fu(\013)19 b Fr(\017)f Fu(\014)27 b Fr(\021)c Fu(\013)2007 1932 y Ft(1)p 2044 1852 52 4 v 2044 1920 a Fu(\014)2095 1940 y Ft(1)2151 1920 y Fw(+)18 b Fu(\013)2287 1932 y Ft(2)p 2324 1852 V 2324 1920 a Fu(\014)2376 1940 y Ft(2)2431 1920 y Fw(+)g Fu(\013)2567 1932 y Ft(3)p 2605 1852 V 2605 1920 a Fu(\014)2656 1940 y Ft(3)3618 1920 y Fw(\(3.15\))648 2069 y(Notice)33 b(the)h(relev)-5 b(an)n(t)33 b(fact)h(that)g Fu(\013)p Fw(\()p Fv(z)p Fw(\))h(is)f(in)g(the)g(quadric)f Fv(Q)2651 2081 y Ft(3)2687 2069 y Fw(.)56 b(This)33 b(suggests)g(that)h Fv(Q)3550 2081 y Ft(3)3620 2069 y Fw(migh)n(t)523 2169 y(ha)n(v)n(e)g(an)g(imp)r (ortan)n(t)h(role)f(to)g(pla)n(y)g(related)g(to)h(our)f(canonical)g (transformation)f Fr(C)3245 2184 y Fs(S)3289 2167 y Fm(2)3325 2169 y Fw(.)59 b(Th)n(us)34 b(let)h(us)523 2268 y(de\014ne)28 b(the)g(follo)n(wing)e(map:)1295 2451 y Fu(\032)1338 2463 y Ft(2)1398 2451 y Fw(:)1455 2431 y Fv(~)1445 2451 y(C)1514 2417 y Ft(2)1569 2451 y Fr(\000)18 b(f)p Fw(0)p Fr(g)82 b(7!)h Fv(Q)2098 2463 y Ft(3)2153 2451 y Fr(\000)18 b(f)p Fw(0)p Fr(g)1443 2615 y Fu(\032)1486 2627 y Ft(2)1524 2615 y Fw(\()p Fu(z)1595 2627 y Ft(1)1632 2615 y Fu(;)c(z)1708 2627 y Ft(2)1744 2615 y Fw(\))93 b(=)2036 2559 y(1)p 2036 2596 42 4 v 2036 2672 a(2)2087 2615 y(\()p Fr(\000)p Fu(z)2227 2580 y Ft(2)2223 2635 y(1)2282 2615 y Fw(+)18 b Fu(z)2408 2580 y Ft(2)2404 2635 y(2)2445 2615 y Fu(;)c({z)2552 2580 y Ft(2)2548 2635 y(1)2606 2615 y Fw(+)k Fu({z)2759 2580 y Ft(2)2755 2635 y(2)2796 2615 y Fu(;)c Fw(2)p Fu(z)2914 2627 y Ft(1)2950 2615 y Fu(z)2989 2627 y Ft(2)3026 2615 y Fw(\))560 b(\(3.16\))523 2841 y(where)19 b(w)n(e)h(de\014ne)1112 2821 y Fv(~)1102 2841 y(C)1171 2811 y Ft(2)1228 2841 y Fw(as)f(the)h(set)g(of)g(equiv)-5 b(alence)19 b(classes)g(of)g Fv(C)2508 2811 y Ft(2)2565 2841 y Fw(iden)n(tifying)h(\()p Fu(z)3041 2853 y Ft(1)3079 2841 y Fu(;)14 b(z)3155 2853 y Ft(2)3191 2841 y Fw(\))20 b(with)h(\()p Fr(\000)p Fu(z)3561 2853 y Ft(1)3597 2841 y Fu(;)14 b Fr(\000)p Fu(z)3738 2853 y Ft(2)3775 2841 y Fw(\).)648 2940 y(No)n(w)24 b(b)r(eing)i(in)f (the)h(quadric)f Fv(Q)1659 2952 y Ft(3)1710 2940 y Fr(\000)13 b(f)p Fw(0)p Fr(g)p Fw(,)25 b(w)n(e)g(can)f(go)h(to)g Fu(T)2500 2910 y Fn(\003)2537 2940 y Fu(S)2593 2910 y Ft(2)2644 2940 y Fr(\000)14 b(f)p Fw(0)p Fr(g)24 b Fw(through)g(the)i (follo)n(wing)e(map)1487 3123 y Fu(\033)1534 3135 y Ft(2)1595 3123 y Fw(:)f Fv(Q)1713 3135 y Ft(3)1769 3123 y Fr(\000)18 b(f)p Fw(0)p Fr(g)81 b(7!)i Fu(T)2286 3089 y Fn(\003)2324 3123 y Fu(S)2380 3089 y Ft(2)2435 3123 y Fr(\000)18 b(f)p Fw(0)p Fr(g)1482 3297 y Fu(\033)1529 3309 y Ft(2)1567 3297 y Fw(\()p Fu(\013)1652 3309 y Ft(1)1689 3297 y Fu(;)c(\013)1779 3309 y Ft(2)1817 3297 y Fu(;)g(\013)1907 3309 y Ft(3)1944 3297 y Fw(\))93 b(=)2225 3180 y Fo(\022)2319 3241 y Fr(<)p Fw(\()p Fu(\013)p Fw(\))p 2296 3278 225 4 v 2296 3354 a Fr(j<)p Fw(\()p Fu(\013)p Fr(j)q Fw(\))2530 3297 y Fu(;)14 b Fr(\000=)p Fw(\()p Fu(\013)p Fw(\))2809 3180 y Fo(\023)3618 3297 y Fw(\(3.17\))523 3530 y(here)27 b Fr(<)p Fw(\()p Fu(\013)p Fw(\))i(and)e Fr(=)p Fw(\()p Fu(\013)p Fw(\))i(are)d(the)i(real)f(and)h(imaginary)d(part)j(of)f Fu(\013)d Fw(=)e(\()p Fu(\013)2794 3542 y Ft(1)2832 3530 y Fu(;)14 b(\013)2922 3542 y Ft(2)2959 3530 y Fu(;)g(\013)3049 3542 y Ft(3)3086 3530 y Fw(\))28 b(resp)r(ectiv)n(ely)-7 b(.)648 3629 y(Th)n(us)27 b(the)h(comp)r(osition)f(map)g(is:)801 3812 y Fu(\033)848 3824 y Ft(2)904 3812 y Fr(\016)18 b Fu(\032)1007 3824 y Ft(2)1067 3812 y Fw(:)1124 3792 y Fv(~)1113 3812 y(C)1182 3778 y Ft(2)1238 3812 y Fr(\000)g(f)p Fw(0)p Fr(g)81 b(7!)i Fu(T)1755 3778 y Fn(\003)1793 3812 y Fu(S)1849 3778 y Ft(2)1904 3812 y Fr(\000)18 b(f)p Fw(0)p Fr(g)949 3990 y Fu(\033)996 4002 y Ft(2)1052 3990 y Fr(\016)g Fu(\032)1155 4002 y Ft(2)1192 3990 y Fw(\()p Fu(z)1263 4002 y Ft(1)1300 3990 y Fu(;)c(z)1376 4002 y Ft(2)1413 3990 y Fw(\))93 b(=)1694 3873 y Fo(\022)1765 3934 y Fr(<)p Fw(\()p Fr(\000)p Fu(z)1965 3904 y Ft(2)1961 3954 y(1)2020 3934 y Fw(+)18 b Fu(z)2146 3904 y Ft(2)2142 3954 y(2)2183 3934 y Fw(\))p 1765 3971 450 4 v 1780 4047 a Fr(j)p Fu(z)1842 4059 y Ft(1)1879 4047 y Fr(j)1902 4023 y Ft(2)1958 4047 y Fw(+)g Fr(j)p Fu(z)2103 4059 y Ft(2)2140 4047 y Fr(j)2163 4023 y Ft(2)2225 3990 y Fu(;)2272 3934 y Fr(<)p Fw(\()p Fu({z)2434 3904 y Ft(2)2430 3954 y(1)2489 3934 y Fw(+)g Fu({z)2642 3904 y Ft(2)2638 3954 y(2)2678 3934 y Fw(\))p 2272 3971 439 4 v 2281 4047 a Fr(j)p Fu(z)2343 4059 y Ft(1)2380 4047 y Fr(j)2403 4023 y Ft(2)2459 4047 y Fw(+)g Fr(j)p Fu(z)2604 4059 y Ft(2)2641 4047 y Fr(j)2664 4023 y Ft(2)2720 3990 y Fu(;)2818 3934 y Fr(<)p Fw(\(2)p Fu(z)2991 3946 y Ft(1)3028 3934 y Fu(z)3067 3946 y Ft(2)3104 3934 y Fw(\))p 2767 3971 420 4 v 2767 4047 a Fr(j)p Fu(z)2829 4059 y Ft(1)2866 4047 y Fr(j)2889 4023 y Ft(2)2945 4047 y Fw(+)g Fr(j)p Fu(z)3090 4059 y Ft(2)3127 4047 y Fr(j)3150 4023 y Ft(2)3197 3990 y Fu(;)1704 4222 y Fr(\000)1779 4166 y Fw(1)p 1779 4203 42 4 v 1779 4279 a(2)1830 4222 y Fr(=)p Fw(\()p Fr(\000)p Fu(z)2030 4188 y Ft(2)2026 4243 y(1)2085 4222 y Fw(+)g Fu(z)2211 4188 y Ft(2)2207 4243 y(2)2248 4222 y Fw(\))p Fu(;)c Fr(\000)2392 4166 y Fw(1)p 2392 4203 V 2392 4279 a(2)2443 4222 y Fr(=)p Fw(\()p Fu({z)2605 4188 y Ft(2)2601 4243 y(1)2660 4222 y Fw(+)k Fu({z)2813 4188 y Ft(2)2809 4243 y(2)2849 4222 y Fw(\))p Fu(;)c Fr(\000)2993 4166 y Fw(1)p 2993 4203 V 2993 4279 a(2)3044 4222 y Fr(=)p Fw(\(2)p Fu(z)3217 4234 y Ft(1)3254 4222 y Fu(z)3293 4234 y Ft(2)3330 4222 y Fw(\))3362 4105 y Fo(\023)3618 4222 y Fw(\(3.18\))648 4460 y(In)27 b(the)h(other)f(hand,) h(since)f(a)h(p)r(oin)n(t)f(in)h Fu(T)1989 4430 y Fn(\003)2026 4460 y Fu(S)2082 4430 y Ft(2)2147 4460 y Fw(can)f(b)r(e)h(describ)r(ed) g(in)f(spherical)g(co)r(ordinates)f(b)n(y)945 4571 y Fo(\022)1006 4688 y Fw(sin\()p Fu(\022)r Fw(\))14 b(cos\()p Fu(\036)p Fw(\))p Fu(;)g Fw(sin)q(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))p Fu(;)g(cos)p Fw(\()p Fu(\022)r Fw(\))p Fu(;)g Fw(cos)q(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))p Fu(p)2720 4700 y Fs(\022)2776 4688 y Fr(\000)2869 4632 y Fw(sin\()p Fu(\036)p Fw(\))p 2869 4669 216 4 v 2873 4745 a(sin\()p Fu(\022)r Fw(\))3095 4688 y Fu(p)3137 4700 y Fs(\036)3181 4688 y Fu(;)1820 4920 y Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin\()p Fu(\036)p Fw(\))p Fu(p)2307 4932 y Fs(\022)2364 4920 y Fw(+)2457 4864 y(cos)o(\()p Fu(\036)p Fw(\))p 2457 4901 226 4 v 2466 4977 a(sin\()p Fu(\022)r Fw(\))2692 4920 y Fu(p)2734 4932 y Fs(\036)2778 4920 y Fu(;)g Fr(\000)g Fw(sin)o(\()p Fu(\022)r Fw(\))p Fu(p)3142 4932 y Fs(\022)3181 4803 y Fo(\023)3618 4920 y Fw(\(3.19\))523 5153 y(then,)33 b(w)n(e)f(\014nd)g(after)g(some)f (computation)g(\(b)n(y)h(using)g(equations)f(\(3.7\))g(and)h(\(3.8\)\)) g(that)g(this)g(p)r(oin)n(t)523 5252 y(in)d Fu(T)682 5222 y Fn(\003)719 5252 y Fu(S)775 5222 y Ft(2)841 5252 y Fw(expressed)f(in)h(terms)g(of)g(\()p Fu(z)1713 5264 y Ft(1)1750 5252 y Fu(;)14 b(z)1826 5264 y Ft(2)1863 5252 y Fu(;)p 1900 5207 43 4 v 14 w(z)1942 5264 y Ft(1)1979 5252 y Fu(;)p 2016 5207 V 14 w(z)2058 5264 y Ft(2)2124 5252 y Fw(is)29 b(exactly)g(the)g(same)f(p)r(oin)n(t)h(that)g(the)h (comp)r(osition)523 5352 y Fu(\033)570 5364 y Ft(2)623 5352 y Fr(\016)15 b Fu(\032)723 5364 y Ft(2)786 5352 y Fw(assigns)25 b(to)h(\()p Fu(z)1234 5364 y Ft(1)1271 5352 y Fu(;)14 b(z)1347 5364 y Ft(2)1384 5352 y Fw(\).)37 b(This)26 b(conclusion)f(allo)n(ws)f(us)i(to)g(extend)h(our)e (canonical)g(transformation)523 5452 y Fr(C)567 5467 y Fs(S)611 5450 y Fm(2)681 5452 y Fw(\(without)34 b(referring)d(to)i (spherical)g(co)r(ordinates\))f(as)g(the)i(map)f(\()p Fu(\033)2832 5464 y Ft(2)2892 5452 y Fr(\016)22 b Fu(\032)2999 5464 y Ft(2)3036 5452 y Fw(\))3068 5422 y Fn(\000)p Ft(1)3157 5452 y Fw(.)54 b(Let)34 b(us)f(state)g(our)523 5551 y(conclusions)27 b(in)g(the)h(follo)n(wing)f(theorem:)p eop %%Page: 18 18 18 17 bop 3747 324 a Fw(18)523 614 y Fv(Theorem)30 b(3.2)41 b Fl(The)26 b(function)1595 582 y Fn(\000)p Fs({)p 1589 596 88 4 v 1589 604 a Fn(p)p 1643 604 34 3 v 1643 653 a Ft(3)1687 614 y Fw(\010)1747 629 y Fs(S)1791 613 y Fm(2)1852 614 y Fl(is)f(a)h(gener)l(ating)f(function)g(of)h(a)f(tr)l (ansformation)h(that)f(c)l(an)g(b)l(e)523 725 y(extende)l(d)k(as)h(a)g (c)l(anonic)l(al)g(tr)l(ansformation)g Fr(C)2012 740 y Fs(S)2056 724 y Fm(2)2122 725 y Fl(which)h(makes)e(the)h(fol)t (lowing)i(triangle)e(of)g(c)l(anonic)l(al)523 825 y(tr)l(ansformations) g(to)g(c)l(ommute:)1886 1597 y Fb(\000)1969 1514 y(\000)2052 1431 y(\000)2135 1348 y(\000)2218 1265 y(\000)2301 1182 y(\000)-83 b(\022)p 1760 1597 4 499 v 1761 1597 a(?)p 1969 1018 416 4 v 1969 1016 a(\033)1594 1701 y Fv(Q)1666 1713 y Ft(3)1721 1701 y Fr(\000)18 b(f)p Fw(0)p Fr(g)1606 1024 y Fv(~)1595 1045 y(C)1664 1014 y Ft(2)1720 1045 y Fr(\000)g(f)p Fw(0)p Fr(g)2425 1039 y Fu(T)2486 1009 y Fn(\003)2523 1039 y Fu(S)2579 1009 y Ft(2)2634 1039 y Fr(\000)g(f)p Fw(0)p Fr(g)2114 913 y(C)2158 928 y Fs(S)2202 911 y Fm(2)1638 1399 y Fu(\032)1681 1411 y Ft(2)2176 1401 y Fu(\033)2223 1413 y Ft(2)523 1902 y Fl(wher)l(e)23 b(\(a\))f(The)i(symple)l(ctic)f(form)g Fu(\027)1670 1914 y Ft(2)1729 1902 y Fl(on)g Fu(T)1902 1872 y Fn(\003)1939 1902 y Fu(S)1995 1872 y Ft(2)2054 1902 y Fl(is)g(given)f(by)h(the)f(r)l (estriction)h(to)f Fu(T)3122 1872 y Fn(\003)3159 1902 y Fu(S)3215 1872 y Ft(2)3274 1902 y Fl(of)i(the)e(c)l(anonic)l(al)523 2002 y(symple)l(ctic)31 b(form)f Fu(!)1174 2014 y Ft(3)1241 2002 y Fl(on)g Fu(T)1421 1972 y Fn(\003)1458 2002 y Fv(R)1530 1972 y Ft(3)1567 2002 y Fl(,)523 2101 y(\(b\))g(The)h(symple)l(ctic)f (form)h Fu(\026)1478 2113 y Ft(2)1545 2101 y Fl(on)1674 2081 y Fv(~)1664 2101 y(C)1733 2071 y Ft(2)1788 2101 y Fr(\000)18 b(f)p Fw(0)p Fr(g)29 b Fl(is)h(given)g(by)1578 2266 y Fu(\026)1628 2278 y Ft(2)1688 2266 y Fw(=)23 b Fr(\000)p Fu({)14 b Fw(\()o Fu(dz)1995 2278 y Ft(1)2051 2266 y Fr(^)19 b Fu(d)p 2168 2220 43 4 v(z)2210 2278 y Ft(1)2266 2266 y Fw(+)f Fu(dz)2431 2278 y Ft(2)2486 2266 y Fr(^)h Fu(d)p 2603 2220 V(z)2646 2278 y Ft(2)2683 2266 y Fw(\))37 b Fu(;)843 b Fw(\(3.20\))523 2431 y Fl(\(c\))30 b(The)h(symple)l(ctic)f(form)h Fu(q)1465 2443 y Ft(3)1532 2431 y Fl(on)f Fv(Q)1723 2443 y Ft(3)1778 2431 y Fr(\000)18 b(f)p Fw(0)p Fr(g)28 b Fl(is)j(given)f(by:)792 2675 y Fu(q)829 2687 y Ft(3)890 2675 y Fw(=)1080 2619 y(1)p 987 2656 228 4 v 987 2732 a(2)p Fu({)p Fr(j<)p Fu(\013)p Fr(j)1282 2571 y Ft(3)1239 2596 y Fo(X)1241 2773 y Fs(j)s Ft(=1)1372 2675 y Fu(d\013)1468 2687 y Fs(j)1522 2675 y Fr(^)19 b Fu(d)p 1639 2629 54 4 v(\013)1693 2687 y Fs(j)1746 2675 y Fr(\000)1951 2619 y Fw(1)p 1839 2656 266 4 v 1839 2732 a(8)p Fu({)p Fr(j<)p Fu(\013)p Fr(j)2067 2708 y Ft(3)2195 2571 y(3)2152 2596 y Fo(X)2128 2775 y Fs(j;k)q Ft(=1)2309 2675 y Fw(\()q Fu(\013)2395 2687 y Fs(j)2448 2675 y Fw(+)p 2531 2629 54 4 v 18 w Fu(\013)2585 2687 y Fs(j)2620 2675 y Fw(\))14 b(\()p Fu(\013)2751 2687 y Fs(k)2811 2675 y Fw(+)p 2894 2629 V 18 w Fu(\013)2947 2687 y Fs(k)2988 2675 y Fw(\))g Fu(d\013)3130 2687 y Fs(j)3184 2675 y Fr(^)k Fu(d)p 3300 2629 V(\013)3354 2687 y Fs(k)3618 2675 y Fw(\(3.21\))523 2947 y Fl(which)33 b(is)f(not)f(the)g(r)l(estriction)h(to)f Fv(Q)1712 2959 y Ft(3)1769 2947 y Fr(\000)19 b(f)p Fw(0)p Fr(g)30 b Fl(of)i(the)f(symple)l(ctic)i(form)2850 2885 y Fo(P)2938 2905 y Ft(3)2938 2972 y Fs(j)s Ft(=1)3071 2947 y Fu(d\013)3167 2959 y Fs(j)3221 2947 y Fr(^)20 b Fu(d)p 3339 2901 V(\013)3393 2959 y Fs(j)3459 2947 y Fl(de\014ne)l(d)32 b(on)523 3059 y(the)e(ambient)g Fv(C)1046 3029 y Ft(3)1083 3059 y Fl(.)648 3159 y(The)25 b(tr)l(ansformation)g Fr(C)1417 3174 y Fs(S)1461 3157 y Fm(2)1522 3159 y Fl(maps)g(the)f(c)l(omp)l(onents)h Fw(\()p Fu(L)2397 3171 y Ft(1)2434 3159 y Fu(;)14 b(L)2528 3171 y Ft(2)2564 3159 y Fu(;)g(L)2658 3171 y Ft(3)2695 3159 y Fw(\))24 b Fl(of)i(the)e(angular)h(momentum)f(and)523 3258 y(the)30 b(ener)l(gy)g(H)f(as)h(fol)t(lows)1141 3466 y Fu(L)1198 3478 y Ft(1)1318 3466 y Fw(=)83 b Fr(\000)14 b Fw(sin)o(\()p Fu(\036)p Fw(\))p Fu(p)1801 3478 y Fs(\022)1858 3466 y Fr(\000)1951 3410 y Fw(cos)o(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))p 1951 3447 457 4 v 2075 3523 a(sin\()p Fu(\022)r Fw(\))2417 3466 y Fu(p)2459 3478 y Fs(\036)2526 3466 y Fw(=)2623 3410 y(1)p 2623 3447 42 4 v 2623 3523 a(2)2675 3466 y(\()p Fu(z)2746 3478 y Ft(1)p 2783 3421 43 4 v 2783 3466 a Fu(z)2825 3478 y Ft(2)2881 3466 y Fw(+)p 2964 3421 V 18 w Fu(z)3006 3478 y Ft(1)3044 3466 y Fu(z)3083 3478 y Ft(2)3119 3466 y Fw(\))24 b Fu(;)1141 3696 y(L)1198 3708 y Ft(2)1318 3696 y Fw(=)83 b(cos)o(\()p Fu(\036)p Fw(\))p Fu(p)1732 3708 y Fs(\022)1789 3696 y Fr(\000)1882 3640 y Fw(cos)o(\()p Fu(\022)r Fw(\))14 b(sin)q(\()p Fu(\036)p Fw(\))p 1882 3677 447 4 v 2001 3753 a(sin\()p Fu(\022)r Fw(\))2339 3696 y Fu(p)2381 3708 y Fs(\036)2447 3696 y Fw(=)2545 3640 y Fr(\000)p Fu({)p 2545 3677 92 4 v 2570 3753 a Fw(2)2646 3696 y(\()p Fu(z)2717 3708 y Ft(1)p 2755 3650 43 4 v 2755 3696 a Fu(z)2797 3708 y Ft(2)2852 3696 y Fr(\000)p 2936 3650 V 19 w Fu(z)2978 3708 y Ft(1)3015 3696 y Fu(z)3054 3708 y Ft(2)3091 3696 y Fw(\))23 b Fu(;)1141 3916 y(L)1198 3928 y Ft(3)1318 3916 y Fw(=)83 b Fu(p)1508 3928 y Fs(\036)1575 3916 y Fw(=)1672 3860 y(1)p 1672 3897 42 4 v 1672 3973 a(2)1724 3916 y(\()p Fr(j)p Fu(z)1818 3928 y Ft(1)1855 3916 y Fr(j)1878 3882 y Ft(2)1934 3916 y Fr(\000)18 b(j)p Fu(z)2079 3928 y Ft(2)2116 3916 y Fr(j)2139 3882 y Ft(2)2176 3916 y Fw(\))23 b Fu(;)1159 4147 y(H)90 b Fw(=)1476 4091 y(1)p 1476 4128 V 1476 4204 a(2)1541 4030 y Fo(\022)1602 4147 y Fu(p)1644 4113 y Ft(2)1644 4168 y Fs(\022)1700 4147 y Fw(+)1895 4091 y(1)p 1793 4128 246 4 v 1793 4210 a(sin)1895 4175 y Ft(2)1932 4210 y Fw(\()p Fu(\022)r Fw(\))2048 4147 y Fu(p)2090 4113 y Ft(2)2090 4168 y Fs(\036)2134 4030 y Fo(\023)2218 4147 y Fw(=)2316 4091 y(1)p 2316 4128 42 4 v 2316 4204 a(2)2381 4030 y Fo(\022)2452 4091 y Fr(j)p Fu(z)2514 4103 y Ft(1)2551 4091 y Fr(j)2574 4061 y Ft(2)2630 4091 y Fw(+)18 b Fr(j)p Fu(z)2775 4103 y Ft(2)2812 4091 y Fr(j)2835 4061 y Ft(2)p 2452 4128 420 4 v 2641 4204 a Fw(2)2882 4030 y Fo(\023)2943 4044 y Ft(2)3618 4147 y Fw(\(3.22\))523 4362 y Fl(which)37 b(is)f(the)g(classic)l(al)h(analo)l(g)g(of)g(what)f(the)g(Bar)l(gmann)g (T)-6 b(r)l(ansform)36 b Fv(B)2935 4377 y Fs(S)2979 4360 y Fm(2)3051 4362 y Fl(do)l(es)g(in)g(quantum)e(me-)523 4462 y(chanics)d(\(se)l(e)f(e)l(quation)g(\(2.23\)\).)523 4698 y Fv(I)s(I)s(I.3)h(Canonical)h(T)-8 b(ransformation)31 b(for)h(a)h(particle)f(mo)m(ving)e(on)h Fu(S)2955 4668 y Ft(3)2992 4698 y Fv(.)648 4845 y Fw(In)23 b(analogy)e(with)i(the)h (cases)e(where)g(the)i(con\014guration)d(space)h(is)h Fv(R)2816 4814 y Fs(n)2884 4845 y Fw(or)f Fu(S)3037 4814 y Ft(2)3074 4845 y Fw(,)i(w)n(e)f(w)n(an)n(t)f(to)h(consider)523 4944 y(the)k(function)1006 4911 y Fn(\000)p Fs({)p 999 4925 88 4 v 999 4934 a Fn(p)p 1054 4934 34 3 v 48 x Ft(2)1097 4944 y Fw(\010)1157 4959 y Fs(S)1201 4942 y Fm(3)1264 4944 y Fw(as)f(a)g(generating)f(function)j(of)e(a)h(canonical)e (transformation)g Fr(C)3297 4959 y Fs(S)3341 4942 y Fm(3)3404 4944 y Fw(whic)n(h)i(maps)523 5065 y Fu(T)584 5035 y Fn(\003)621 5065 y Fu(S)677 5035 y Ft(3)738 5065 y Fr(\000)c(f)p Fw(0)p Fr(g)34 b Fw(on)n(to)g(a)h(space)f(in)n(v)n(olving)g(four)h (complex)g(v)-5 b(ariables)34 b(and)h(their)g(conjugates)f(\(w)n(e)h (used)523 5165 y(four)29 b(complex)g(v)-5 b(ariables)28 b(when)i(w)n(e)f(de\014ned)h(the)g(Bargmann)e(transform)g(for)h Fu(L)3123 5135 y Ft(2)3160 5165 y Fw(\()p Fu(S)3248 5135 y Ft(3)3285 5165 y Fw(\)\).)43 b(Notice)30 b(that)523 5264 y Fu(T)584 5234 y Fn(\003)621 5264 y Fu(S)677 5234 y Ft(3)722 5264 y Fr(\000)8 b(f)p Fw(0)p Fr(g)20 b Fw(is)i(a)f(six)h (\(real\))g(dimensional)g(manifold,)h(whereas)e(to)h(determine)g(four)g (complex)f(v)-5 b(ariables)523 5364 y(w)n(e)29 b(need)g(eigh)n(t)g (real)f(parameters.)41 b(Th)n(us)29 b(w)n(e)f(exp)r(ect)i(to)f(ha)n(v)n (e)f(some)h(restrictions)f(b)r(et)n(w)n(een)h(the)h(four)523 5464 y(complex)25 b(v)-5 b(ariables.)34 b(The)25 b(\014rst)g (restriction)f(w)n(e)g(exp)r(ect)i(is)f(the)g(follo)n(wing)f(\(in)h (analogy)e(with)j(equation)523 5563 y(\(2.32\))h(in)h(quan)n(tum)f(mec) n(hanics\):)1595 5728 y Fr(j)p Fu(z)1657 5740 y Ft(1)1694 5728 y Fr(j)1717 5694 y Ft(2)1773 5728 y Fw(+)18 b Fr(j)p Fu(z)1918 5740 y Ft(2)1955 5728 y Fr(j)1978 5694 y Ft(2)2038 5728 y Fw(=)23 b Fr(j)p Fu(z)2188 5740 y Ft(3)2225 5728 y Fr(j)2248 5694 y Ft(2)2304 5728 y Fw(+)18 b Fr(j)p Fu(z)2449 5740 y Ft(4)2485 5728 y Fr(j)2508 5694 y Ft(2)2569 5728 y Fu(:)1026 b Fw(\(3.23\))p eop %%Page: 19 19 19 18 bop 3747 324 a Fw(19)523 614 y(This)27 b(equation)g(is)g (actually)g(con)n(tained)f(in)i(the)f(equations)g(determined)g(b)n(y)g (the)h(generating)e(function)539 681 y Fn(\000)p Fs({)p 533 695 88 4 v 533 704 a Fn(p)p 588 704 34 3 v 48 x Ft(2)631 714 y Fw(\010)691 729 y Fs(S)735 712 y Fm(3)805 714 y Fw(as)33 b(w)n(e)h(sho)n(w)f(it)h(b)r(elo)n(w.)56 b(The)34 b(second)f(restriction)g(comes)g(from)h(a)f Fu(S)3070 684 y Ft(1)3141 714 y Fw(in)n(v)-5 b(ariance)33 b(of)h(those)523 823 y(equations.)648 923 y(Th)n(us)41 b(let)g(us)g(consider)g(the)g (canonical)f(transformation)g(generated)g(b)n(y)3114 890 y Fn(\000)p Fs({)p 3108 904 88 4 v 3108 912 a Fn(p)p 3162 912 34 3 v 3162 961 a Ft(2)3205 923 y Fw(\010)3265 938 y Fs(S)3309 921 y Fm(3)3387 923 y Fw(through)h(the)523 1034 y(follo)n(wing)27 b(equations:)1000 1251 y Fu(p)1042 1263 y Fs(\022)1102 1251 y Fw(=)c Fr(\000)1285 1195 y Fu(@)p 1265 1232 90 4 v 1265 1308 a(@)5 b(\022)1384 1195 y Fr(\000)p Fu({)p 1374 1232 111 4 v 1374 1249 a Fr(p)p 1443 1249 42 4 v 68 x Fw(2)1495 1251 y(\010)1555 1266 y Fs(S)1599 1249 y Fm(3)1635 1251 y Fu(;)108 b(p)1808 1263 y Fs(\036)1935 1251 y Fw(=)83 b Fr(\000)2182 1195 y Fu(@)p 2158 1232 99 4 v 2158 1308 a(@)5 b(\036)2285 1195 y Fr(\000)p Fu({)p 2276 1232 111 4 v 2276 1249 a Fr(p)p 2345 1249 42 4 v 68 x Fw(2)2396 1251 y(\010)2456 1266 y Fs(S)2500 1249 y Fm(3)2537 1251 y Fu(;)108 b(p)2710 1263 y Fs(\025)2776 1251 y Fw(=)23 b Fr(\000)2963 1195 y Fu(@)p 2939 1232 98 4 v 2939 1308 a(@)5 b(\025)3065 1195 y Fr(\000)p Fu({)p 3056 1232 111 4 v 3056 1249 a Fr(p)p 3125 1249 42 4 v 68 x Fw(2)3176 1251 y(\010)3236 1266 y Fs(S)3280 1249 y Fm(3)3317 1251 y Fu(;)1758 1475 y(w)1817 1487 y Fs(j)1935 1475 y Fw(=)83 b Fu({)2156 1419 y(@)p 2120 1456 123 4 v 2120 1532 a(@)5 b(z)2208 1544 y Fs(j)2272 1419 y Fr(\000)p Fu({)p 2262 1456 111 4 v 2262 1473 a Fr(p)p 2331 1473 42 4 v 68 x Fw(2)2383 1475 y(\010)2443 1490 y Fs(S)2487 1473 y Fm(3)2617 1475 y Fu(j)28 b Fw(=)23 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)p Fu(;)g Fw(4)p Fu(:)549 b Fw(\(3.24\))648 1700 y(Since)729 1879 y(\010)789 1894 y Fs(S)833 1877 y Fm(3)870 1879 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))84 b(=)1295 1806 y Fr(p)p 1364 1806 V 73 x Fw(2)13 b([)p Fu(z)1481 1891 y Ft(1)1518 1879 y Fu(z)1557 1891 y Ft(4)1594 1879 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))20 b Fr(\000)e Fu({)c Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin\()p Fu(\025)p Fw(\)\))20 b Fr(\000)e Fu(z)2611 1891 y Ft(2)2648 1879 y Fu(z)2687 1891 y Ft(3)2724 1879 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))i(+)e Fu({)c Fw(cos)n(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\)\))1295 2003 y(+)23 b Fu({z)1449 2015 y Ft(1)1486 2003 y Fu(z)1525 2015 y Ft(3)1575 2003 y Fw(sin\()p Fu(\022)r Fw(\))14 b(sin)q(\()p Fu(\025)p Fw(\))g(exp\()p Fr(\000)p Fu({\036)p Fw(\))19 b Fr(\000)f Fu({z)2525 2015 y Ft(2)2562 2003 y Fu(z)2601 2015 y Ft(4)2652 2003 y Fw(sin\()p Fu(\022)r Fw(\))c(sin\()p Fu(\025)p Fw(\))g(exp)q(\()p Fu({\036)p Fw(\)])38 b Fu(;)523 2182 y Fw(then)28 b(w)n(e)f(ha)n(v)n(e)853 2361 y Fu(p)895 2373 y Fs(\022)1015 2361 y Fw(=)83 b Fr(\000)p Fu(z)1267 2373 y Ft(1)1303 2361 y Fu(z)1342 2373 y Ft(4)1393 2361 y Fw(sin\()p Fu(\022)r Fw(\))14 b(sin)q(\()p Fu(\025)p Fw(\))19 b Fr(\000)f Fu(z)1970 2373 y Ft(2)2007 2361 y Fu(z)2046 2373 y Ft(3)2097 2361 y Fw(sin\()p Fu(\022)r Fw(\))c(sin\()p Fu(\025)p Fw(\))20 b Fr(\000)1301 2485 y(\000)p Fu(z)1405 2497 y Ft(1)1442 2485 y Fu(z)1481 2497 y Ft(3)1531 2485 y Fw(cos\()p Fu(\022)r Fw(\))14 b(sin\()p Fu(\025)p Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))19 b(+)f Fu(z)2464 2497 y Ft(2)2501 2485 y Fu(z)2540 2497 y Ft(4)2590 2485 y Fw(cos)o(\()p Fu(\022)r Fw(\))c(sin)q(\()p Fu(\025)p Fw(\))g(exp)q(\()p Fu({\036)p Fw(\))24 b Fu(;)846 2610 y(p)888 2622 y Fs(\036)1015 2610 y Fw(=)83 b Fu({)14 b Fw(sin)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\))g([)q Fu(z)1716 2622 y Ft(1)1753 2610 y Fu(z)1792 2622 y Ft(3)1843 2610 y Fw(exp)o(\()p Fr(\000)p Fu({\036)p Fw(\))19 b(+)f Fu(z)2315 2622 y Ft(2)2352 2610 y Fu(z)2391 2622 y Ft(4)2442 2610 y Fw(exp\()p Fu({\036)p Fw(\)])37 b Fu(;)847 2734 y(p)889 2746 y Fs(\025)1015 2734 y Fw(=)83 b Fr(\000)p Fu({z)1294 2746 y Ft(1)1330 2734 y Fu(z)1369 2746 y Ft(4)1406 2734 y Fw(\(sin\()p Fu(\025)p Fw(\))20 b(+)e Fu({)c Fw(cos)n(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\025)p Fw(\)\))20 b Fr(\000)e Fu({z)2450 2746 y Ft(2)2487 2734 y Fu(z)2526 2746 y Ft(3)2563 2734 y Fw(\()p Fr(\000)c Fw(sin)o(\()p Fu(\025)p Fw(\))20 b(+)e Fu({)c Fw(cos)n(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\025)p Fw(\)\))1301 2859 y Fr(\000)p Fu(z)1405 2871 y Ft(1)1442 2859 y Fu(z)1481 2871 y Ft(3)1531 2859 y Fw(sin\()p Fu(\022)r Fw(\))g(cos\()p Fu(\025)p Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))19 b(+)f Fu(z)2464 2871 y Ft(2)2501 2859 y Fu(z)2540 2871 y Ft(4)2590 2859 y Fw(sin\()p Fu(\022)r Fw(\))c(cos\()p Fu(\025)p Fw(\))g(exp)q(\()p Fu({\036)p Fw(\))24 b Fu(;)836 2983 y(w)895 2995 y Ft(1)1015 2983 y Fw(=)83 b Fu(z)1202 2995 y Ft(4)1239 2983 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))20 b Fr(\000)e Fu({)c Fw(cos)n(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\)\))20 b(+)e Fu({z)2283 2995 y Ft(3)2333 2983 y Fw(sin\()p Fu(\022)r Fw(\))c(sin)q(\()p Fu(\025)p Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))23 b Fu(;)836 3108 y(w)895 3120 y Ft(2)1015 3108 y Fw(=)83 b Fr(\000)p Fu(z)1267 3120 y Ft(3)1303 3108 y Fw(\(cos\()p Fu(\025)p Fw(\))19 b(+)f Fu({)c Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\)\))19 b Fr(\000)f Fu({z)2347 3120 y Ft(4)2398 3108 y Fw(sin\()p Fu(\022)r Fw(\))c(sin\()p Fu(\025)p Fw(\))g(exp)q(\()p Fu({\036)p Fw(\))24 b Fu(;)836 3233 y(w)895 3245 y Ft(3)1015 3233 y Fw(=)83 b Fr(\000)p Fu(z)1267 3245 y Ft(2)1303 3233 y Fw(\(cos\()p Fu(\025)p Fw(\))19 b(+)f Fu({)c Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\)\))19 b(+)f Fu({z)2347 3245 y Ft(1)2398 3233 y Fw(sin\()p Fu(\022)r Fw(\))c(sin\()p Fu(\025)p Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))24 b Fu(;)836 3357 y(w)895 3369 y Ft(4)1015 3357 y Fw(=)83 b Fu(z)1202 3369 y Ft(1)1239 3357 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))20 b Fr(\000)e Fu({)c Fw(cos)n(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\)\))20 b Fr(\000)e Fu({z)2283 3369 y Ft(2)2333 3357 y Fw(sin\()p Fu(\022)r Fw(\))c(sin)q(\()p Fu(\025)p Fw(\))g(exp)q(\()p Fu({\036)p Fw(\))23 b Fu(:)521 b Fw(\(3.25\))648 3536 y(As)27 b(in)h(the)g(case)f(for)g Fu(S)1369 3506 y Ft(2)1406 3536 y Fw(,)h(let)f(us)h(state)f(the)h(realit)n(y)f (condition:)1649 3715 y Fu(\022)e Fw(=)p 1801 3647 V 23 w Fu(\022)r(;)83 b(\036)h Fw(=)p 2228 3647 50 4 v 82 w Fu(\036)q(;)f(\025)23 b Fw(=)p 2543 3647 49 4 v 23 w Fu(\025;)1536 3839 y(p)1578 3851 y Fs(\022)1639 3839 y Fw(=)p 1726 3793 42 4 v 22 w Fu(p)1768 3859 y Fs(\022)1806 3839 y Fu(;)83 b(p)1954 3851 y Fs(\036)2081 3839 y Fw(=)p 2228 3793 V 82 w Fu(p)2270 3859 y Fs(\036)2314 3839 y Fu(;)g(p)2462 3851 y Fs(\025)2529 3839 y Fw(=)p 2616 3793 V 22 w Fu(p)2658 3859 y Fs(\025)2725 3839 y Fu(:)870 b Fw(\(3.26\))648 4018 y(This)27 b(condition)h(implies:)1060 4323 y(sin\()p Fu(\022)r Fw(\))83 b(=)2155 4267 y Fr(j)23 b Fu(z)2240 4279 y Ft(1)2277 4267 y Fu(z)2316 4279 y Ft(3)2371 4267 y Fw(+)p 2454 4221 43 4 v 18 w Fu(z)2497 4279 y Ft(2)p 2534 4221 V 2534 4267 a Fu(z)2576 4279 y Ft(4)2636 4267 y Fr(j)p 1508 4304 1799 4 v 1508 4321 a Fo(p)p 1591 4321 1716 4 v 71 x Fw(\()p Fr(j)p Fu(z)1685 4404 y Ft(1)1722 4392 y Fr(j)1745 4368 y Ft(2)1801 4392 y Fw(+)18 b Fr(j)p Fu(z)1946 4404 y Ft(2)1983 4392 y Fr(j)2006 4368 y Ft(2)2043 4392 y Fw(\))c(\()q Fr(j)p Fu(z)2184 4404 y Ft(3)2221 4392 y Fr(j)2244 4368 y Ft(2)2299 4392 y Fw(+)k Fr(j)p Fu(z)2444 4404 y Ft(4)2481 4392 y Fr(j)2504 4368 y Ft(2)2542 4392 y Fw(\))g Fr(\000)g Fw(\()p Fr(<)p Fw(\()p Fu(z)2838 4404 y Ft(1)2876 4392 y Fu(z)2915 4404 y Ft(4)2970 4392 y Fr(\000)g Fu(z)3092 4404 y Ft(2)3129 4392 y Fu(z)3168 4404 y Ft(3)3205 4392 y Fw(\)\))3269 4368 y Ft(2)3317 4323 y Fu(;)1050 4582 y Fw(cos\()p Fu(\022)r Fw(\))83 b(=)2139 4486 y Ft(+)2139 4542 y Fn(\000)2195 4522 y Fr(=)p Fw(\()p Fu(z)2326 4534 y Ft(1)2363 4522 y Fu(z)2402 4534 y Ft(4)2458 4522 y Fw(+)18 b Fu(z)2580 4534 y Ft(3)2617 4522 y Fu(z)2656 4534 y Ft(2)2693 4522 y Fw(\))p 1508 4563 1799 4 v 1508 4579 a Fo(p)p 1591 4579 1716 4 v 71 x Fw(\()p Fr(j)p Fu(z)1685 4662 y Ft(1)1722 4650 y Fr(j)1745 4626 y Ft(2)1801 4650 y Fw(+)g Fr(j)p Fu(z)1946 4662 y Ft(2)1983 4650 y Fr(j)2006 4626 y Ft(2)2043 4650 y Fw(\))c(\()q Fr(j)p Fu(z)2184 4662 y Ft(3)2221 4650 y Fr(j)2244 4626 y Ft(2)2299 4650 y Fw(+)k Fr(j)p Fu(z)2444 4662 y Ft(4)2481 4650 y Fr(j)2504 4626 y Ft(2)2542 4650 y Fw(\))g Fr(\000)g Fw(\()p Fr(<)p Fw(\()p Fu(z)2838 4662 y Ft(1)2876 4650 y Fu(z)2915 4662 y Ft(4)2970 4650 y Fr(\000)g Fu(z)3092 4662 y Ft(2)3129 4650 y Fu(z)3168 4662 y Ft(3)3205 4650 y Fw(\)\))3269 4626 y Ft(2)3317 4582 y Fu(;)1000 4827 y Fw(exp\()p Fu({\036)p Fw(\))83 b(=)1548 4792 y Ft(+)1548 4848 y Fn(\000)1614 4771 y Fu({)22 b Fr(j)i Fu(z)1749 4783 y Ft(2)1785 4771 y Fu(z)1824 4783 y Ft(4)1880 4771 y Fw(+)p 1963 4725 43 4 v 18 w Fu(z)2005 4783 y Ft(1)p 2042 4725 V 2042 4771 a Fu(z)2085 4783 y Ft(3)2145 4771 y Fr(j)p 1614 4808 555 4 v 1685 4884 a Fu(z)1724 4896 y Ft(2)1761 4884 y Fu(z)1800 4896 y Ft(4)1855 4884 y Fw(+)p 1938 4838 43 4 v 18 w Fu(z)1980 4896 y Ft(1)p 2017 4838 V 2017 4884 a Fu(z)2060 4896 y Ft(3)2178 4827 y Fu(;)1052 5065 y Fw(sin\()p Fu(\025)p Fw(\))84 b(=)1508 4936 y Fo(p)p 1591 4936 1716 4 v 71 x Fw(\()p Fr(j)p Fu(z)1685 5019 y Ft(1)1722 5007 y Fr(j)1745 4983 y Ft(2)1801 5007 y Fw(+)18 b Fr(j)p Fu(z)1946 5019 y Ft(2)1983 5007 y Fr(j)2006 4983 y Ft(2)2043 5007 y Fw(\))c(\()q Fr(j)p Fu(z)2184 5019 y Ft(3)2221 5007 y Fr(j)2244 4983 y Ft(2)2299 5007 y Fw(+)k Fr(j)p Fu(z)2444 5019 y Ft(4)2481 5007 y Fr(j)2504 4983 y Ft(2)2542 5007 y Fw(\))g Fr(\000)g Fw(\()p Fr(<)p Fw(\()p Fu(z)2838 5019 y Ft(1)2876 5007 y Fu(z)2915 5019 y Ft(4)2970 5007 y Fr(\000)g Fu(z)3092 5019 y Ft(2)3129 5007 y Fu(z)3168 5019 y Ft(3)3205 5007 y Fw(\)\))3269 4983 y Ft(2)p 1508 5046 1799 4 v 1874 5063 a Fo(p)p 1957 5063 983 4 v 71 x Fw(\()q Fr(j)p Fu(z)2052 5146 y Ft(1)2089 5134 y Fr(j)2112 5110 y Ft(2)2167 5134 y Fw(+)g Fr(j)p Fu(z)2312 5146 y Ft(2)2349 5134 y Fr(j)2372 5110 y Ft(2)2410 5134 y Fw(\))c(\()p Fr(j)p Fu(z)2550 5146 y Ft(3)2587 5134 y Fr(j)2610 5110 y Ft(2)2666 5134 y Fw(+)k Fr(j)p Fu(z)2811 5146 y Ft(4)2848 5134 y Fr(j)2871 5110 y Ft(2)2908 5134 y Fw(\))3317 5065 y Fu(;)1043 5311 y Fw(cos)o(\()p Fu(\025)p Fw(\))84 b(=)1548 5275 y Ft(+)1548 5331 y Fn(\000)1882 5254 y Fr(<)p Fw(\()p Fu(z)2013 5266 y Ft(1)2050 5254 y Fu(z)2089 5266 y Ft(4)2144 5254 y Fr(\000)18 b Fu(z)2266 5266 y Ft(2)2303 5254 y Fu(z)2342 5266 y Ft(3)2379 5254 y Fw(\))p 1614 5291 1066 4 v 1614 5308 a Fo(p)p 1697 5308 983 4 v 71 x Fw(\()p Fr(j)p Fu(z)1791 5391 y Ft(1)1828 5379 y Fr(j)1851 5355 y Ft(2)1907 5379 y Fw(+)g Fr(j)p Fu(z)2052 5391 y Ft(2)2089 5379 y Fr(j)2112 5355 y Ft(2)2149 5379 y Fw(\))c(\()p Fr(j)p Fu(z)2289 5391 y Ft(3)2326 5379 y Fr(j)2349 5355 y Ft(2)2405 5379 y Fw(+)k Fr(j)p Fu(z)2550 5391 y Ft(4)2587 5379 y Fr(j)2610 5355 y Ft(2)2647 5379 y Fw(\))2712 5311 y Fu(:)883 b Fw(\(3.27\))648 5549 y(These)27 b(equations)g(in)g(turn)h(imply:)1458 5728 y(\()p Fu(w)1549 5740 y Ft(1)1587 5728 y Fu(;)14 b(w)1683 5740 y Ft(2)1720 5728 y Fu(;)g(w)1816 5740 y Ft(3)1854 5728 y Fu(;)g(w)1950 5740 y Ft(4)1987 5728 y Fw(\))24 b(=)2180 5693 y Ft(+)2180 5749 y Fn(\000)2236 5728 y Fw(\()p 2268 5682 43 4 v Fu(z)2311 5740 y Ft(1)2348 5728 y Fu(;)p 2385 5682 V 14 w(z)2427 5740 y Ft(2)2464 5728 y Fu(;)p 2501 5682 V 14 w(z)2543 5740 y Ft(3)2581 5728 y Fu(;)p 2618 5682 V 14 w(z)2660 5740 y Ft(4)2697 5728 y Fw(\))889 b(\(3.28\))p eop %%Page: 20 20 20 19 bop 3747 324 a Fw(20)648 614 y(W)-7 b(e)27 b(shall)f(c)n(ho)r (ose)g(the)h(plus)g(sign)g(wherev)n(er)e(w)n(e)i(ha)n(v)n(e)e(the)j(c)n (hoice)e(b)r(et)n(w)n(een)g(either)h(plus)g(or)f(min)n(us)523 714 y(sign)33 b(in)i(the)f(ab)r(o)n(v)n(e)e(equations.)55 b(Th)n(us)34 b(w)n(e)f(ha)n(v)n(e)g(the)h(relation)f(\()p Fu(w)2711 726 y Ft(1)2749 714 y Fu(;)14 b(w)2845 726 y Ft(2)2883 714 y Fu(;)g(w)2979 726 y Ft(3)3016 714 y Fu(;)g(w)3112 726 y Ft(4)3150 714 y Fw(\))34 b(=)f(\()p 3346 668 43 4 v Fu(z)3388 726 y Ft(1)3426 714 y Fu(;)p 3463 668 V 14 w(z)3505 726 y Ft(2)3542 714 y Fu(;)p 3579 668 V 14 w(z)3621 726 y Ft(3)3658 714 y Fu(;)p 3695 668 V 14 w(z)3738 726 y Ft(4)3775 714 y Fw(\).)523 814 y(Using)27 b(this)h(relation)f(in)h(the)g(last)f(four)g(equations)g(w)n(e)g (obtain)h(the)g(exp)r(ected)f(restriction)1595 990 y Fr(j)p Fu(z)1657 1002 y Ft(1)1694 990 y Fr(j)1717 956 y Ft(2)1773 990 y Fw(+)18 b Fr(j)p Fu(z)1918 1002 y Ft(2)1955 990 y Fr(j)1978 956 y Ft(2)2038 990 y Fw(=)23 b Fr(j)p Fu(z)2188 1002 y Ft(3)2225 990 y Fr(j)2248 956 y Ft(2)2304 990 y Fw(+)18 b Fr(j)p Fu(z)2449 1002 y Ft(4)2485 990 y Fr(j)2508 956 y Ft(2)2569 990 y Fu(:)1026 b Fw(\(3.29\))523 1167 y(This)34 b(restriction)g(will)g(app)r(ear)g(as)f(part)h(of)g(the) h(de\014nition)g(of)f(the)h(manifold)f(where)g(our)f(canonical)523 1267 y(transformation)26 b Fr(C)1133 1282 y Fs(S)1177 1265 y Fm(3)1241 1267 y Fw(should)h(tak)n(e)g(v)-5 b(alues)27 b(in.)648 1367 y(In)g(the)h(other)f(hand,)h(the)g(expressions)e(for)h Fu(p)2087 1379 y Fs(\022)2124 1367 y Fu(;)14 b(p)2203 1379 y Fs(\036)2247 1367 y Fu(;)g(p)2326 1379 y Fs(\025)2397 1367 y Fw(in)28 b(terms)f(of)h(\()p Fu(z)2891 1379 y Ft(1)2928 1367 y Fu(;)14 b(z)3004 1379 y Ft(2)3040 1367 y Fu(;)g(z)3116 1379 y Ft(3)3153 1367 y Fu(;)g(z)3229 1379 y Ft(4)3266 1367 y Fw(\))28 b(are:)1127 1684 y Fu(p)1169 1696 y Fs(\022)1290 1684 y Fw(=)82 b Fr(\000)1512 1622 y(<)p Fw(\()p Fu(z)1643 1634 y Ft(1)1680 1622 y Fu(z)1719 1634 y Ft(4)1756 1622 y Fw(\))1802 1555 y Fo(\000)1840 1622 y Fr(j)p Fu(z)1902 1634 y Ft(3)1939 1622 y Fr(j)1962 1592 y Ft(2)2018 1622 y Fw(+)18 b Fr(j)p Fu(z)2163 1634 y Ft(2)2200 1622 y Fr(j)2223 1592 y Ft(2)2260 1555 y Fo(\001)2316 1622 y Fw(+)g Fr(<)p Fw(\()p Fu(z)2530 1634 y Ft(2)2567 1622 y Fu(z)2606 1634 y Ft(3)2643 1622 y Fw(\))2689 1555 y Fo(\000)2728 1622 y Fr(j)p Fu(z)2790 1634 y Ft(1)2826 1622 y Fr(j)2849 1592 y Ft(2)2905 1622 y Fw(+)g Fr(j)p Fu(z)3050 1634 y Ft(4)3087 1622 y Fr(j)3110 1592 y Ft(2)3147 1555 y Fo(\001)p 1512 1665 1674 4 v 2119 1741 a Fr(j)q Fu(z)2182 1753 y Ft(1)2218 1741 y Fu(z)2257 1753 y Ft(3)2313 1741 y Fw(+)p 2396 1695 43 4 v 18 w Fu(z)2438 1753 y Ft(2)p 2475 1695 V 2475 1741 a Fu(z)2517 1753 y Ft(4)2555 1741 y Fr(j)3195 1684 y Fu(;)1121 1904 y(p)1163 1916 y Fs(\036)1290 1904 y Fw(=)1447 1848 y(1)p 1447 1885 42 4 v 1447 1961 a(2)1513 1837 y Fo(\000)1551 1904 y Fr(j)p Fu(z)1613 1916 y Ft(1)1650 1904 y Fr(j)1673 1870 y Ft(2)1728 1904 y Fr(\000)g(j)p Fu(z)1873 1916 y Ft(2)1910 1904 y Fr(j)1933 1870 y Ft(2)1989 1904 y Fw(+)g Fr(j)p Fu(z)2134 1916 y Ft(3)2171 1904 y Fr(j)2194 1870 y Ft(2)2250 1904 y Fr(\000)g(j)p Fu(z)2395 1916 y Ft(4)2432 1904 y Fr(j)2455 1870 y Ft(2)2492 1837 y Fo(\001)2544 1904 y Fu(;)1121 2126 y(p)1163 2138 y Fs(\025)1290 2126 y Fw(=)1593 1998 y Fo(\000)1631 2065 y Fr(j)p Fu(z)1693 2077 y Ft(3)1730 2065 y Fr(j)1753 2035 y Ft(2)1808 2065 y Fw(+)g Fr(j)p Fu(z)1953 2077 y Ft(4)1990 2065 y Fr(j)2013 2035 y Ft(2)2051 1998 y Fo(\001)2103 2065 y Fr(=)p Fw(\()p Fu(z)2234 2077 y Ft(1)2271 2065 y Fu(z)2310 2077 y Ft(4)2365 2065 y Fw(+)p 2448 2019 43 4 v 18 w Fu(z)2490 2077 y Ft(2)p 2527 2019 V 2527 2065 a Fu(z)2570 2077 y Ft(3)2607 2065 y Fw(\))p 1447 2107 1338 4 v 1447 2124 a Fo(q)p 1530 2124 1255 4 v 104 x Fw(\()q Fr(j)p Fu(z)1625 2240 y Ft(3)1661 2228 y Fr(j)1684 2204 y Ft(2)1740 2228 y Fw(+)g Fr(j)p Fu(z)1885 2240 y Ft(4)1922 2228 y Fr(j)1945 2204 y Ft(2)1982 2228 y Fw(\))2015 2186 y Ft(2)2070 2228 y Fr(\000)g Fw(\()q Fr(<)p Fw(\()p Fu(z)2317 2240 y Ft(1)2354 2228 y Fu(z)2393 2240 y Ft(4)2448 2228 y Fr(\000)g Fu(z)2570 2240 y Ft(2)2607 2228 y Fu(z)2646 2240 y Ft(3)2683 2228 y Fw(\)\))2747 2186 y Ft(2)3618 2126 y Fw(\(3.30\))648 2418 y(No)n(w)44 b(notice)g(the)i(follo)n(wing)d(relev)-5 b(an)n(t)44 b(prop)r(ert)n(y)g(of)h(equations)f(\(3.25\):)70 b(Supp)r(ose)45 b(the)g(p)r(oin)n(t)523 2517 y(\()p Fu(\022)r(;)14 b(\036;)g(\025;)g(p) 846 2529 y Fs(\022)885 2517 y Fu(;)g(p)964 2529 y Fs(\036)1007 2517 y Fu(;)g(p)1086 2529 y Fs(\025)1129 2517 y Fw(\))32 b(is)f(giv)n(en.)46 b(If)32 b(\()p Fu(z)1697 2529 y Ft(1)1734 2517 y Fu(;)14 b(z)1810 2529 y Ft(2)1847 2517 y Fu(;)g(z)1923 2529 y Ft(3)1959 2517 y Fu(;)g(z)2035 2529 y Ft(4)2072 2517 y Fw(\))32 b(is)e(a)h(solution)g(of)g(equations)f(\(3.25\);)i (then,)g(for)f(an)n(y)523 2617 y(real)20 b(n)n(um)n(b)r(er)h Fu( )s Fw(,)i(the)f(v)n(ector)d(\()p Fu(z)1527 2629 y Ft(1)1578 2617 y Fw(exp\()p Fu({ )s Fw(\))p Fu(;)14 b(z)1929 2629 y Ft(2)1980 2617 y Fw(exp\()p Fu({ )s Fw(\))p Fu(;)g(z)2331 2629 y Ft(3)2382 2617 y Fw(exp\()p Fr(\000)p Fu({ )s Fw(\))p Fu(;)g(z)2798 2629 y Ft(4)2848 2617 y Fw(exp\()p Fr(\000)p Fu({ )s Fw(\)\))22 b(is)f(a)g(solution)f(to)r(o.)523 2717 y(Th)n(us)j(w)n(e)f(de\014ne)h(the)g(follo)n(wing)f(equiv)-5 b(alence)23 b(relation)e Fr(\030)i Fw(in)g Fv(C)2547 2687 y Ft(4)2607 2717 y Fw(in)g(order)f(to)g(consider)g(the)i(in)n(v)-5 b(ariance)523 2816 y(of)28 b(equations)e(\(3.25\):)1352 2993 y(\()p Fu(z)1423 3005 y Ft(1)1460 2993 y Fu(;)14 b(z)1536 3005 y Ft(2)1573 2993 y Fu(;)g(z)1649 3005 y Ft(3)1686 2993 y Fu(;)g(z)1762 3005 y Ft(4)1799 2993 y Fw(\))23 b Fr(\030)g Fw(\()p Fu(z)2013 3005 y Ft(1)2050 2993 y Fr(0)p Fu(;)14 b(z)2149 3005 y Ft(2)2185 2993 y Fr(0)p Fu(;)g(z)2284 3005 y Ft(3)2321 2993 y Fr(0)p Fu(;)g(z)2420 3005 y Ft(4)2456 2993 y Fr(0)p Fw(\))47 b(i\013)30 b Fr(9)23 b Fu( )j Fr(2)d Fv(R)g Fw(suc)n(h)g(that)871 3118 y(\()p Fu(z)942 3130 y Ft(1)979 3118 y Fr(0)p Fu(;)14 b(z)1078 3130 y Ft(2)1115 3118 y Fr(0)p Fu(;)g(z)1214 3130 y Ft(3)1250 3118 y Fr(0)p Fu(;)g(z)1349 3130 y Ft(4)1386 3118 y Fr(0)p Fw(\))23 b(=)g(\()p Fu(z)1623 3130 y Ft(1)1674 3118 y Fw(exp\()p Fu({ )s Fw(\))p Fu(;)14 b(z)2025 3130 y Ft(2)2076 3118 y Fw(exp)o(\()p Fu({ )s Fw(\))p Fu(;)g(z)2426 3130 y Ft(3)2477 3118 y Fw(exp\()p Fr(\000)p Fu({ )s Fw(\))p Fu(;)g(z)2893 3130 y Ft(4)2944 3118 y Fw(exp\()p Fr(\000)p Fu({ )s Fw(\)\))302 b(\(3.31\))523 3294 y(W)-7 b(e)28 b(shall)f(denote)h(b)n(y)f([\()p Fu(z)1337 3306 y Ft(1)1374 3294 y Fu(;)14 b(z)1450 3306 y Ft(2)1487 3294 y Fu(;)g(z)1563 3306 y Ft(3)1600 3294 y Fu(;)g(z)1676 3306 y Ft(4)1712 3294 y Fw(\)])29 b(the)e(equiv)-5 b(alence)28 b(class)e(of)i(\()p Fu(z)2737 3306 y Ft(1)2774 3294 y Fu(;)14 b(z)2850 3306 y Ft(2)2887 3294 y Fu(;)g(z)2963 3306 y Ft(3)3000 3294 y Fu(;)g(z)3076 3306 y Ft(4)3112 3294 y Fw(\).)648 3394 y(Th)n(us)38 b(our)g(canonical)g(transformation) f Fr(C)2025 3409 y Fs(S)2069 3392 y Fm(3)2144 3394 y Fw(should)i(tak)n(e)f(v)-5 b(alues)39 b(in)g(the)g(follo)n(wing)f(six)h (\(real\))523 3494 y(dimensional)27 b(set:)1037 3650 y Fv(~)1026 3670 y(C)1095 3636 y Ft(4)1155 3670 y Fr(\021)c(f)p Fw(\()p Fu(z)1356 3682 y Ft(1)1393 3670 y Fu(;)14 b(z)1469 3682 y Ft(2)1505 3670 y Fu(;)g(z)1581 3682 y Ft(3)1618 3670 y Fu(;)g(z)1694 3682 y Ft(4)1731 3670 y Fw(\))23 b Fr(2)h Fv(C)1934 3636 y Ft(4)1994 3670 y Fr(j)f(j)p Fu(z)2102 3682 y Ft(1)2139 3670 y Fr(j)2162 3636 y Ft(2)2218 3670 y Fw(+)18 b Fr(j)p Fu(z)2363 3682 y Ft(2)2400 3670 y Fr(j)2423 3636 y Ft(2)2483 3670 y Fw(=)23 b Fr(j)p Fu(z)2633 3682 y Ft(3)2669 3670 y Fr(j)2692 3636 y Ft(2)2748 3670 y Fw(+)18 b Fr(j)p Fu(z)2893 3682 y Ft(4)2930 3670 y Fr(j)2953 3636 y Ft(2)2990 3670 y Fr(g)p Fu(=)23 b Fr(\030)456 b Fw(\(3.32\))648 3847 y(W)-7 b(e)41 b(shall)f(regard)f (the)i(manifold)1807 3827 y Fv(~)1796 3847 y(C)1865 3817 y Ft(4)1929 3847 y Fr(\000)27 b(f)p Fw(0)p Fr(g)40 b Fw(as)g(a)g(symplectic)h(manifold)g(endo)n(w)n(ed)f(with)h(the)523 3947 y(follo)n(wing)27 b(symplectic)g(form:)1172 4124 y Fu(\026)1222 4136 y Ft(3)1283 4124 y Fw(=)22 b Fr(\000)p Fu({)14 b Fw(\()p Fu(dz)1590 4136 y Ft(1)1645 4124 y Fr(^)19 b Fu(d)p 1762 4078 43 4 v(z)1805 4136 y Ft(1)1860 4124 y Fw(+)f Fu(dz)2025 4136 y Ft(2)2081 4124 y Fr(^)h Fu(d)p 2198 4078 V(z)2240 4136 y Ft(2)2296 4124 y Fw(+)f Fu(dz)2461 4136 y Ft(3)2516 4124 y Fr(^)h Fu(d)p 2633 4078 V(z)2676 4136 y Ft(3)2731 4124 y Fw(+)f Fu(dz)2896 4136 y Ft(4)2952 4124 y Fr(^)h Fu(d)p 3069 4078 V(z)3111 4136 y Ft(4)3148 4124 y Fw(\))438 b(\(3.33\))523 4371 y Fv(I)s(I)s(I.3.1)30 b(Canonical)i(transformation)f(for)h Fu(S)2116 4341 y Ft(3)2185 4371 y Fv(and)g(quadric.)648 4518 y Fw(No)n(w)27 b(w)n(e)g(w)n(an)n(t)g(to)g(extend)h(the)g (assignmen)n(t)f(\()p Fu(\022)r(;)14 b(\036;)g(\025;)g(p)2427 4530 y Fs(\022)2465 4518 y Fu(;)g(p)2544 4530 y Fs(\036)2588 4518 y Fw(\))23 b Fr(7!)g Fw([\()p Fu(z)2843 4530 y Ft(1)2881 4518 y Fu(;)14 b(z)2957 4530 y Ft(2)2993 4518 y Fu(;)g(z)3069 4530 y Ft(3)3106 4518 y Fu(;)g(z)3182 4530 y Ft(4)3219 4518 y Fw(\)])28 b(giv)n(en)f(b)n(y)g(equa-)523 4618 y(tions)c(\(3.25\))f(to)h(the)h(whole)e(space)h Fu(T)1702 4588 y Fn(\003)1739 4618 y Fu(S)1795 4588 y Ft(3)1841 4618 y Fr(\000)9 b(f)p Fw(0)p Fr(g)p Fw(.)34 b(T)-7 b(o)23 b(that)g(end,)h(let)g(us)f(write)g(the)g(generating)f(function)539 4685 y Fn(\000)p Fs({)p 533 4699 88 4 v 533 4707 a Fn(p)p 588 4707 34 3 v 49 x Ft(2)631 4718 y Fw(\010)691 4733 y Fs(S)735 4716 y Fm(3)799 4718 y Fw(as)27 b(a)g(inner)g(pro)r(duct:) 1700 4897 y Fr(\000)p Fu({)p 1690 4934 111 4 v 1690 4950 a Fr(p)p 1759 4950 42 4 v 69 x Fw(2)1811 4953 y(\010)1871 4968 y Fs(S)1915 4951 y Fm(3)1974 4953 y Fw(=)c Fr(\000)p Fu({)p Fw(\()p Fu(\013)p Fw(\()p Fv(z)p Fw(\))c Fr(\017)f Fv(x)p Fw(\))3618 5125 y(\(3.34\))523 5302 y(where)885 5479 y Fu(\013)p Fw(\()p Fv(z)p Fw(\))24 b(=)f(\()p Fu({z)1254 5491 y Ft(1)1291 5479 y Fu(z)1330 5491 y Ft(3)1385 5479 y Fr(\000)18 b Fu({z)1534 5491 y Ft(2)1571 5479 y Fu(z)1610 5491 y Ft(4)1647 5479 y Fu(;)c(z)1723 5491 y Ft(1)1759 5479 y Fu(z)1798 5491 y Ft(3)1854 5479 y Fw(+)k Fu(z)1976 5491 y Ft(2)2013 5479 y Fu(z)2052 5491 y Ft(4)2088 5479 y Fu(;)c Fr(\000)p Fu({z)2256 5491 y Ft(1)2293 5479 y Fu(z)2332 5491 y Ft(4)2387 5479 y Fr(\000)k Fu({z)2536 5491 y Ft(2)2572 5479 y Fu(z)2611 5491 y Ft(3)2648 5479 y Fu(;)c(z)2724 5491 y Ft(1)2761 5479 y Fu(z)2800 5491 y Ft(4)2855 5479 y Fr(\000)k Fu(z)2977 5491 y Ft(2)3014 5479 y Fu(z)3053 5491 y Ft(3)3090 5479 y Fw(\))47 b(and)954 5604 y Fv(x)23 b Fw(=)g(\(sin\()p Fu(\025)p Fw(\))14 b(sin)q(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))p Fu(;)g Fw(sin)q(\()p Fu(\025)p Fw(\))g(sin)q(\()p Fu(\022)r Fw(\))g(sin\()p Fu(\036)p Fw(\))p Fu(;)g Fw(sin)q(\()p Fu(\025)p Fw(\))g(cos\()p Fu(\022)r Fw(\))p Fu(;)g Fw(cos\()p Fu(\025)p Fw(\)\))3618 5728 y(\(3.35\))p eop %%Page: 21 21 21 20 bop 3747 324 a Fw(21)523 614 y(No)n(w)27 b(notice)h(that,)g(for)f (all)g(\()p Fu(z)1473 626 y Ft(1)1510 614 y Fu(;)14 b(z)1586 626 y Ft(2)1623 614 y Fu(;)g(z)1699 626 y Ft(3)1736 614 y Fu(;)g(z)1812 626 y Ft(4)1848 614 y Fw(\))24 b Fr(2)f Fv(C)2051 584 y Ft(4)2088 614 y Fw(,)28 b(the)g(v)n(ector)e Fu(\013)p Fw(\()p Fv(z)p Fw(\))j(b)r(elongs)e(to)h(the)g(quadric)1140 797 y Fv(Q)1212 809 y Ft(4)1272 797 y Fw(=)22 b Fr(f)p Fw(\()p Fu(\013)1486 809 y Ft(1)1523 797 y Fu(;)14 b(\013)1613 809 y Ft(2)1651 797 y Fu(;)g(\013)1741 809 y Ft(3)1778 797 y Fu(;)g(\013)1868 809 y Ft(4)1905 797 y Fw(\))24 b Fr(2)f Fv(C)2108 763 y Ft(4)2145 797 y Fu(=\013)2240 763 y Ft(2)2240 818 y(1)2296 797 y Fw(+)18 b Fu(\013)2432 763 y Ft(2)2432 818 y(2)2488 797 y Fw(+)g Fu(\013)2624 763 y Ft(2)2624 818 y(3)2680 797 y Fw(+)g Fu(\013)2816 763 y Ft(2)2816 818 y(4)2877 797 y Fw(=)k(0)p Fr(g)570 b Fw(\(3.36\))523 980 y(This)28 b(suggests)e(to)h(de\014ne)h(the)g (follo)n(wing)f(map)804 1162 y Fu(\032)847 1174 y Ft(3)907 1162 y Fw(:)964 1142 y Fv(~)953 1162 y(C)1022 1128 y Ft(4)1078 1162 y Fr(\000)18 b(f)p Fw(0)p Fr(g)128 b(7!)83 b Fv(Q)1653 1174 y Ft(4)1708 1162 y Fr(\000)18 b(f)p Fw(0)p Fr(g)727 1287 y Fu(\032)770 1299 y Ft(3)807 1287 y Fw(\([)p Fu(z)901 1299 y Ft(1)938 1287 y Fu(;)c(z)1014 1299 y Ft(2)1051 1287 y Fu(;)g(z)1127 1299 y Ft(3)1163 1287 y Fu(;)g(z)1239 1299 y Ft(4)1276 1287 y Fw(]\))93 b(=)f(\()p Fu({z)1679 1299 y Ft(1)1716 1287 y Fu(z)1755 1299 y Ft(3)1810 1287 y Fr(\000)18 b Fu({z)1959 1299 y Ft(2)1995 1287 y Fu(z)2034 1299 y Ft(4)2071 1287 y Fu(;)c(z)2147 1299 y Ft(1)2184 1287 y Fu(z)2223 1299 y Ft(3)2278 1287 y Fw(+)k Fu(z)2400 1299 y Ft(2)2437 1287 y Fu(z)2476 1299 y Ft(4)2513 1287 y Fu(;)c Fr(\000)p Fu({z)2681 1299 y Ft(1)2717 1287 y Fu(z)2756 1299 y Ft(4)2812 1287 y Fr(\000)k Fu({z)2961 1299 y Ft(2)2997 1287 y Fu(z)3036 1299 y Ft(3)3073 1287 y Fu(;)c(z)3149 1299 y Ft(1)3186 1287 y Fu(z)3225 1299 y Ft(4)3280 1287 y Fr(\000)k Fu(z)3402 1299 y Ft(2)3439 1287 y Fu(z)3478 1299 y Ft(3)3515 1287 y Fw(\))71 b(\(3.37\))523 1469 y(with)33 b(\()p Fu(z)788 1481 y Ft(1)825 1469 y Fu(;)14 b(z)901 1481 y Ft(2)937 1469 y Fu(;)g(z)1013 1481 y Ft(3)1050 1469 y Fu(;)g(z)1126 1481 y Ft(4)1163 1469 y Fw(\))32 b(a)g(represen)n(tativ)n(e)e(of)i(the) g(equiv)-5 b(alen)n(t)32 b(class)f([)p Fu(z)2745 1481 y Ft(1)2782 1469 y Fu(;)14 b(z)2858 1481 y Ft(2)2894 1469 y Fu(;)g(z)2970 1481 y Ft(3)3007 1469 y Fu(;)g(z)3083 1481 y Ft(4)3120 1469 y Fw(].)50 b(F)-7 b(rom)32 b(no)n(w)f(on)h(w)n(e) 523 1569 y(shall)27 b(denote)h Fu(\032)1028 1581 y Ft(3)1065 1569 y Fw(\([)p Fu(z)1159 1581 y Ft(1)1196 1569 y Fu(;)14 b(z)1272 1581 y Ft(2)1309 1569 y Fu(;)g(z)1385 1581 y Ft(3)1422 1569 y Fu(;)g(z)1498 1581 y Ft(4)1534 1569 y Fw(]\))28 b(just)h(b)n(y)e Fu(\032)1940 1581 y Ft(3)1977 1569 y Fw(\()p Fu(z)2048 1581 y Ft(1)2085 1569 y Fu(;)14 b(z)2161 1581 y Ft(2)2198 1569 y Fu(;)g(z)2274 1581 y Ft(3)2311 1569 y Fu(;)g(z)2387 1581 y Ft(4)2423 1569 y Fw(\).)648 1669 y(Let)27 b(us)h(also)e(de\014ne)i(the)g(map)1576 1851 y Fu(\033)1623 1863 y Ft(3)1684 1851 y Fw(:)23 b Fv(Q)1802 1863 y Ft(4)1857 1851 y Fr(\000)18 b(f)p Fw(0)p Fr(g)k(7!)h Fu(T)2255 1817 y Fn(\003)2292 1851 y Fu(S)2348 1817 y Ft(3)2404 1851 y Fr(\000)18 b(f)p Fw(0)p Fr(g)1781 2026 y Fu(\033)1828 2038 y Ft(3)1866 2026 y Fw(\()p Fu(\013)p Fw(\))24 b(=)2095 1909 y Fo(\022)2189 1970 y Fr(<)p Fu(\013)p 2166 2007 160 4 v 2166 2083 a Fr(j<)p Fu(\013)p Fr(j)2335 2026 y Fu(;)14 b Fr(\000=)p Fu(\013)2550 1909 y Fo(\023)3618 2026 y Fw(\(3.38\))523 2264 y(As)39 b(in)g(the)h(case)e(for)g Fu(S)1301 2233 y Ft(2)1338 2264 y Fw(,)k(w)n(e)d(shall)f(extend)i(our)e (assignmen)n(t)g(\()p Fu(\022)r(;)14 b(\036;)g(\025;)g(p)2945 2276 y Fs(\022)2983 2264 y Fu(;)g(p)3062 2276 y Fs(\036)3106 2264 y Fw(\))42 b Fr(7!)g Fw([\()p Fu(z)3399 2276 y Ft(1)3437 2264 y Fu(;)14 b(z)3513 2276 y Ft(2)3549 2264 y Fu(;)g(z)3625 2276 y Ft(3)3662 2264 y Fu(;)g(z)3738 2276 y Ft(4)3775 2264 y Fw(\)])523 2363 y(determined)31 b(b)n(y)g(equations)f(\(3.25\))g (to)h(the)g(whole)g(space)f Fu(T)2473 2333 y Fn(\003)2510 2363 y Fu(S)2566 2333 y Ft(3)2624 2363 y Fr(\000)20 b(f)p Fw(0)p Fr(g)p Fw(.)46 b(F)-7 b(or)30 b(this)i(purp)r(ose,)f(w)n(e)f (\014rst)523 2463 y(c)n(hec)n(k)d(that)h(if)g(w)n(e)f(consider)g(a)g(p) r(oin)n(t)g(in)h Fu(T)1892 2433 y Fn(\003)1930 2463 y Fu(S)1986 2433 y Ft(3)2041 2463 y Fr(\000)18 b(f)p Fw(0)p Fr(g)26 b Fw(giv)n(en)h(in)h(spherical)e(co)r(ordinates)h(b)n(y)938 2645 y(\()q(sin\()p Fu(\025)p Fw(\))14 b(sin\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))p Fu(;)g Fw(sin)q(\()p Fu(\025)p Fw(\))g(sin)q(\()p Fu(\022)r Fw(\))g(sin\()p Fu(\036)p Fw(\))p Fu(;)g Fw(sin)q(\()p Fu(\025)p Fw(\))g(cos\()p Fu(\022)r Fw(\))p Fu(;)g Fw(cos\()p Fu(\025)p Fw(\))p Fu(;)953 2818 y Fw(cos)o(\()p Fu(\025)p Fw(\))g(sin)q(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))p Fu(p)1679 2830 y Fs(\025)1741 2818 y Fw(+)k(cos\()p Fu(\022)r Fw(\))c(cos)o(\()p Fu(\036)p Fw(\))p Fu(p)2321 2830 y Fs(\022)2360 2818 y Fu(=)g Fw(sin)o(\()p Fu(\025)p Fw(\))20 b Fr(\000)2809 2762 y Fw(sin\()p Fu(\036)p Fw(\))p Fu(p)3066 2774 y Fs(\036)p 2742 2799 437 4 v 2742 2875 a Fw(sin\()p Fu(\025)p Fw(\))14 b(sin\()p Fu(\022)r Fw(\))3188 2818 y Fu(;)971 3047 y Fw(cos\()p Fu(\025)p Fw(\))g(sin\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))p Fu(p)1688 3059 y Fs(\025)1751 3047 y Fw(+)k(cos)o(\()p Fu(\022)r Fw(\))c(sin\()p Fu(\036)p Fw(\))p Fu(p)2321 3059 y Fs(\022)2360 3047 y Fu(=)g Fw(sin)o(\()p Fu(\025)p Fw(\))20 b Fr(\000)2804 2991 y Fw(cos)o(\()p Fu(\036)p Fw(\))p Fu(p)3070 3003 y Fs(\036)p 2742 3028 V 2742 3104 a Fw(sin\()p Fu(\025)p Fw(\))14 b(sin\()p Fu(\022)r Fw(\))3188 3047 y Fu(;)1602 3220 y Fw(cos)o(\()p Fu(\025)p Fw(\))g(cos\()p Fu(\022)r Fw(\))p Fu(p)2098 3232 y Fs(\025)2161 3220 y Fr(\000)k Fw(sin\()p Fu(\022)r Fw(\))p Fu(p)2493 3232 y Fs(\022)2531 3220 y Fu(=)c Fw(sin)o(\()p Fu(\025)p Fw(\))p Fu(;)g Fr(\000)g Fw(sin\()p Fu(\025)p Fw(\))p Fu(p)3172 3232 y Fs(\025)3217 3220 y Fw(\))369 b(\(3.39\))523 3403 y(and)26 b(w)n(e)f(use)h(equations)f(\(3.27\),)h(then)g(w)n(e)g(obtain)f(a)h (map)g(whic)n(h)g(assigns)e(to)i(\()p Fu(z)3075 3415 y Ft(1)3112 3403 y Fu(;)14 b(z)3188 3415 y Ft(2)3225 3403 y Fu(;)g(z)3301 3415 y Ft(3)3337 3403 y Fu(;)g(z)3413 3415 y Ft(4)3450 3403 y Fw(\))26 b(the)h(same)523 3503 y(elemen)n(t)k(in)g Fu(T)994 3472 y Fn(\003)1031 3503 y Fu(S)1087 3472 y Ft(3)1155 3503 y Fw(as)f(the)i(ev)-5 b(aluation)30 b(of)g(the)i(function)f Fu(\033)2428 3515 y Ft(3)2486 3503 y Fr(\016)21 b Fu(\032)2592 3515 y Ft(3)2660 3503 y Fw(in)31 b(\()p Fu(z)2831 3515 y Ft(1)2868 3503 y Fu(;)14 b(z)2944 3515 y Ft(2)2981 3503 y Fu(;)g(z)3057 3515 y Ft(3)3093 3503 y Fu(;)g(z)3169 3515 y Ft(4)3206 3503 y Fw(\).)47 b(This)31 b(allo)n(ws)e(us)523 3602 y(to)f(extend)f(our)g(canonical)g(transformation)e Fr(C)2016 3617 y Fs(S)2060 3600 y Fm(3)2124 3602 y Fw(as)i(the)h(map)g(\()p Fu(\033)2633 3614 y Ft(3)2689 3602 y Fr(\016)18 b Fu(\032)2792 3614 y Ft(3)2829 3602 y Fw(\))2861 3572 y Fn(\000)p Ft(1)2951 3602 y Fw(.)648 3702 y(Let)27 b(us)h(state)f(our)g(conclusions)g(in)g (the)h(follo)n(wing)523 3868 y Fv(Theorem)i(3.3)41 b Fl(The)30 b(function)1602 3835 y Fn(\000)p Fs({)p 1596 3849 88 4 v 1596 3858 a Fn(p)p 1651 3858 34 3 v 48 x Ft(2)1694 3868 y Fw(\010)1754 3883 y Fs(S)1798 3866 y Fm(3)1863 3868 y Fl(gener)l(ates)e(a)h(tr)l(ansformation)h(which)g(c)l (an)f(b)l(e)f(extende)l(d)h(as)g(a)523 3979 y(c)l(anonic)l(al)f(tr)l (ansformation)g Fr(C)1493 3994 y Fs(S)1537 3977 y Fm(3)1601 3979 y Fl(that)f(makes)g(the)h(fol)t(lowing)h(triangle)f(of)g(c)l (anonic)l(al)g(tr)l(ansformations)523 4079 y(to)i(c)l(ommute:)1886 4851 y Fb(\000)1969 4768 y(\000)2052 4685 y(\000)2135 4602 y(\000)2218 4519 y(\000)2301 4436 y(\000)-83 b(\022)p 1760 4851 4 499 v 1761 4851 a(?)p 1969 4271 416 4 v 1969 4269 a(\033)1594 4955 y Fv(Q)1666 4967 y Ft(4)1721 4955 y Fr(\000)18 b(f)p Fw(0)p Fr(g)1606 4278 y Fv(~)1595 4298 y(C)1664 4268 y Ft(4)1720 4298 y Fr(\000)g(f)p Fw(0)p Fr(g)2425 4293 y Fu(T)2486 4263 y Fn(\003)2523 4293 y Fu(S)2579 4263 y Ft(3)2634 4293 y Fr(\000)g(f)p Fw(0)p Fr(g)2114 4166 y(C)2158 4181 y Fs(S)2202 4164 y Fm(3)1638 4653 y Fu(\032)1681 4665 y Ft(3)2176 4655 y Fu(\033)2223 4667 y Ft(3)523 5166 y Fl(wher)l(e)23 b(\(a\))f(The)i(symple)l(ctic)f (form)g Fu(\027)1670 5178 y Ft(3)1729 5166 y Fl(on)g Fu(T)1902 5136 y Fn(\003)1939 5166 y Fu(S)1995 5136 y Ft(3)2054 5166 y Fl(is)g(given)f(by)h(the)f(r)l(estriction)h(to)f Fu(T)3122 5136 y Fn(\003)3159 5166 y Fu(S)3215 5136 y Ft(3)3274 5166 y Fl(of)i(the)e(c)l(anonic)l(al)523 5266 y(symple)l(ctic)31 b(form)f Fu(!)1174 5278 y Ft(4)1241 5266 y Fl(on)g Fu(T)1421 5236 y Fn(\003)1458 5266 y Fv(R)1530 5236 y Ft(4)1567 5266 y Fl(,)523 5365 y(\(b\))g(The)h(symple)l(ctic)f (form)h Fu(\026)1478 5377 y Ft(3)1545 5365 y Fl(on)1674 5345 y Fv(~)1664 5365 y(C)1733 5335 y Ft(4)1788 5365 y Fr(\000)18 b(f)p Fw(0)p Fr(g)29 b Fl(is)h(given)g(by)1193 5548 y Fu(\026)1243 5560 y Ft(3)1304 5548 y Fw(=)22 b Fr(\000)p Fu({)14 b Fw(\()p Fu(dz)1611 5560 y Ft(1)1666 5548 y Fr(^)19 b Fu(d)p 1783 5502 43 4 v(z)1825 5560 y Ft(1)1881 5548 y Fw(+)f Fu(dz)2046 5560 y Ft(2)2102 5548 y Fr(^)g Fu(d)p 2218 5502 V(z)2261 5560 y Ft(2)2298 5548 y Fu(dz)2380 5560 y Ft(3)2436 5548 y Fr(^)g Fu(d)p 2552 5502 V(z)2595 5560 y Ft(3)2651 5548 y Fw(+)g Fu(dz)2816 5560 y Ft(4)2871 5548 y Fr(^)h Fu(d)p 2988 5502 V(z)3030 5560 y Ft(4)3068 5548 y Fw(\))37 b Fu(;)458 b Fw(\(3.40\))p eop %%Page: 22 22 22 21 bop 3747 324 a Fw(22)523 614 y Fl(\(c\))30 b(The)h(symple)l(ctic) f(form)h Fu(q)1465 626 y Ft(4)1532 614 y Fl(on)f Fv(Q)1723 626 y Ft(4)1778 614 y Fr(\000)18 b(f)p Fw(0)p Fr(g)28 b Fl(is)j(given)f(by:)792 876 y Fu(q)829 888 y Ft(4)890 876 y Fw(=)1080 820 y(1)p 987 857 228 4 v 987 933 a(2)p Fu({)p Fr(j<)p Fu(\013)p Fr(j)1282 772 y Ft(4)1239 797 y Fo(X)1241 974 y Fs(j)s Ft(=1)1372 876 y Fu(d\013)1468 888 y Fs(j)1522 876 y Fr(^)19 b Fu(d)p 1639 830 54 4 v(\013)1693 888 y Fs(j)1746 876 y Fr(\000)1951 820 y Fw(1)p 1839 857 266 4 v 1839 933 a(8)p Fu({)p Fr(j<)p Fu(\013)p Fr(j)2067 909 y Ft(3)2195 772 y(4)2152 797 y Fo(X)2128 976 y Fs(j;k)q Ft(=1)2309 876 y Fw(\()q Fu(\013)2395 888 y Fs(j)2448 876 y Fw(+)p 2531 830 54 4 v 18 w Fu(\013)2585 888 y Fs(j)2620 876 y Fw(\))14 b(\()p Fu(\013)2751 888 y Fs(k)2811 876 y Fw(+)p 2894 830 V 18 w Fu(\013)2947 888 y Fs(k)2988 876 y Fw(\))g Fu(d\013)3130 888 y Fs(j)3184 876 y Fr(^)k Fu(d)p 3300 830 V(\013)3354 888 y Fs(k)3618 876 y Fw(\(3.41\))523 1166 y Fl(which)33 b(is)f(not)f(the)g(r)l (estriction)h(to)f Fv(Q)1712 1178 y Ft(4)1769 1166 y Fr(\000)19 b(f)p Fw(0)p Fr(g)30 b Fl(of)i(the)f(symple)l(ctic)i(form) 2850 1104 y Fo(P)2938 1124 y Ft(4)2938 1191 y Fs(j)s Ft(=1)3071 1166 y Fu(d\013)3167 1178 y Fs(j)3221 1166 y Fr(^)20 b Fu(d)p 3339 1120 V(\013)3393 1178 y Fs(j)3459 1166 y Fl(de\014ne)l(d)32 b(on)523 1278 y(the)e(ambient)g Fv(C)1046 1248 y Ft(4)1083 1278 y Fl(.)39 b(The)31 b(c)l(omp)l(onents)e (of)i(the)f(angular)g(momentum)f(and)h(R)n(unge-L)l(enz)e(ve)l(ctors)i (ar)l(e:)1446 1495 y Fu(L)1503 1507 y Ft(1)1622 1495 y Fw(=)1780 1439 y(1)p 1780 1476 42 4 v 1780 1552 a(2)1845 1495 y(\()q Fu(z)1917 1507 y Ft(1)p 1953 1449 43 4 v 1953 1495 a Fu(z)1996 1507 y Ft(2)2051 1495 y Fw(+)18 b Fu(z)2173 1507 y Ft(3)p 2210 1449 V 2210 1495 a Fu(z)2253 1507 y Ft(4)2308 1495 y Fw(+)g Fu(z)2430 1507 y Ft(2)p 2467 1449 V 2467 1495 a Fu(z)2509 1507 y Ft(1)2565 1495 y Fw(+)g Fu(z)2687 1507 y Ft(4)p 2724 1449 V 2724 1495 a Fu(z)2766 1507 y Ft(3)2803 1495 y Fw(\))d Fu(;)1446 1695 y(L)1503 1707 y Ft(2)1622 1695 y Fw(=)1793 1639 y(1)p 1780 1676 69 4 v 1780 1752 a(2)p Fu({)1872 1695 y Fw(\()p Fu(z)1943 1707 y Ft(1)p 1980 1649 43 4 v 1980 1695 a Fu(z)2022 1707 y Ft(2)2078 1695 y Fw(+)j Fu(z)2200 1707 y Ft(3)p 2237 1649 V 2237 1695 a Fu(z)2279 1707 y Ft(4)2335 1695 y Fr(\000)g Fu(z)2457 1707 y Ft(2)p 2494 1649 V 2494 1695 a Fu(z)2536 1707 y Ft(1)2592 1695 y Fr(\000)g Fu(z)2714 1707 y Ft(4)p 2751 1649 V 2751 1695 a Fu(z)2793 1707 y Ft(3)2830 1695 y Fw(\))c Fu(;)1446 1895 y(L)1503 1907 y Ft(3)1622 1895 y Fw(=)1780 1839 y(1)p 1780 1876 42 4 v 1780 1952 a(2)1845 1895 y(\()q Fu(z)1917 1907 y Ft(1)p 1953 1849 43 4 v 1953 1895 a Fu(z)1996 1907 y Ft(1)2051 1895 y Fw(+)k Fu(z)2173 1907 y Ft(3)p 2210 1849 V 2210 1895 a Fu(z)2253 1907 y Ft(3)2308 1895 y Fr(\000)g Fu(z)2430 1907 y Ft(2)p 2467 1849 V 2467 1895 a Fu(z)2509 1907 y Ft(2)2565 1895 y Fr(\000)g Fu(z)2687 1907 y Ft(4)p 2724 1849 V 2724 1895 a Fu(z)2766 1907 y Ft(4)2803 1895 y Fw(\))d Fu(;)1440 2095 y(A)1502 2107 y Ft(1)1622 2095 y Fw(=)1780 2038 y(1)p 1780 2076 42 4 v 1780 2152 a(2)1845 2095 y(\()q Fu(z)1917 2107 y Ft(1)p 1953 2049 43 4 v 1953 2095 a Fu(z)1996 2107 y Ft(2)2051 2095 y Fw(+)j Fu(z)2173 2107 y Ft(2)p 2210 2049 V 2210 2095 a Fu(z)2253 2107 y Ft(1)2308 2095 y Fr(\000)g Fu(z)2430 2107 y Ft(3)p 2467 2049 V 2467 2095 a Fu(z)2509 2107 y Ft(4)2565 2095 y Fr(\000)g Fu(z)2687 2107 y Ft(4)p 2724 2049 V 2724 2095 a Fu(z)2766 2107 y Ft(3)2803 2095 y Fw(\))d Fu(;)1440 2294 y(A)1502 2306 y Ft(2)1622 2294 y Fw(=)1793 2238 y(1)p 1780 2275 69 4 v 1780 2351 a(2)p Fu({)1872 2294 y Fw(\()p Fu(z)1943 2306 y Ft(1)p 1980 2249 43 4 v 1980 2294 a Fu(z)2022 2306 y Ft(2)2078 2294 y Fw(+)j Fu(z)2200 2306 y Ft(4)p 2237 2249 V 2237 2294 a Fu(z)2279 2306 y Ft(3)2335 2294 y Fr(\000)g Fu(z)2457 2306 y Ft(2)p 2494 2249 V 2494 2294 a Fu(z)2536 2306 y Ft(1)2592 2294 y Fw(+)g Fu(z)2714 2306 y Ft(3)p 2751 2249 V 2751 2294 a Fu(z)2793 2306 y Ft(4)2830 2294 y Fw(\))c Fu(;)1440 2494 y(A)1502 2506 y Ft(3)1622 2494 y Fw(=)1780 2438 y(1)p 1780 2475 42 4 v 1780 2551 a(2)1845 2494 y(\()q Fu(z)1917 2506 y Ft(1)p 1953 2449 43 4 v 1953 2494 a Fu(z)1996 2506 y Ft(1)2051 2494 y Fw(+)k Fu(z)2173 2506 y Ft(4)p 2210 2449 V 2210 2494 a Fu(z)2253 2506 y Ft(4)2308 2494 y Fr(\000)g Fu(z)2430 2506 y Ft(2)p 2467 2449 V 2467 2494 a Fu(z)2509 2506 y Ft(2)2565 2494 y Fw(+)g Fu(z)2687 2506 y Ft(3)p 2724 2449 V 2724 2494 a Fu(z)2766 2506 y Ft(3)2803 2494 y Fw(\))d Fu(:)745 b Fw(\(3.42\))523 2710 y Fl(The)31 b(expr)l(essions)f (for)h Fu(L)1317 2680 y Ft(2)1376 2710 y Fw(=)23 b Fu(L)1521 2680 y Ft(2)1521 2731 y(1)1576 2710 y Fw(+)18 b Fu(L)1716 2680 y Ft(2)1716 2731 y(2)1771 2710 y Fw(+)g Fu(L)1911 2680 y Ft(2)1911 2731 y(3)1971 2710 y Fl(,)30 b Fu(A)2088 2680 y Ft(2)2149 2710 y Fw(=)23 b Fu(A)2299 2680 y Ft(2)2299 2731 y(1)2355 2710 y Fw(+)18 b Fu(A)2500 2680 y Ft(2)2500 2731 y(2)2556 2710 y Fw(+)g Fu(A)2701 2680 y Ft(2)2701 2731 y(3)2768 2710 y Fl(and)30 b(the)g(ener)l(gy)g(H)f(ar)l(e:)631 2932 y Fu(L)688 2897 y Ft(2)748 2932 y Fw(=)846 2876 y(1)p 846 2913 42 4 v 846 2989 a(4)897 2932 y(\()p Fu(z)968 2944 y Ft(1)p 1005 2886 43 4 v 1005 2932 a Fu(z)1047 2944 y Ft(1)1103 2932 y Fw(+)18 b Fu(z)1225 2944 y Ft(2)p 1262 2886 V 1262 2932 a Fu(z)1304 2944 y Ft(2)1341 2932 y Fw(\))1373 2897 y Ft(2)1429 2932 y Fw(+)1522 2876 y(1)p 1522 2913 42 4 v 1522 2989 a(4)1574 2932 y(\()p Fu(z)1645 2944 y Ft(3)p 1682 2886 43 4 v 1682 2932 a Fu(z)1724 2944 y Ft(3)1780 2932 y Fw(+)g Fu(z)1902 2944 y Ft(4)p 1939 2886 V 1939 2932 a Fu(z)1981 2944 y Ft(4)2018 2932 y Fw(\))2050 2897 y Ft(2)1769 3132 y Fw(+)1844 3075 y(1)p 1844 3112 42 4 v 1844 3189 a(2)1909 3064 y Fo(\000)1947 3132 y Fw(4)p Fr(<)p Fw(\()p Fu(z)2120 3144 y Ft(1)2157 3132 y Fu(z)2196 3144 y Ft(4)p 2233 3086 43 4 v 2233 3132 a Fu(z)2275 3144 y Ft(2)p 2312 3086 V 2312 3132 a Fu(z)2354 3144 y Ft(3)2392 3132 y Fw(\))g(+)g(\()p Fr(j)p Fu(z)2619 3144 y Ft(2)2657 3132 y Fr(j)2680 3097 y Ft(2)2735 3132 y Fr(\000)g(j)p Fu(z)2880 3144 y Ft(1)2917 3132 y Fr(j)2940 3097 y Ft(2)2978 3132 y Fw(\)\()p Fr(j)p Fu(z)3104 3144 y Ft(4)3141 3132 y Fr(j)3164 3097 y Ft(2)3220 3132 y Fr(\000)g(j)p Fu(z)3365 3144 y Ft(3)3402 3132 y Fr(j)3425 3097 y Ft(2)3462 3132 y Fw(\))3494 3064 y Fo(\001)3569 3132 y Fu(;)625 3331 y(A)687 3297 y Ft(2)748 3331 y Fw(=)846 3275 y(1)p 846 3312 42 4 v 846 3388 a(4)897 3331 y(\()p Fu(z)968 3343 y Ft(1)p 1005 3286 43 4 v 1005 3331 a Fu(z)1047 3343 y Ft(1)1103 3331 y Fw(+)g Fu(z)1225 3343 y Ft(2)p 1262 3286 V 1262 3331 a Fu(z)1304 3343 y Ft(2)1341 3331 y Fw(\))1373 3297 y Ft(2)1429 3331 y Fw(+)1522 3275 y(1)p 1522 3312 42 4 v 1522 3388 a(4)1574 3331 y(\()p Fu(z)1645 3343 y Ft(3)p 1682 3286 43 4 v 1682 3331 a Fu(z)1724 3343 y Ft(3)1780 3331 y Fw(+)g Fu(z)1902 3343 y Ft(4)p 1939 3286 V 1939 3331 a Fu(z)1981 3343 y Ft(4)2018 3331 y Fw(\))2050 3297 y Ft(2)1769 3531 y Fr(\000)1844 3475 y Fw(1)p 1844 3512 42 4 v 1844 3588 a(2)1909 3464 y Fo(\000)1947 3531 y Fw(4)p Fr(<)p Fw(\()p Fu(z)2120 3543 y Ft(1)2157 3531 y Fu(z)2196 3543 y Ft(4)p 2233 3486 43 4 v 2233 3531 a Fu(z)2275 3543 y Ft(2)p 2312 3486 V 2312 3531 a Fu(z)2354 3543 y Ft(3)2392 3531 y Fw(\))g(+)g(\()p Fr(j)p Fu(z)2619 3543 y Ft(2)2657 3531 y Fr(j)2680 3497 y Ft(2)2735 3531 y Fr(\000)g(j)p Fu(z)2880 3543 y Ft(1)2917 3531 y Fr(j)2940 3497 y Ft(2)2978 3531 y Fw(\)\()p Fr(j)p Fu(z)3104 3543 y Ft(4)3141 3531 y Fr(j)3164 3497 y Ft(2)3220 3531 y Fr(\000)g(j)p Fu(z)3365 3543 y Ft(3)3402 3531 y Fr(j)3425 3497 y Ft(2)3462 3531 y Fw(\))3494 3464 y Fo(\001)3569 3531 y Fu(;)581 3767 y(H)30 b Fw(=)777 3711 y(1)p 777 3748 42 4 v 777 3824 a(2)843 3625 y Fo( )908 3767 y Fu(p)950 3732 y Ft(2)950 3787 y Fs(\025)1012 3767 y Fw(+)1191 3711 y Fu(p)1233 3680 y Ft(2)1233 3734 y Fs(\022)p 1105 3748 253 4 v 1105 3830 a Fw(sin)1207 3795 y Ft(2)1244 3830 y Fw(\()p Fu(\025)p Fw(\))1386 3767 y(+)1691 3699 y Fu(p)1733 3669 y Ft(2)1733 3723 y Fs(\036)p 1479 3748 512 4 v 1479 3830 a Fw(sin)1581 3795 y Ft(2)1618 3830 y Fw(\()p Fu(\025)p Fw(\))14 b(sin)1847 3795 y Ft(2)1884 3830 y Fw(\()p Fu(\022)r Fw(\))2000 3625 y Fo(!)2088 3767 y Fw(=)2186 3711 y(1)p 2186 3748 42 4 v 2186 3824 a(2)2251 3650 y Fo(\022)2322 3711 y Fu(z)2361 3723 y Ft(1)p 2398 3665 43 4 v 2398 3711 a Fu(z)2440 3723 y Ft(1)2496 3711 y Fw(+)k Fu(z)2618 3723 y Ft(2)p 2655 3665 V 2655 3711 a Fu(z)2697 3723 y Ft(2)2753 3711 y Fw(+)g Fu(z)2875 3723 y Ft(3)p 2912 3665 V 2912 3711 a Fu(z)2954 3723 y Ft(3)3010 3711 y Fw(+)g Fu(z)3132 3723 y Ft(4)p 3169 3665 V 3169 3711 a Fu(z)3211 3723 y Ft(4)p 2322 3748 926 4 v 2765 3824 a Fw(2)3258 3650 y Fo(\023)3319 3667 y Ft(2)3393 3767 y Fu(:)202 b Fw(\(3.43\))p eop %%Page: 23 23 23 22 bop 3747 324 a Fw(23)523 614 y Fv(IV.)32 b(QUANTIZA)-8 b(TIONS)33 b(OF)f(THE)f(MAPS)h Fu(\032)2296 626 y Ft(2)2333 614 y Fu(;)14 b(\033)2417 626 y Ft(2)2487 614 y Fv(and)32 b Fu(\032)2714 626 y Ft(3)2751 614 y Fu(;)14 b(\033)2835 626 y Ft(3)2873 614 y Fv(.)648 714 y Fw(Once)19 b(w)n(e)g(ha)n(v)n(e)f (the)i(comm)n(uting)f(triangles)f(of)i(canonical)e(transformations)g (in)h(theorems)g(\(3.2\))g(and)523 814 y(\(3.3\),)24 b(and)e(the)i(analogy)d(b)r(et)n(w)n(een)h Fv(B)1721 829 y Fs(S)1765 812 y Fm(2)1802 814 y Fw(,)i Fv(B)1917 829 y Fs(S)1961 812 y Fm(3)2020 814 y Fw(and)f(their)g(corresp)r (onding)d(canonical)i(transformations)523 913 y Fr(C)567 928 y Fs(S)611 911 y Fm(2)647 913 y Fw(,)i Fr(C)738 928 y Fs(S)782 911 y Fm(3)818 913 y Fw(,)g(a)f(natural)f(question)g(to)h (ask)f(is)g(ab)r(out)h(the)g(description)f(of)h(unitary)f (transformations)f(whic)n(h)523 1013 y(pla)n(y)38 b(the)i(role)e(of)h (quan)n(tizations)e(of)i(the)h(maps)e Fu(\032)2203 1025 y Ft(2)2240 1013 y Fu(;)14 b(\033)2324 1025 y Ft(2)2401 1013 y Fw(and)39 b Fu(\032)2617 1025 y Ft(3)2654 1013 y Fu(;)14 b(\033)2738 1025 y Ft(3)2814 1013 y Fw(\(for)39 b(the)g(case)g(of)f Fu(S)3488 983 y Ft(2)3564 1013 y Fw(and)h Fu(S)3793 983 y Ft(3)523 1112 y Fw(resp)r(ectiv)n(ely\).)d (Suc)n(h)28 b(a)f(description)g(is)h(the)g(goal)e(of)h(this)h(section.) 523 1283 y Fv(IV.1)k(Quan)m(tizations)f(of)h Fu(\032)1497 1295 y Ft(2)1566 1283 y Fv(and)h Fu(\033)1798 1295 y Ft(2)648 1430 y Fw(Let)27 b Fu(d\026)889 1442 y Ft(2)927 1430 y Fw(\()p Fu(\013)p Fw(\))h(b)r(e)g(the)g(measure)f(on)g Fv(Q)1839 1442 y Ft(3)1895 1430 y Fr(\000)18 b(f)p Fw(0)p Fr(g)26 b Fw(obtained)h(from)h(the)g(Gaussian)e(measure)1479 1691 y Fu(d\027)1563 1703 y Ft(2)1600 1691 y Fw(\()p Fv(z)p Fw(\))e(=)1851 1635 y(1)p 1828 1672 88 4 v 1828 1748 a Fu(\031)1878 1724 y Ft(2)1939 1691 y Fw(exp\()p Fr(\000j)p Fv(z)p Fr(j)2251 1657 y Ft(2)2289 1691 y Fw(\))2376 1588 y Ft(2)2339 1613 y Fo(Y)2335 1789 y Fs(j)s Ft(=1)2464 1691 y Fu(dx)2554 1703 y Fs(j)2589 1691 y Fu(dy)2673 1703 y Fs(j)3659 1691 y Fw(\(4.1\))523 1968 y(through)i(the)g(map)h Fu(\032)1204 1980 y Ft(2)1267 1968 y Fw(\(here)f(w)n(e)g(are)g (actually)f(regarding)f(the)j(map)f Fu(\032)2787 1980 y Ft(2)2851 1968 y Fw(as)g(a)g(function)g(from)g Fv(C)3607 1938 y Ft(2)3671 1968 y Fw(on)n(to)523 2068 y Fv(Q)595 2080 y Ft(3)632 2068 y Fw(\).)648 2167 y(Giv)n(en)36 b Fu(`)h Fr(\025)h Fw(0)e(an)g(in)n(teger)f(n)n(um)n(b)r(er,)k(let)d Fr(X)2081 2179 y Fs(`)2150 2167 y Fw(b)r(e)h(the)g(complex)f(v)n(ector) f(space)h(of)g(homogeneous)523 2267 y(p)r(olynomials)25 b(of)g(order)f Fu(`)h Fw(in)h(three)f(complex)g(v)-5 b(ariables)24 b(\()p Fu(\013)2405 2279 y Ft(1)2442 2267 y Fu(;)14 b(\013)2532 2279 y Ft(2)2570 2267 y Fu(;)g(\013)2660 2279 y Ft(3)2697 2267 y Fw(\))26 b(with)g(the)f(restriction)g Fu(\013)3531 2237 y Ft(2)3531 2287 y(1)3582 2267 y Fw(+)14 b Fu(\013)3714 2237 y Ft(2)3714 2287 y(2)3766 2267 y Fw(+)523 2366 y Fu(\013)576 2336 y Ft(2)576 2387 y(3)644 2366 y Fw(=)31 b(0.)50 b(The)32 b(space)f Fr(X)1315 2378 y Fs(`)1380 2366 y Fw(has)g(dimension)h(2)p Fu(`)21 b Fw(+)g(1)32 b(and)g(a)g(particular)f(basis)g(for)h(it)g(can)g(b)r(e)g (obtained)523 2497 y(b)n(y)j(considering)f(the)i(monomials)f Fu(y)1704 2509 y Fs(`m)1830 2497 y Fw(=)2027 2451 y Fs(z)2061 2424 y Fp(`)p Fm(+)p Fp(m)2059 2471 y Fm(1)2183 2451 y Fs(z)2217 2424 y Fp(`)p Fk(\000)p Fp(m)2215 2471 y Fm(2)p 1941 2478 487 4 v 1941 2488 a Fr(p)p 2010 2488 418 4 v 56 x Ft(\()p Fs(`)p Ft(+)p Fs(m)p Ft(\)!\()p Fs(`)p Fn(\000)p Fs(m)p Ft(\)!)2473 2497 y Fw(and)h(the)g(map)f Fu(\032)3029 2461 y Fn(\000)p Ft(1)3029 2519 y(2)3118 2497 y Fw(.)61 b(That)35 b(is,)j(w)n(e)d(shall)523 2633 y(consider)27 b(the)h(follo)n(wing)e(functions)i(as)f(a)g(basis)g(for)g Fr(X)2259 2645 y Fs(`)2291 2633 y Fw(:)954 2816 y(\007)1019 2828 y Fs(`m)1109 2816 y Fw(\()p Fu(\013)p Fw(\))d(=)f Fu(y)1379 2828 y Fs(`m)1469 2816 y Fw(\()p Fv(z)p Fw(\()p Fu(\013)p Fw(\)\))72 b Fu(m)23 b Fw(=)f Fr(\000)p Fu(`;)14 b Fr(\000)p Fu(`)j Fw(+)h(1)c Fu(:)g(:)g(:)f(`;)106 b(\013)23 b Fw(=)g(\()p Fu(\013)2863 2828 y Ft(1)2900 2816 y Fu(;)14 b(\013)2990 2828 y Ft(2)3028 2816 y Fu(;)g(\013)3118 2828 y Ft(3)3155 2816 y Fw(\))23 b Fu(;)426 b Fw(\(4.2\))523 2999 y(with)28 b Fv(z)p Fw(\()p Fu(\013)p Fw(\))h(determined)f(b)n(y)f (the)h(equations:)1280 3181 y Fu(z)1323 3147 y Ft(2)1319 3202 y(1)1382 3181 y Fw(=)23 b Fr(\000)p Fu(\013)1588 3193 y Ft(1)1643 3181 y Fr(\000)18 b Fu({\013)1806 3193 y Ft(2)1843 3181 y Fu(;)60 b(z)1969 3147 y Ft(2)1965 3202 y(2)2029 3181 y Fw(=)23 b Fu(\013)2170 3193 y Ft(1)2225 3181 y Fr(\000)18 b Fu({\013)2388 3193 y Ft(2)2426 3181 y Fu(;)60 b(z)2548 3193 y Ft(1)2584 3181 y Fu(z)2623 3193 y Ft(2)2683 3181 y Fw(=)23 b Fu(\013)2824 3193 y Ft(3)2884 3181 y Fu(:)752 b Fw(\(4.3\))523 3364 y(Note)35 b(that)f(the)h(functions)g Fu(y)1473 3376 y Fs(`m)1598 3364 y Fw(dep)r(end)h(on)e Fv(z)h Fw(=)f(\()p Fu(z)2263 3376 y Ft(1)2300 3364 y Fu(;)14 b(z)2376 3376 y Ft(2)2413 3364 y Fw(\))35 b(through)f(in)n(teger)f(p)r(o)n(w)n(ers)g(of)i Fu(z)3509 3334 y Ft(2)3505 3385 y(1)3545 3364 y Fu(;)14 b(z)3625 3334 y Ft(2)3621 3385 y(2)3696 3364 y Fw(and)523 3464 y Fu(z)562 3476 y Ft(1)599 3464 y Fu(z)638 3476 y Ft(2)675 3464 y Fw(,)27 b(so)g(equations\(4.3\))g(are)f(enough)h(in)h (order)e(to)i(express)e Fu(y)2540 3476 y Fs(`m)2659 3464 y Fw(in)h(terms)h(of)f Fu(\013)3134 3476 y Ft(1)3172 3464 y Fu(;)14 b(\013)3262 3476 y Ft(2)3327 3464 y Fw(and)27 b Fu(\013)3541 3476 y Ft(3)3578 3464 y Fw(.)648 3563 y(Explicitly)-7 b(,)27 b(the)h(functions)g(\007)1609 3575 y Fs(`m)1727 3563 y Fw(are)f(giv)n(en)f(b)n(y)1206 3814 y(\007)1271 3826 y Fs(`m)1361 3814 y Fw(\()p Fu(\013)1446 3826 y Ft(1)1484 3814 y Fu(;)14 b(\013)1574 3826 y Ft(2)1611 3814 y Fu(;)g(\013)1701 3826 y Ft(3)1738 3814 y Fw(\))24 b(=)1891 3757 y(\()p Fr(\000)p Fw(sgn)o(\()p Fu(m)p Fw(\))p Fu(\013)2298 3769 y Ft(1)2355 3757 y Fr(\000)18 b Fu({\013)2518 3769 y Ft(2)2555 3757 y Fw(\))2587 3727 y Fn(j)p Fs(m)p Fn(j)2690 3757 y Fu(\013)2743 3714 y Fs(`)p Fn(\000j)p Fs(m)p Fn(j)2743 3780 y Ft(3)p 1891 3794 1035 4 v 2070 3811 a Fo(p)p 2153 3811 594 4 v 71 x Fw(\()p Fu(`)h Fw(+)f Fu(m)p Fw(\)!\()p Fu(`)g Fr(\000)g Fu(m)p Fw(\)!)2958 3814 y Fu(;)678 b Fw(\(4.4\))523 4061 y(with)28 b Fu(sg)s(n)p Fw(\()p Fu(m)p Fw(\))23 b(=)g(1)k(if)h Fu(m)23 b Fr(\025)g Fw(0)k(and)g Fu(sg)s(n)p Fw(\()p Fu(m)p Fw(\))c(=)g Fr(\000)p Fw(1)j(if)i Fu(m)23 b(<)g Fw(0.)648 4161 y(Moreo)n(v)n(er,)34 b(for)h(eac)n(h)f Fu(`)h Fr(\025)g Fw(0,)i(the)f(set)f(of)g(functions)h Fr(f)p Fw(\007)2509 4173 y Fs(`m)2598 4161 y Fr(j)p Fu(m)g Fw(=)f Fr(\000)p Fu(`;)14 b Fr(\000)p Fu(`)22 b Fw(+)h(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(`)p Fr(g)34 b Fw(is)i(an)e(or-)523 4260 y(thonormal)27 b(basis)f(of)i Fr(X)1282 4272 y Fs(`)1342 4260 y Fw(with)g(inner)f(pro)r(duct:)1476 4564 y Fr(h)p Fu(F)r(;)14 b(G)p Fr(i)25 b Fw(=)1809 4451 y Fo(Z)1856 4640 y Fs(\013)p Fn(2)p Fq(Q)2000 4648 y Fm(3)2050 4564 y Fu(F)12 b Fw(\()p Fu(\013)p Fw(\))p 2232 4492 184 4 v Fu(G)p Fw(\()p Fu(\013)p Fw(\))r Fu(d\026)2509 4576 y Ft(2)2547 4564 y Fw(\()p Fu(\013)p Fw(\))24 b Fu(:)948 b Fw(\(4.5\))648 4800 y(Let)22 b Fr(E)835 4812 y Ft(3)895 4800 y Fw(b)r(e)h(the)g(Hilb)r(ert)h(space)e(generated)f(b)n(y)h(all)h (the)g(spaces)f Fr(X)2681 4812 y Fs(`)2735 4800 y Fw(with)i(the)f (inner)f(pro)r(duct)h(de\014ned)523 4900 y(as)k(in)h(equation)f (\(4.5\).)648 4999 y(Th)n(us)c(the)i(quan)n(tization)e(of)h(the)h(map)f Fu(\032)1925 5011 y Ft(2)1986 4999 y Fw(is)g(the)h(unitary)e (transformation)g Fv(U)3131 5011 y Fs(\032)3165 5019 y Fm(2)3225 4999 y Fw(:)h Fr(B)3327 5014 y Fs(S)3371 4998 y Fm(2)3430 4999 y Fr(7!)f(E)3580 5011 y Ft(3)3641 4999 y Fw(giv)n(en)523 5099 y(b)n(y)1503 5282 y Fv(U)1576 5294 y Fs(\032)1610 5302 y Fm(2)1648 5282 y Fw(\011\()p Fu(\013)p Fw(\))g(=)g(\011\()p Fv(z)p Fw(\()p Fu(\013)p Fw(\)\))p Fu(;)85 b Fw(\011)22 b Fr(2)i(B)2558 5297 y Fs(S)2602 5280 y Fm(2)2661 5282 y Fu(:)975 b Fw(\(4.6\))523 5464 y(where)30 b Fv(z)p Fw(\()p Fu(\013)p Fw(\))j(is)d(determined)h(b) n(y)g(equations)f(4.3.)46 b(Notice)31 b(that)g(all)f(the)i(functions)f (in)g(the)g(space)f Fr(B)3750 5479 y Fs(S)3794 5462 y Fm(2)523 5564 y Fw(are)25 b(in\014nite)i(linear)e(com)n(binations)g(of) g(the)i(functions)f Fu(y)2304 5576 y Fs(`m)2395 5564 y Fw(,)g(so)f(the)h(righ)n(t)g(hand)f(side)h(of)g(equation\(4.6\))523 5664 y(mak)n(es)h(sense.)p eop %%Page: 24 24 24 23 bop 3747 324 a Fw(24)648 614 y(The)24 b(quan)n(tization)f(of)g Fu(\033)1428 626 y Ft(2)1490 614 y Fw(can)h(b)r(e)g(obtained)g(in)g (the)g(follo)n(wing)f(w)n(a)n(y)-7 b(.)35 b(Let)24 b(us)f(consider)g (the)i(unitary)523 714 y(transformation)h Fv(U)1162 726 y Fs(\033)1200 734 y Fm(2)1261 714 y Fw(:)d Fr(E)1351 726 y Ft(3)1411 714 y Fr(7!)g Fu(L)1574 684 y Ft(2)1611 714 y Fw(\()p Fu(S)1699 684 y Ft(2)1736 714 y Fw(\))28 b(giv)n(en)f(b)n(y)712 979 y Fv(U)785 991 y Fs(\033)823 999 y Fm(2)861 979 y Fw(\011\()p Fv(x)p Fw(\))c(=)66 b(lim)1151 1032 y Fs(M)6 b Fn(7!1)1367 866 y Fo(Z)1413 1054 y Fn(j)p Fs(\013)p Fn(j\024)p Fs(M)1635 837 y Fo( )1727 875 y Fn(1)1700 900 y Fo(X)1705 1078 y Fs(`)p Ft(=0)1915 875 y Fs(`)1869 900 y Fo(X)1834 1078 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 2038 912 156 4 v 2038 979 a Fw(\007)2103 991 y Fs(`m)2193 979 y Fw(\()p Fu(\013)p Fw(\))p Fu(Y)2358 991 y Fs(`m)2450 979 y Fw(\()p Fv(x)p Fw(\))2564 837 y Fo(!)2645 979 y Fw(\011\()p Fu(\013)p Fw(\))p Fu(d\026)2920 991 y Ft(2)2958 979 y Fw(\()p Fu(\013)p Fw(\))p Fu(;)84 b Fw(\011)22 b Fr(2)i(E)3392 991 y Ft(3)3452 979 y Fu(:)184 b Fw(\(4.7\))648 1243 y(By)27 b(using)g(the)i(analog)d(argumen)n(ts)g (to)i(those)f(giv)n(en)g(in)h(App)r(endix)h(A,)f(the)g(k)n(ernel)f(of)h Fv(U)3489 1255 y Fs(\033)3527 1263 y Fm(2)3592 1243 y Fw(can)g(b)r(e)523 1342 y(written)g(as)f(a)g(p)r(o)n(w)n(er)f(series)h (of)g(the)h(function:)949 1525 y(\010)1009 1537 y Fs(\033)1047 1545 y Fm(2)1084 1525 y Fw(\()p Fu(\013;)14 b Fv(x)p Fw(\))84 b(=)p 1520 1453 273 4 v 83 w(\007)1585 1537 y Ft(1)p Fs(;)p Ft(1)1675 1525 y Fw(\()p Fu(\013)p Fw(\))q Fu(Y)1841 1537 y Ft(1)p Fs(;)p Ft(1)1931 1525 y Fw(\()p Fv(x)p Fw(\))19 b(+)p 2147 1453 V 18 w(\007)2212 1537 y Ft(1)p Fs(;)p Ft(0)2302 1525 y Fw(\()p Fu(\013)p Fw(\))q Fu(Y)2468 1537 y Ft(1)p Fs(;)p Ft(0)2558 1525 y Fw(\()p Fv(x)p Fw(\))g(+)p 2775 1453 325 4 v 19 w(\007)2840 1537 y Ft(1)p Fs(;)p Fn(\000)p Ft(1)2981 1525 y Fw(\()p Fu(\013)p Fw(\))q Fu(Y)3147 1537 y Ft(1)p Fs(;)p Fn(\000)p Ft(1)3289 1525 y Fw(\()p Fv(x)p Fw(\))1372 1661 y(=)1520 1588 y Fr(p)p 1589 1588 42 4 v 73 x Fw(3)p Fv(x)f Fr(\017)g Fu(\013;)1824 b Fw(\(4.8\))648 1844 y(Since)1457 1901 y Fs(`)1411 1926 y Fo(X)1376 2104 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 1580 1938 156 4 v 1580 2005 a Fw(\007)1645 2017 y Fs(`m)1735 2005 y Fw(\()p Fu(\013)p Fw(\))p Fu(Y)1900 2017 y Fs(`m)1992 2005 y Fw(\()p Fv(x)p Fw(\))24 b(=)2228 1881 y Fr(p)p 2297 1881 220 4 v 67 x Fw(2)p Fu(`)18 b Fw(+)g(1)p 2228 1985 289 4 v 2240 2071 a(\()2272 2002 y Fr(p)p 2341 2002 42 4 v 69 x Fw(3\))2415 2047 y Fs(`)2447 2071 y Fu(`)p Fw(!)2540 2005 y(\(\010)2632 2017 y Fs(\033)2670 2025 y Fm(2)2707 2005 y Fw(\()p Fu(\013;)c Fv(x)p Fw(\)\))2945 1963 y Fs(`)3659 2005 y Fw(\(4.9\))648 2235 y(then)28 b(w)n(e)f(ha)n(v)n(e:)807 2572 y Fv(U)880 2584 y Fs(\033)918 2592 y Fm(2)956 2572 y Fw(\011\()p Fv(x)p Fw(\))c(=)66 b(lim)1246 2625 y Fs(M)6 b Fn(7!1)1462 2430 y Fo( )1527 2459 y(Z)1573 2647 y Fn(j)p Fs(\013)p Fn(j\024)p Fs(M)1822 2468 y Fn(1)1795 2493 y Fo(X)1800 2672 y Fs(`)p Ft(=0)1939 2448 y Fr(p)p 2008 2448 220 4 v 67 x Fw(2)p Fu(`)18 b Fw(+)g(1)p 1939 2553 289 4 v 2055 2629 a Fu(`)p Fw(!)2237 2572 y(\()p Fv(x)i Fr(\017)e Fu(\013)p Fw(\))2484 2537 y Fs(`)2516 2430 y Fo(!)2596 2572 y Fw(\011\()p Fu(\013)p Fw(\))p Fu(d\026)2871 2584 y Ft(2)2909 2572 y Fw(\()p Fu(\013)p Fw(\))p Fu(;)84 b Fw(\011)22 b Fr(2)i(E)3343 2584 y Ft(3)3618 2572 y Fw(\(4.10\))648 2829 y(Let)j(us)h(summarize)f (our)f(conclusions)h(of)h(this)f(section)h(in)f(the)h(follo)n(wing)523 3011 y Fv(Theorem)i(4.1)41 b Fl(The)31 b(fol)t(lowing)h(triangle)e(of)h (unitary)f(tr)l(ansformations)g(c)l(ommutes:)1886 3784 y Fb(\000)1969 3701 y(\000)2052 3617 y(\000)2135 3534 y(\000)2218 3451 y(\000)2301 3368 y(\000)-83 b(\022)p 1760 3784 4 499 v 1761 3784 a(?)p 1969 3204 416 4 v 1969 3202 a(\033)1721 3889 y Fr(E)1765 3901 y Ft(3)1694 3223 y Fr(B)1749 3238 y Fs(S)1793 3222 y Fm(2)2508 3226 y Fu(L)2565 3196 y Ft(2)2601 3226 y Fw(\()p Fu(S)2689 3196 y Ft(2)2727 3226 y Fw(\))2102 3099 y Fv(B)2170 3114 y Fs(S)2214 3097 y Fm(2)1606 3593 y Fv(U)1679 3605 y Fs(\032)1713 3613 y Fm(2)2144 3594 y Fv(U)2217 3606 y Fs(\033)2255 3614 y Fm(2)648 4132 y Fw(This)25 b(triangle)f(is)i(the)g(quan)n(tum)f (v)n(ersion)f(of)h(the)h(comm)n(uting)f(triangle)g(of)g(canonical)f (transforma-)523 4232 y(tions)j(in)h(theorem)f(\(3.2\).)523 4402 y Fv(IV.2)32 b(Quan)m(tizations)f(of)h Fu(\032)1497 4414 y Ft(3)1566 4402 y Fv(and)h Fu(\033)1798 4414 y Ft(3)648 4549 y Fw(As)21 b(in)g(the)h(previous)e(section,)i(let)g Fu(d\026)1817 4561 y Ft(4)1854 4549 y Fw(\()p Fu(\013)p Fw(\))g(b)r(e)g(the)g(measure)e(on)h Fv(Q)2735 4561 y Ft(4)2777 4549 y Fr(\000)6 b(f)p Fw(0)p Fr(g)19 b Fw(obtained)i(from)g (the)h(mea-)523 4657 y(sure)27 b Fu(d\027)783 4669 y Ft(4)820 4657 y Fw(\()p Fv(z)p Fw(\))d(=)1068 4625 y Ft(1)p 1048 4639 74 4 v 1048 4686 a Fs(\031)1089 4669 y Fm(4)1145 4657 y Fw(exp\()p Fr(\000j)p Fv(z)p Fr(j)1457 4627 y Ft(2)1494 4657 y Fw(\))1540 4595 y Fo(Q)1619 4616 y Ft(4)1619 4682 y Fs(j)s Ft(=1)1752 4657 y Fu(dx)1842 4669 y Fs(j)1878 4657 y Fu(dy)1962 4669 y Fs(j)2024 4657 y Fw(through)i(the)i(map)f Fu(\032)2707 4669 y Ft(3)2772 4657 y Fw(\(again,)f(here)h(w)n(e)g(are)f(regarding)523 4770 y(the)i(map)g Fu(\032)894 4782 y Ft(3)958 4770 y Fw(as)f(a)g(function)i(from)e Fv(C)1720 4739 y Ft(4)1785 4770 y Fw(on)n(to)g Fv(Q)2044 4782 y Ft(4)2081 4770 y Fw(\).)648 4869 y(No)n(w,)e(giv)n(en)g Fu(n)e Fr(\025)g Fw(1)i(a)h(natural)f(n)n(um)n(b)r(er,)h(let)g Fr(Z)2160 4881 y Fs(n)2232 4869 y Fw(b)r(e)g(the)g(space)f(of)h(homogeneous)e(p)r (olynomials)h(of)523 4969 y(degree)20 b Fu(n)t Fr(\000)t Fw(1)g(in)h(four)f(complex)g(v)-5 b(ariables)19 b Fu(\013)1924 4981 y Ft(1)1961 4969 y Fu(;)14 b(\013)2051 4981 y Ft(2)2089 4969 y Fu(;)g(\013)2179 4981 y Ft(3)2216 4969 y Fu(;)g(\013)2306 4981 y Ft(4)2364 4969 y Fw(with)21 b(the)g(restriction)e Fu(\013)3125 4939 y Ft(2)3125 4989 y(1)3167 4969 y Fw(+)t Fu(\013)3289 4939 y Ft(2)3289 4989 y(2)3331 4969 y Fw(+)t Fu(\013)3453 4939 y Ft(2)3453 4989 y(3)3495 4969 y Fw(+)t Fu(\013)3617 4939 y Ft(2)3617 4989 y(4)3678 4969 y Fw(=)k(0.)523 5068 y(W)-7 b(e)28 b(shall)f(consider)g(the)h(follo)n(wing)e(inner)i (pro)r(duct)f(in)h Fr(Z)2358 5080 y Fs(n)1476 5289 y Fr(h)p Fu(F)r(;)14 b(G)p Fr(i)25 b Fw(=)1809 5176 y Fo(Z)1856 5365 y Fs(\013)p Fn(2)p Fq(Q)2000 5373 y Fm(4)2050 5289 y Fu(F)12 b Fw(\()p Fu(\013)p Fw(\))p 2232 5217 184 4 v Fu(G)p Fw(\()p Fu(\013)p Fw(\))r Fu(d\026)2509 5301 y Ft(4)2547 5289 y Fw(\()p Fu(\013)p Fw(\))24 b Fu(:)907 b Fw(\(4.11\))648 5535 y(The)27 b(space)g Fr(Z)1100 5547 y Fs(n)1173 5535 y Fw(has)g(dimension)h Fu(n)1764 5505 y Ft(2)1828 5535 y Fw(and)g(the)g(follo)n(wing)e(particular)h (orthonormal)e(basis)1741 5717 y(\007)1806 5729 y Fs(n`m)1961 5717 y Fw(=)d Fu(y)2089 5729 y Fs(n`m)2221 5717 y Fw(\()p Fv(z)p Fw(\()p Fu(\013)p Fw(\)\))1174 b(\(4.12\))p eop %%Page: 25 25 25 24 bop 3747 324 a Fw(25)523 614 y(where,)31 b(since)g(the)h (functions)f Fu(y)1546 626 y Fs(n`m)1708 614 y Fw(dep)r(end)h(on)f Fv(z)g Fw(through)f(the)i(v)-5 b(ariables)30 b Fu(z)3043 626 y Ft(1)3079 614 y Fu(z)3118 626 y Ft(3)3155 614 y Fu(;)43 b(z)3260 626 y Ft(1)3297 614 y Fu(z)3336 626 y Ft(4)3372 614 y Fu(;)g(z)3477 626 y Ft(2)3514 614 y Fu(z)3553 626 y Ft(3)3590 614 y Fu(;)f(z)3694 626 y Ft(2)3731 614 y Fu(z)3770 626 y Ft(4)3807 614 y Fw(,)523 714 y(w)n(e)27 b(mean)h(b)n(y)f Fv(z)p Fw(\()p Fu(\013)p Fw(\))i(the)f(expressions:) 743 931 y Fu(z)782 943 y Ft(1)819 931 y Fu(z)858 943 y Ft(3)917 931 y Fw(=)1015 874 y Fr(\000)p Fu({\013)1160 886 y Ft(1)1215 874 y Fw(+)18 b Fu(\013)1351 886 y Ft(2)p 1015 912 374 4 v 1181 988 a Fw(2)1399 931 y Fu(;)60 b(z)1521 943 y Ft(1)1557 931 y Fu(z)1596 943 y Ft(4)1656 931 y Fw(=)1754 874 y Fu({\013)1834 886 y Ft(3)1889 874 y Fw(+)18 b Fu(\013)2025 886 y Ft(4)p 1754 912 309 4 v 1888 988 a Fw(2)2073 931 y Fu(;)60 b(z)2195 943 y Ft(2)2232 931 y Fu(z)2271 943 y Ft(3)2331 931 y Fw(=)2428 874 y Fu({\013)2508 886 y Ft(3)2564 874 y Fr(\000)18 b Fu(\013)2700 886 y Ft(4)p 2428 912 V 2562 988 a Fw(2)2747 931 y Fu(;)60 b(z)2869 943 y Ft(2)2906 931 y Fu(z)2945 943 y Ft(4)3005 931 y Fw(=)3102 874 y Fu({\013)3182 886 y Ft(1)3238 874 y Fw(+)18 b Fu(\013)3374 886 y Ft(2)p 3102 912 V 3236 988 a Fw(2)3421 931 y Fu(:)174 b Fw(\(4.13\))648 1137 y(Let)30 b Fr(E)843 1149 y Ft(4)911 1137 y Fw(b)r(e)h(the)g(Hilb)r(ert) g(space)f(generated)f(b)n(y)h(the)h(spaces)f Fr(Z)2650 1149 y Fs(n)2695 1137 y Fw(.)46 b(The)31 b(quan)n(tization)e(of)i Fu(\032)3561 1149 y Ft(3)3628 1137 y Fw(is)g(the)523 1236 y(unitary)e(op)r(erator)f Fv(U)1228 1248 y Fs(\032)1262 1256 y Fm(3)1328 1236 y Fw(giv)n(en)h(b)n(y)g(the)h(linear)e(extension) h(of)h(the)f(assignmen)n(t)g Fu(y)3122 1248 y Fs(n`m)3279 1236 y Fr(7!)d Fw(\007)3453 1248 y Fs(n`m)3585 1236 y Fw(.)42 b(That)523 1336 y(is,)991 1518 y Fv(U)1064 1530 y Fs(\032)1098 1538 y Fm(3)1239 1518 y Fw(:)104 b Fr(F)1426 1530 y Ft(4)1486 1518 y Fr(7!)23 b(E)1636 1530 y Ft(4)809 1728 y Fv(U)882 1740 y Fs(\032)916 1748 y Fm(3)953 1728 y Fw(\011\()p Fu(\013)p Fw(\))84 b(=)1366 1615 y Fo(Z)1412 1804 y Fq(z)p Fn(2)p Fq(C)1544 1787 y Fm(4)1595 1586 y Fo( )1690 1624 y Fn(1)1663 1649 y Fo(X)1661 1825 y Fs(n)p Ft(=1)1800 1624 y Fs(n)p Fn(\000)p Ft(1)1803 1649 y Fo(X)1807 1828 y Fs(`)p Ft(=)p Fs(o)2021 1624 y(`)1974 1649 y Fo(X)1940 1828 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 2143 1656 280 4 v 2143 1728 a Fu(y)2184 1740 y Fs(n`m)2316 1728 y Fw(\()p Fv(z)p Fw(\))q(\007)2488 1740 y Fs(n`m)2619 1728 y Fw(\()p Fu(\013)2704 1586 y Fo(!)2784 1728 y Fw(\011\()p Fv(z)p Fw(\))p Fu(d\027)5 b Fw(\()p Fv(z)p Fw(\))p Fu(;)86 b Fw(\011)22 b Fr(2)i(F)3485 1740 y Ft(4)3521 1728 y Fu(:)74 b Fw(\(4.14\))523 1992 y(Since,)28 b(for)f Fu(n)c Fr(\025)f Fw(1)28 b(a)f(natural)g(n)n(um)n(b)r(er,)1070 2148 y Fs(n)p Fn(\000)p Ft(1)1073 2173 y Fo(X)1077 2352 y Fs(`)p Ft(=)p Fs(o)1291 2148 y(`)1245 2173 y Fo(X)1210 2352 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 1413 2180 V 1413 2252 a Fu(y)1454 2264 y Fs(n`m)1586 2252 y Fw(\()p Fv(z)p Fw(\))q(\007)1758 2264 y Fs(n`m)1889 2252 y Fw(\()p Fu(\013)p Fw(\))d(=)2298 2196 y(1)p 2128 2233 383 4 v 2128 2309 a(\(\()p Fu(n)19 b Fr(\000)f Fw(1\)!\))2473 2285 y Ft(2)2534 2252 y Fw(\(\010)2626 2264 y Fs(\032)2660 2272 y Fm(3)2697 2252 y Fw(\()p Fv(z)p Fu(;)c(\013)p Fw(\)\))2927 2210 y Fs(n)p Fn(\000)p Ft(1)3094 2252 y Fu(;)501 b Fw(\(4.15\))523 2516 y(where)1192 2760 y(\010)1252 2772 y Fs(\032)1286 2780 y Fm(3)1323 2760 y Fw(\()p Fv(z)p Fu(;)14 b(\013)p Fw(\))24 b(=)1674 2656 y Ft(1)1631 2681 y Fo(X)1635 2860 y Fs(`)p Ft(=)p Fs(o)1845 2656 y(`)1799 2681 y Fo(X)1764 2860 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 1968 2687 272 4 v 1968 2760 a Fu(y)2009 2772 y Ft(2)p Fs(`m)2132 2760 y Fw(\()p Fv(z)p Fw(\))q(\007)2304 2772 y Ft(2)p Fs(`m)2428 2760 y Fw(\()p Fu(\013)p Fw(\))g(=)2667 2704 y Fu(\013)18 b Fr(\017)g Fu(\032)2841 2716 y Ft(3)2879 2704 y Fw(\()p Fv(z)p Fw(\))p 2667 2741 320 4 v 2805 2817 a(2)3618 2760 y(\(4.16\))523 3024 y(then)28 b(w)n(e)f(ha)n(v)n(e)743 3260 y Fv(U)816 3272 y Fs(\032)850 3280 y Fm(3)887 3260 y Fw(\011\()p Fu(\013)p Fw(\))d(=)1180 3147 y Fo(Z)1226 3336 y Fq(z)p Fn(2)p Fq(C)1358 3319 y Fm(4)1409 3118 y Fo( )1504 3157 y Fn(1)1478 3181 y Fo(X)1475 3357 y Fs(n)p Ft(=1)1794 3204 y Fw(1)p 1624 3241 383 4 v 1624 3317 a(\(\()p Fu(n)19 b Fr(\000)f Fw(1\)!\))1969 3293 y Ft(2)2026 3204 y Fw(\()p Fu(\013)h Fr(\017)f Fu(\032)2233 3216 y Ft(3)2270 3204 y Fw(\()p Fv(z)p Fw(\)\))2410 3162 y Fs(n)p Fn(\000)p Ft(1)p 2026 3241 514 4 v 2197 3317 a Fw(2)2239 3293 y Fs(n)p Fn(\000)p Ft(1)2550 3118 y Fo(!)2629 3260 y Fw(\011\()p Fv(z)p Fw(\))p Fu(d\027)2884 3272 y Ft(4)2922 3260 y Fw(\()p Fv(z)p Fw(\))p Fu(;)84 b Fw(\011)23 b Fr(2)g(F)3361 3272 y Ft(4)3421 3260 y Fu(:)174 b Fw(\(4.17\))648 3525 y(The)39 b(quan)n(tization)f(of)i Fu(\033)1475 3537 y Ft(3)1552 3525 y Fw(is)f(the)h(Unitary)f(map)g Fv(U)2392 3537 y Fs(\033)2430 3545 y Fm(3)2511 3525 y Fw(:)j Fr(E)2620 3537 y Ft(4)2700 3525 y Fr(7!)h Fu(L)2883 3494 y Ft(2)2920 3525 y Fw(\()p Fu(S)3008 3494 y Ft(3)3045 3525 y Fw(\))d(giv)n(en)e(b)n(y)h(the)h(linear)523 3624 y(extension)27 b(of)h(the)g(assignmen)n(t)e(\007)1620 3636 y Fs(n`m)1775 3624 y Fr(7!)d Fu(Y)1929 3636 y Fs(n`m)2061 3624 y Fw(:)798 3884 y Fv(U)871 3896 y Fs(\033)909 3904 y Fm(3)947 3884 y Fw(\011\()p Fv(x)p Fw(\))g(=)66 b(lim)1237 3938 y Fs(M)6 b Fn(7!1)1453 3771 y Fo(Z)1499 3960 y Fn(j)p Fs(\013)p Fn(j\024)p Fs(M)1721 3742 y Fo( )1816 3780 y Fn(1)1789 3805 y Fo(X)1787 3981 y Fs(n)p Ft(=1)1926 3780 y Fs(n)p Fn(\000)p Ft(1)1929 3805 y Fo(X)1933 3984 y Fs(`)p Ft(=)p Fs(o)2146 3780 y(`)2100 3805 y Fo(X)2065 3984 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 2269 3812 315 4 v 2269 3884 a Fw(\007)2334 3896 y Fs(n`m)2465 3884 y Fw(\()p Fu(\013)p Fw(\))q Fu(Y)2631 3896 y Fs(n`m)2763 3884 y Fw(\()p Fv(x)p Fw(\))2877 3742 y Fo(!)2958 3884 y Fw(\011\()p Fu(\013)p Fw(\))p Fu(d\026)3233 3896 y Ft(4)3271 3884 y Fw(\()p Fu(\013)p Fw(\))230 b(\(4.18\))523 4148 y(Since,)28 b(for)f Fu(n)c Fr(\025)f Fw(1)28 b(a)f(natural)g(n)n (um)n(b)r(er,)972 4306 y Fs(n)p Fn(\000)p Ft(1)975 4331 y Fo(X)979 4509 y Fs(`)p Ft(=)p Fs(o)1192 4306 y(`)1146 4331 y Fo(X)1111 4509 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 1315 4337 V 1315 4410 a Fw(\007)1380 4422 y Fs(n`m)1511 4410 y Fw(\()p Fu(\013)p Fw(\))q Fu(Y)1677 4422 y Fs(n`m)1809 4410 y Fw(\()p Fv(x)p Fw(\))d(=)2126 4294 y Fr(p)p 2195 4294 50 4 v 59 x Fu(n)p 2045 4390 281 4 v 2045 4466 a Fw(\()p Fu(n)19 b Fr(\000)f Fw(1\)!)2349 4292 y Fo(\022)2420 4353 y Fw(1)p 2420 4390 42 4 v 2420 4466 a(2)2472 4292 y Fo(\023)2543 4283 y Fp(n)p Fk(\000)p Fm(1)p 2543 4297 110 4 v 2584 4330 a(2)2681 4410 y Fw(\(\010)2773 4422 y Fs(\033)2811 4430 y Fm(3)2848 4410 y Fw(\()p Fu(\013;)c Fv(x)p Fw(\)\))3085 4368 y Fs(n)p Fn(\000)p Ft(1)3618 4410 y Fw(\(4.19\))523 4674 y(with)1169 4917 y(\010)1229 4929 y Fs(\033)1267 4937 y Fm(3)1304 4917 y Fw(\()p Fu(\013;)g Fv(x)p Fw(\))24 b(=)1663 4814 y Ft(1)1620 4838 y Fo(X)1624 5017 y Fs(`)p Ft(=)p Fs(o)1834 4814 y(`)1788 4838 y Fo(X)1753 5017 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 1957 4845 307 4 v 1957 4917 a Fw(\007)2022 4929 y Ft(2)p Fs(`m)2145 4917 y Fw(\()p Fu(\013)p Fw(\))q Fu(Y)2311 4929 y Ft(2)p Fs(`m)2435 4917 y Fw(\()p Fv(x)p Fw(\))g(=)2661 4845 y Fr(p)p 2730 4845 42 4 v 72 x Fw(2\()p Fv(x)19 b Fr(\017)f Fu(\013)p Fw(\))600 b(\(4.20\))523 5181 y(then)736 5418 y Fv(U)809 5430 y Fs(\033)847 5438 y Fm(3)885 5418 y Fw(\011\()p Fv(x)p Fw(\))23 b(=)66 b(lim)1175 5472 y Fs(M)6 b Fn(7!1)1391 5305 y Fo(Z)1437 5494 y Fn(j)p Fs(\013)p Fn(j\024)p Fs(M)1659 5276 y Fo( )1754 5314 y Fn(1)1727 5339 y Fo(X)1725 5515 y Fs(n)p Ft(=1)1954 5302 y Fr(p)p 2023 5302 50 4 v 60 x Fu(n)p 1874 5399 281 4 v 1874 5475 a Fw(\()p Fu(n)18 b Fr(\000)g Fw(1\)!)2164 5418 y(\()p Fv(x)h Fr(\017)f Fu(\013)p Fw(\))2410 5384 y Fs(n)p Fn(\000)p Ft(1)2541 5276 y Fo(!)2620 5418 y Fw(\011\()p Fu(\013)p Fw(\))p Fu(d\026)2895 5430 y Ft(4)2934 5418 y Fw(\()p Fu(\013)p Fw(\))p Fu(;)84 b Fw(\011)22 b Fr(2)i(E)3368 5430 y Ft(4)3428 5418 y Fu(:)167 b Fw(\(4.21\))p eop %%Page: 26 26 26 25 bop 3747 324 a Fw(26)523 614 y Fv(Theorem)30 b(4.2)41 b Fl(The)31 b(fol)t(lowing)h(triangle)e(of)h(unitary)f(tr)l (ansformations)g(c)l(ommutes:)1886 1386 y Fb(\000)1969 1303 y(\000)2052 1220 y(\000)2135 1137 y(\000)2218 1054 y(\000)2301 971 y(\000)-83 b(\022)p 1760 1386 4 499 v 1761 1386 a(?)p 1969 807 416 4 v 1969 805 a(\033)1721 1492 y Fr(E)1765 1504 y Ft(4)1713 827 y Fr(F)1773 839 y Ft(4)2508 829 y Fu(L)2565 799 y Ft(2)2601 829 y Fw(\()p Fu(S)2689 799 y Ft(3)2727 829 y Fw(\))2102 702 y Fv(B)2170 717 y Fs(S)2214 700 y Fm(3)1606 1196 y Fv(U)1679 1208 y Fs(\032)1713 1216 y Fm(3)2144 1197 y Fv(U)2217 1209 y Fs(\033)2255 1217 y Fm(3)648 1735 y Fw(This)25 b(triangle)f(is)i(the) g(quan)n(tum)f(v)n(ersion)f(of)h(the)h(comm)n(uting)f(triangle)g(of)g (canonical)f(transforma-)523 1835 y(tions)j(in)h(theorem)f(\(3.3\).)p eop %%Page: 27 27 27 26 bop 3747 324 a Fw(27)523 614 y Fv(V.)44 b(THE)g(BAR)m(GMANN)f (TRANSF)m(ORM)h(AS)g(A)h(COHERENT)d(ST)-8 b(A)g(TE)45 b(TRANS-)523 714 y(F)m(ORM)648 861 y Fw(It)37 b(is)g(kno)n(wn)f(that)h (the)h(Bargmann)d(transform)h Fv(B)2327 873 y Fq(R)2383 857 y Fp(n)2465 861 y Fw(can)g(b)r(e)i(iden)n(ti\014ed)f(as)g(a)f (coheren)n(t)g(state)523 960 y(transform.)54 b(That)33 b(is,)i(for)e(an)n(y)g Fu(F)45 b Fr(2)33 b Fu(L)1821 930 y Ft(2)1858 960 y Fw(\()p Fv(R)1962 930 y Fs(n)2007 960 y Fw(\),)i(the)f(function)g Fv(B)2645 972 y Fq(R)2701 956 y Fp(n)2746 960 y Fu(F)12 b Fw(\()p Fv(z)p Fw(\))34 b(can)g(b)r(e)g(expressed)e(as)h(the)523 1060 y(inner)26 b(pro)r(duct)f(of)h(F)g(with)g(a)f(canonical)g(coheren)n(t)g(state,)h (lab)r(eled)f(b)n(y)h Fv(z)d Fr(2)h Fv(C)2999 1030 y Fs(n)3044 1060 y Fw(,)i(of)g(the)g(n-dimensional)523 1160 y(harmonic)h(oscillator.)648 1259 y(In)21 b(this)h(section)f(w)n (e)g(sho)n(w)g(that)h(b)r(oth)f(Bargmann)f(transforms)g Fv(B)2726 1274 y Fs(S)2770 1258 y Fm(2)2828 1259 y Fw(and)i Fv(B)3052 1274 y Fs(S)3096 1258 y Fm(3)3154 1259 y Fw(can)f(also)f(b)r (e)i(written)523 1359 y(as)32 b(coheren)n(t)f(states)h(transforms.)49 b(Here)32 b(sets)g(of)h(coheren)n(t)e(states)h(in)h Fu(L)2877 1329 y Ft(2)2913 1359 y Fw(\()p Fu(S)3001 1329 y Ft(2)3038 1359 y Fw(\))g(\()p Fu(L)3192 1329 y Ft(2)3229 1359 y Fw(\()p Fu(S)3317 1329 y Ft(3)3354 1359 y Fw(\)\))g(app)r(ear)f(for)523 1459 y(eac)n(h)22 b(irreducible)g(represen)n(tation)f(of)h Fu(S)5 b(O)r Fw(\(3\))23 b(\()p Fu(S)5 b(O)r Fw(\(4\)\))24 b(considered)e(when)h(w)n(e)f(de\014ned)h(the)g(Bargmann)523 1558 y(transform)j Fv(B)972 1573 y Fs(S)1016 1556 y Fm(2)1081 1558 y Fw(\()p Fv(B)1181 1573 y Fs(S)1225 1556 y Fm(3)1261 1558 y Fw(\).)523 1729 y Fv(V.1)31 b(B)769 1741 y Fq(R)825 1725 y Fp(n)902 1729 y Fv(as)h(a)g(coheren)m(t)g(state)g(transform)648 1876 y Fw(Let)g(us)f(note)h(that)g(the)h(k)n(ernel)e(of)g Fv(B)1847 1888 y Fq(R)1903 1871 y Fp(n)1980 1876 y Fw(can)g(b)r(e)i (expressed)d(in)j(the)f(follo)n(wing)f(w)n(a)n(y:)44 b(let)32 b(us)g(tak)n(e)523 1975 y Fv(z)24 b Fw(=)e(\()p Fv(q)d Fr(\000)f Fu({)p Fv(p)p Fw(\))p Fu(=)1014 1907 y Fr(p)p 1083 1907 42 4 v 68 x Fw(2)27 b(with)h Fv(x)p Fu(;)14 b Fv(p)24 b Fr(2)f Fv(R)1655 1945 y Fs(n)1700 1975 y Fw(,)28 b(then)532 2198 y Fu(A)p Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)f(exp)1040 2081 y Fo(\022)1101 2198 y Fr(\000)1176 2142 y Fw(1)p 1176 2179 V 1176 2255 a(2)1227 2198 y(\()p Fv(z)1301 2164 y Ft(2)1357 2198 y Fw(+)18 b Fv(x)1490 2164 y Ft(2)1528 2198 y Fw(\))h(+)1662 2126 y Fr(p)p 1731 2126 V 72 x Fw(2)o Fv(z)g Fr(\017)f Fv(x)1943 2081 y Fo(\023)2042 2198 y Fw(=)k(exp)2270 2081 y Fo(\022)2331 2198 y Fr(\000)2406 2142 y Fw(\()p Fv(x)d Fr(\000)f Fv(q)p Fw(\))2672 2112 y Ft(2)p 2405 2179 305 4 v 2537 2255 a Fw(2)2738 2198 y(+)2831 2142 y Fu({)p Fv(p)g Fr(\017)g Fv(q)p 2831 2179 209 4 v 2914 2255 a Fw(2)3068 2198 y Fr(\000)g Fu({)p Fv(p)g Fr(\017)g Fv(x)h Fw(+)3471 2142 y Fv(q)3521 2112 y Ft(2)3577 2142 y Fw(+)f Fv(p)3713 2112 y Ft(2)p 3471 2179 280 4 v 3590 2255 a Fw(4)3760 2081 y Fo(\023)3659 2348 y Fw(\(5.1\))648 2447 y(No)n(w)27 b(let)h(us)f(denote)h(b)n(y)966 2652 y(\010)1026 2618 y Ft(\()p Fq(z)p Ft(\))1116 2652 y Fw(\()p Fv(x)p Fw(\))84 b(=)1532 2596 y(1)p 1471 2633 163 4 v 1471 2710 a Fu(\031)1521 2686 y Fs(n=)p Ft(4)p 1644 2580 257 4 v 1644 2652 a Fu(A)p Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))1314 2867 y(=)1532 2811 y(1)p 1471 2848 163 4 v 1471 2926 a Fu(\031)1521 2902 y Fs(n=)p Ft(4)1658 2867 y Fw(exp)1798 2750 y Fo(\022)1859 2867 y Fr(\000)1934 2811 y Fw(\()p Fv(x)19 b Fr(\000)f Fv(q)q Fw(\))2201 2781 y Ft(2)p 1934 2848 305 4 v 2065 2924 a Fw(2)2266 2867 y Fr(\000)2359 2811 y Fu({)p Fv(p)h Fr(\017)f Fv(q)p 2359 2848 209 4 v 2443 2924 a Fw(2)2596 2867 y(+)h Fu({)p Fv(p)f Fr(\017)g Fv(x)h Fw(+)3000 2811 y Fv(q)3050 2781 y Ft(2)3106 2811 y Fw(+)f Fv(p)3242 2781 y Ft(2)p 3000 2848 280 4 v 3118 2924 a Fw(4)3289 2750 y Fo(\023)3364 2867 y Fu(;)272 b Fw(\(5.2\))648 3098 y(The)32 b(functions)h(\010)1246 3068 y Ft(\()p Fq(z)p Ft(\))1366 3098 y Fr(2)f Fu(L)1510 3068 y Ft(2)1546 3098 y Fw(\()p Fv(R)1650 3068 y Fs(n)1695 3098 y Fw(\),)i(lab)r(eled)f(b)n(y)f Fv(z)g Fr(2)f Fv(C)2422 3068 y Fs(n)2467 3098 y Fu(;)i Fw(are)e(the)i(canonical)e(coheren)n(t)g (states)h(of)523 3198 y(the)c(n-dimensional)f(harmonic)g(oscillator.) 648 3298 y(Since)1171 3433 y(1)p 1111 3470 163 4 v 1111 3548 a Fu(\031)1161 3524 y Fs(n=)p Ft(2)1297 3376 y Fo(Z)1343 3565 y Fq(x)p Fn(2)p Fq(R)1484 3548 y Fp(n)1542 3489 y Fu(A)p Fw(\()p Fv(x)p Fu(;)14 b Fv(w)q Fw(\))p 1825 3417 257 4 v Fu(A)p Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))s Fu(d)p Fv(x)24 b Fw(=)f(exp)o(\()p 2446 3442 43 4 v Fv(z)q(w)q Fw(\))p Fu(;)107 b Fv(z)p Fu(;)14 b Fv(w)25 b Fr(2)e Fv(C)3041 3455 y Fs(n)3659 3489 y Fw(\(5.3\))523 3706 y(then)30 b(w)n(e)e(ha)n(v)n(e)g(that)h(the)h(Bargmann)d(transform)h (sends)h(the)g(coheren)n(t)f(states)h(of)g(the)g(n-dimensional)523 3805 y(harmonic)e(oscillator)e(o)n(v)n(er)h(in)n(to)i(the)g(coheren)n (t)e(states)h(de\014ned)h(in)g(section)f(I)r(I:)1682 3975 y Fv(B)1750 3987 y Fq(R)1806 3971 y Fp(n)1850 3975 y Fw(\010)1910 3941 y Ft(\()p Fq(z)p Ft(\))2000 3975 y Fw(\()p Fv(w)q Fw(\))d(=)e(\011)2310 3987 y Fq(z)2348 3975 y Fw(\()p Fv(w)q Fw(\))p Fu(;)1154 b Fw(\(5.4\))523 4145 y(Th)n(us)29 b(the)h(Bargmann)e(transform)g Fv(B)1738 4157 y Fq(R)1794 4141 y Fp(n)1869 4145 y Fw(acting)h(on)g(a)g(function) h Fu(F)38 b Fr(2)27 b Fu(L)2865 4115 y Ft(2)2901 4145 y Fw(\()p Fv(R)3005 4115 y Fs(n)3051 4145 y Fw(\))i(can)h(b)r(e)g (expressed)e(as)523 4245 y(a)f(coheren)n(t)g(state)g(transform:)1268 4436 y Fv(B)1336 4448 y Fq(R)1392 4432 y Fp(n)1436 4436 y Fu(F)12 b Fw(\()p Fv(z)p Fw(\))24 b(=)f Fr(h)p Fu(F)r(;)14 b Fw(\010)1903 4402 y Ft(\()p Fq(z)p Ft(\))1993 4436 y Fr(i)24 b Fw(=)2136 4323 y Fo(Z)2182 4512 y Fq(x)p Fn(2)p Fq(R)2323 4495 y Fp(n)2381 4436 y Fu(F)12 b Fw(\()p Fv(x)p Fw(\))p 2560 4359 265 4 v(\010)2620 4412 y Ft(\()p Fq(z)p Ft(\))2711 4436 y Fw(\()p Fv(x)p Fw(\))q Fu(d)p Fv(x)740 b Fw(\(5.5\))648 4669 y(It)28 b(is)h(also)e(kno)n(wn)h(that)h (the)g(set)f(of)h(coheren)n(t)e(states)h(\010)2437 4638 y Ft(\()p Fq(z)p Ft(\))2555 4669 y Fw(giv)n(es)g(a)g(resolution)f(of)h (the)h(iden)n(tit)n(y)g(in)523 4768 y(the)f(sense)f(that)h(ev)n(ery)e (function)i Fu(F)35 b Fr(2)24 b Fu(L)1827 4738 y Ft(2)1864 4768 y Fw(\()p Fv(R)1968 4738 y Fs(n)2013 4768 y Fw(\))k(can)f(b)r(e)h (written)g(as)1264 4981 y Fu(F)12 b Fw(\()p Fv(x)p Fw(\))24 b(=)66 b(lim)1555 5034 y Fs(M)6 b Fn(7!1)1770 4868 y Fo(Z)1816 5056 y Fn(j)p Fq(z)p Fn(j\024)p Fs(M)2015 4981 y Fr(h)p Fu(F)r(;)14 b Fw(\010)2199 4946 y Ft(\()p Fq(z)p Ft(\))2289 4981 y Fr(i)2321 4993 y Fq(R)2377 4976 y Fp(n)2422 4981 y Fw(\010)2482 4946 y Ft(\()p Fq(z)p Ft(\))2572 4981 y Fw(\()p Fv(x)p Fw(\))p Fu(d\027)2770 4993 y Fs(n)2816 4981 y Fw(\()p Fv(z)p Fw(\))737 b(\(5.6\))523 5212 y(where)27 b Fu(d\027)847 5224 y Fs(n)893 5212 y Fw(\()p Fv(z)p Fw(\))h(is)g(the)g(measure)e(giv)n(en)h(in)h(equation)f(\(2.1\).)523 5382 y(V.2)h Fv(B)746 5397 y Fs(S)790 5380 y Fm(2)858 5382 y Fv(as)k(a)g(coheren)m(t)g(state)g(transform.)648 5529 y Fw(In)g(section)f(I)r(I)r(I)h(w)n(e)g(sho)n(w)n(ed)e(that)i(the) h(generating)d(function)i(\010)2711 5544 y Fs(S)2755 5527 y Fm(2)2824 5529 y Fw(can)f(b)r(e)h(written)g(as)f(the)h(inner)523 5629 y(pro)r(duct)i(of)g(the)g(v)n(ectors)e Fu(\032)1422 5641 y Ft(2)1459 5629 y Fw(\()p Fv(z)p Fw(\))j Fr(2)f Fv(Q)1761 5641 y Ft(3)1831 5629 y Fw(and)g Fv(x)24 b Fr(2)f Fu(S)2207 5598 y Ft(2)2278 5629 y Fw(\(see)34 b(equation\(3.13\)\).)54 b(Th)n(us)34 b(w)n(e)f(are)g(able)g(to)523 5728 y(establish)27 b(the)h(follo)n(wing)p eop %%Page: 28 28 28 27 bop 3747 324 a Fw(28)523 614 y Fv(Theorem)30 b(5.1)41 b Fl(The)25 b(Bar)l(gmann)g(tr)l(ansform)f Fv(B)2098 629 y Fs(S)2142 613 y Fm(2)2202 614 y Fl(is)h(a)f(c)l(oher)l(ent)h (state)e(tr)l(ansform)i(in)f(the)g(sense)g(that)523 714 y Fv(B)591 729 y Fs(S)635 712 y Fm(2)701 714 y Fl(acting)30 b(on)g(a)g(function)g Fw(\011)23 b Fr(2)g Fu(L)1690 684 y Ft(2)1727 714 y Fw(\()p Fu(S)1815 684 y Ft(2)1852 714 y Fw(\))30 b Fl(c)l(an)g(b)l(e)g(written)f(in)h(the)g(fol)t(lowing)i (way)1402 1042 y Fv(B)1470 1057 y Fs(S)1514 1040 y Fm(2)1550 1042 y Fw(\011\()p Fv(z)p Fw(\))24 b(=)1860 938 y Fn(1)1833 963 y Fo(X)1837 1142 y Fs(`)p Ft(=0)1976 919 y Fr(p)p 2046 919 220 4 v 2046 986 a Fw(2)p Fu(`)17 b Fw(+)h(1)p 1976 1023 289 4 v 2092 1099 a Fu(`)p Fw(!)2275 1042 y Fr(h)p Fw(\011)p Fu(;)c Fw(\010)2469 1057 y Fs(\032)2503 1065 y Fm(2)2535 1057 y Ft(\()p Fq(z)p Ft(\))p Fs(;`)2672 1042 y Fr(i)2704 1057 y Fs(S)2748 1040 y Fm(2)3659 1042 y Fw(\(5.7\))523 1304 y Fl(wher)l(e,)31 b(for)g(e)l(ach)f Fu(\013)24 b Fr(2)f Fv(Q)1329 1316 y Ft(3)1385 1304 y Fr(\000)18 b(f)p Fw(0)p Fr(g)p Fl(,)28 b(we)i(de\014ne)1764 1498 y Fw(\010)1824 1510 y Fs(\013;`)1919 1498 y Fw(\()p Fv(x)p Fw(\))24 b(=)e(\()q Fv(x)d Fr(\017)e Fu(\013)p Fw(\))2391 1456 y Fs(`)3659 1498 y Fw(\(5.8\))523 1681 y Fl(The)31 b(functions)e Fw(\010)1113 1693 y Fs(\013;`)1238 1681 y Fl(ar)l(e)h(c)l(oher)l(ent)g(states)f(in)h(the)g(sense)g (describ)l(e)l(d)h(b)l(elow.)648 1863 y Fw(F)-7 b(or)34 b(eac)n(h)h(in)n(teger)f(n)n(um)n(b)r(er)h Fu(`)h Fr(\025)f Fw(1,)i(the)f(states)f(\010)2319 1875 y Fs(\013;`)2413 1863 y Fw(,)j(with)e Fu(\013)g Fr(2)g Fv(Q)2923 1875 y Ft(3)2984 1863 y Fr(\000)23 b(f)p Fw(0)p Fr(g)p Fw(,)36 b(are)e(a)h(system)g(of)523 1963 y(coheren)n(t)27 b(states)h(for)g(the) h(eigenspace)e Fu(V)1822 1975 y Fs(`)1883 1963 y Fw(of)h(the)h (Laplacian)e(\001)2571 1978 y Fs(S)2615 1961 y Fm(2)2680 1963 y Fw(\(see)h(reference)3169 1933 y Ft(8)3205 1963 y Fw(\))h(in)g(the)f(follo)n(wing)523 2063 y(sense:)523 2162 y(\(a\))36 b(The)g(map)f Fu(\033)1083 2174 y Ft(2)1156 2162 y Fw(establishes)g(a)h(bijectiv)n(e)f(iden)n(ti\014cation)h(b)r (et)n(w)n(een)f Fv(Q)2900 2174 y Ft(3)2961 2162 y Fr(\000)24 b(f)p Fw(0)p Fr(g)34 b Fw(and)h Fu(T)3440 2132 y Fn(\003)3477 2162 y Fu(S)3533 2132 y Ft(2)3594 2162 y Fr(\000)24 b(f)p Fw(0)p Fr(g)p Fw(.)523 2262 y(Th)n(us)h(the)h(states)f Fr(f)p Fw(\010)1208 2274 y Fs(\013;`)1302 2262 y Fr(g)g Fw(are)g(lab)r(eled)g(b)n(y)g(elemen)n(ts)h(in)f(the)h(cotangen)n(t)f (bundle)h(of)f(the)h(con\014guration)523 2361 y(space)h Fu(S)801 2331 y Ft(2)866 2361 y Fw(\(min)n(us)g(the)h(zero)f (section\).)523 2461 y(\(b\))k(F)-7 b(or)29 b(large)g Fu(`)p Fw(,)h(the)g(state)g(\010)1520 2473 y Fs(\013;`)1645 2461 y Fw(is)g(concen)n(trated)e(around)h(the)i(geo)r(desic)e(in)h Fu(S)3137 2431 y Ft(2)3204 2461 y Fw(\(classical)f(orbit)g(of)523 2561 y(the)h(geo)r(desic)e(\015o)n(w\))h(generated)f(b)n(y)h(the)h(v)n (ectors)d Fr(<)p Fw(\()p Fu(\013)p Fw(\))p Fu(=)p Fr(k<)p Fw(\()p Fu(\013)p Fw(\))p Fr(k)j Fw(in)g Fu(S)2792 2531 y Ft(2)2858 2561 y Fw(and)f Fr(\000=)p Fw(\()p Fu(\013)p Fw(\))h(in)f Fu(T)3452 2531 y Fn(\003)3489 2561 y Fu(S)3545 2531 y Ft(2)3612 2561 y Fw(at)g(the)523 2660 y(p)r(oin)n(t)c Fr(<)p Fw(\()p Fu(\013)p Fw(\))p Fu(=)p Fr(k<)p Fw(\()p Fu(\013)p Fw(\))p Fr(k)p Fw(.)36 b(The)24 b(concen)n(tration)f(is)i(in) f(the)h(sense)f(that)h(most)f(of)g(the)h Fu(L)3148 2630 y Ft(2)3185 2660 y Fw(\()p Fu(S)3273 2630 y Ft(2)3310 2660 y Fw(\)-norm)f(of)g(\010)3735 2672 y Fs(\013;`)523 2760 y Fw(comes)j(from)g(a)g(small)h(neigh)n(b)r(orho)r(o)r(d)e(of)i (the)g(geo)r(desic)e(\(see)i(reference)2824 2730 y Ft(7)2860 2760 y Fw(\).)523 2860 y(\(c\))35 b(The)g(states)f Fr(f)p Fw(\010)1182 2872 y Fs(\013;`)1277 2860 y Fr(g)g Fw(giv)n(e)g(a)g (resolution)f(of)i(the)g(iden)n(tit)n(y)g(in)g(the)g(space)f Fu(V)3094 2872 y Fs(`)3126 2860 y Fw(.)58 b(Namely)-7 b(,)37 b(for)d(eac)n(h)523 2959 y(function)28 b(\011)23 b Fr(2)g Fu(V)1062 2971 y Fs(`)1095 2959 y Fw(,)k(the)h(follo)n(wing)f (equation)g(holds:)1266 3180 y(\011\()p Fv(x)p Fw(\))d(=)f Fu(r)r Fw(\()p Fu(`)p Fw(\))1709 3067 y Fo(Z)1755 3255 y Fq(Q)1811 3263 y Fm(3)1844 3255 y Fn(\000f)p Ft(0)p Fn(g)2001 3180 y Fr(h)p Fw(\011)p Fu(;)14 b Fw(\010)2195 3192 y Fs(\013;`)2290 3180 y Fr(i)2322 3195 y Fs(S)2366 3178 y Fm(2)2403 3180 y Fw(\010)2463 3192 y Fs(\013;`)2557 3180 y Fw(\()p Fv(x)p Fw(\))p Fu(d\026)2764 3192 y Ft(2)2803 3180 y Fw(\()p Fu(\013)p Fw(\))739 b(\(5.9\))523 3423 y(for)27 b(some)g(constan)n(t)g Fu(r)r Fw(\()p Fu(`)p Fw(\))i(whic)n(h)e(will)h(b)r(e)g(ev)-5 b(aluated)27 b(b)r(elo)n(w.)648 3523 y(This)f(equation)f(is)h(a)g(consequence)f(of)h (the)h(follo)n(wing)e(three)h(facts:)36 b(\(i\))27 b(the)f(op)r(erator) f(on)g(the)i(righ)n(t)523 3623 y(hand)e(side)f(of)g(equation)g(\(5.9\)) h(comm)n(utes)f(with)h(all)f(the)h(rotations)e(in)i Fu(S)5 b(O)r Fw(\(3\))25 b(acting)f(as)g(op)r(erators)e(on)523 3722 y Fu(V)571 3734 y Fs(`)603 3722 y Fw(,)k(\(ii\))h(the)e (irreducibilit)n(y)g(of)h(the)f(represen)n(tation)f(of)h Fu(S)5 b(O)r Fw(\(3\))26 b(acting)f(on)g Fu(V)2948 3734 y Fs(`)3006 3722 y Fw(and)g(\(iii\))i(the)f(in)n(v)-5 b(ariance)523 3822 y(of)28 b(the)g(measure)e Fu(d\026)1177 3834 y Ft(2)1215 3822 y Fw(\()p Fu(\013)p Fw(\))i(under)g(rotations)e (in)i Fu(S)5 b(O)r Fw(\(3\),)28 b(whic)n(h)g(w)n(e)f(state)g(as)g (follo)n(ws:)523 3988 y Fv(Prop)s(osition)j(5.2)41 b Fl(L)l(et)29 b Fu(R)24 b Fr(2)f Fu(S)5 b(O)r Fw(\(3\))q Fl(,)30 b(then)f(for)i(any)f Fu(d\026)2363 4000 y Ft(2)2401 3988 y Fr(\000)p Fl(inte)l(gr)l(able)f(function)h Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))30 b Fl(we)h(have:)1281 4100 y Fo(Z)1327 4289 y Fq(Q)1383 4297 y Fa(3)1419 4289 y Fn(\000f)p Ft(0)p Fn(g)1590 4213 y Fu(f)9 b Fw(\()p Fu(R)q(\013)p Fw(\))p Fu(d\026)1914 4225 y Ft(2)1952 4213 y Fw(\()p Fu(\013)p Fw(\))24 b(=)2181 4100 y Fo(Z)2227 4289 y Fq(Q)2283 4297 y Fa(3)2319 4289 y Fn(\000f)p Ft(0)p Fn(g)2490 4213 y Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))p Fu(d\026)2750 4225 y Ft(2)2788 4213 y Fw(\()p Fu(\013)p Fw(\))713 b(\(5.10\))523 4457 y Fl(her)l(e)30 b Fu(R)q(\013)g Fl(me)l(ans)g Fu(R)q Fr(<)p Fw(\()p Fu(\013)p Fw(\))19 b(+)f Fu({R)q Fr(=)p Fw(\()p Fu(\013)p Fw(\))p Fl(.)523 4623 y(Pr)l(o)l(of)29 b Fw(By)e(de\014nition)h(of)g(the)g(measure)e Fu(d\026)1897 4635 y Ft(2)1023 4730 y Fo(Z)1069 4919 y Fq(Q)1125 4927 y Fa(3)1161 4919 y Fn(\000f)p Ft(0)p Fn(g)1332 4843 y Fu(f)9 b Fw(\()p Fu(R)q(\013)p Fw(\))p Fu(d\026)1656 4855 y Ft(2)1694 4843 y Fw(\()p Fu(\013)p Fw(\))24 b(=)1923 4730 y Fo(Z)1969 4919 y Fq(z)p Fn(2)p Fq(C)2101 4902 y Fm(2)2152 4843 y Fu(f)9 b Fw(\()p Fu(R)q(\032)2341 4855 y Ft(2)2377 4843 y Fw(\()p Fv(z)p Fw(\)\))14 b(exp)q(\()p Fr(\000j)p Fv(z)p Fr(j)2842 4809 y Ft(2)2880 4843 y Fw(\))p Fu(d\027)2996 4855 y Ft(2)3034 4843 y Fw(\()p Fv(z)p Fw(\))p Fu(:)455 b Fw(\(5.11\))523 5087 y(Since)31 b(for)g(an)n(y)g (rotation)f Fu(R)f Fr(2)h Fu(S)5 b(O)r Fw(\(3\))32 b(there)f(exists)f Fu(U)38 b Fr(2)30 b Fu(S)5 b(U)k Fw(\(2\))30 b(suc)n(h)h(that)h Fu(R)q(\032)3161 5099 y Ft(2)3198 5087 y Fw(\()p Fv(z)p Fw(\))e(=)e Fu(\032)3470 5099 y Ft(2)3508 5087 y Fw(\()p Fu(U)9 b Fv(z)p Fw(\),)33 b(w)n(e)523 5187 y(just)g(need)f(to)g(mak)n (e)g(the)g(c)n(hange)f(of)h(v)-5 b(ariables)31 b Fv(w)h Fw(=)f Fu(U)9 b Fv(z)32 b Fw(and)g(use)g(the)h(in)n(v)-5 b(ariance)31 b(of)h(the)g(measure)523 5286 y Fu(d\027)607 5298 y Ft(2)672 5286 y Fw(under)c(the)g(unitary)f(transformation)f(U.) 1773 b Fj(2)648 5386 y Fw(F)-7 b(or)40 b(eac)n(h)f Fu(`)p Fw(,)44 b(the)d(image)f(of)g(the)h(coheren)n(t)f(states)g(\010)2440 5398 y Fs(\013;`)2575 5386 y Fw(under)g(the)h(transformation)e Fv(U)3631 5356 y Fn(\000)p Ft(1)3631 5406 y Fs(\033)3669 5414 y Fm(2)3766 5386 y Fw(=)523 5486 y Fv(B)591 5501 y Fs(S)635 5484 y Fm(2)688 5486 y Fr(\016)17 b Fv(U)820 5498 y Fs(\032)854 5506 y Fm(2)918 5486 y Fw(pro)n(vide)26 b(a)h(repro)r(ducing)e(k)n(ernel)h(for)h(the)g(space)f Fr(X)2533 5498 y Fs(`)2565 5486 y Fw(.)37 b(The)27 b(pro)r(of)f(of)h (this)g(fact)g(is)g(based)f(on)523 5585 y(the)i(t)n(w)n(o)f(follo)n (wing)f(prop)r(ositions:)p eop %%Page: 29 29 29 28 bop 3747 324 a Fw(29)523 614 y Fv(Prop)s(osition)30 b(5.3)41 b Fl(The)30 b(inner)g(pr)l(o)l(duct)g(of)h(two)f(c)l(oher)l (ent)f(states)h Fw(\010)2764 626 y Fs(\014)s(;`)2886 614 y Fl(and)g Fw(\010)3107 626 y Fs(\013;k)3240 614 y Fl(in)g Fu(L)3399 584 y Ft(2)3436 614 y Fw(\()p Fu(S)3524 584 y Ft(2)3561 614 y Fw(\))g Fl(is:)1498 821 y Fr(h)p Fw(\010)1590 833 y Fs(\014)s(;`)1683 821 y Fu(;)14 b Fw(\010)1780 833 y Fs(\013;k)1883 821 y Fr(i)1915 836 y Fs(S)1959 820 y Fm(2)2019 821 y Fw(=)2158 765 y Fu(\015)2201 777 y Fs(`)p 2117 802 158 4 v 2117 878 a Fw(2)2159 854 y Fs(`)p Ft(+1)2284 821 y Fw(\()p Fu(\013)19 b Fr(\017)f Fu(\014)t Fw(\))2531 787 y Fs(`)2564 821 y Fu(\016)2601 833 y Fs(`;k)3618 821 y Fw(\(5.12\))523 1053 y Fl(with)41 b Fu(\015)757 1065 y Fs(`)830 1053 y Fr(\021)937 987 y Fo(R)992 1007 y Fs(\031)976 1083 y Ft(0)1051 1053 y Fw(sin)1153 1019 y Ft(2)p Fs(`)p Ft(+1)1302 1053 y Fw(\()p Fu(\022)r Fw(\))p Fu(d\022)k Fw(=)c(2)1734 1013 y Ft(\(2)p Fs(`)p Ft(\)!!)p 1692 1034 235 4 v 1692 1082 a(\(2)p Fs(`)p Ft(+1\)!!)1976 1053 y Fl(\(her)l(e)g(the)f(notation)g Fw(\(2)p Fu(`)p Fw(\)!!)g Fl(me)l(ans)g(the)g(pr)l(o)l(duct)g(of)h(al)t (l)523 1167 y(p)l(ositive)30 b(inte)l(ger)f(even)g(numb)l(ers)f(lesser) i(or)f(e)l(qual)g(than)g Fw(2)p Fu(`)p Fl(,)f(the)i(notation)f(for)g Fw(\(2)p Fu(`)16 b Fw(+)g(1\)!!)29 b Fl(is)g(similar\).)523 1333 y(Pr)l(o)l(of)p Fw(:)648 1433 y(Giv)n(en)19 b Fu(\014)27 b Fr(2)d Fv(Q)1105 1445 y Ft(3)1144 1433 y Fr(\000)s Fw(0,)d(there)e(exists)g Fu(\025)24 b(>)f Fw(0)c(and)g(a)h(rotation)e Fu(R)24 b Fr(2)f Fu(S)5 b(O)r Fw(\(3\))20 b(suc)n(h)g(that)g Fu(\014)27 b Fw(=)c Fu(\025R)q Fw(\()p Fv(e)3583 1445 y Ft(1)3623 1433 y Fr(\000)s Fu({)p Fv(e)3762 1445 y Ft(2)3798 1433 y Fw(\))523 1532 y(\(with)28 b Fr(f)p Fv(e)830 1544 y Fs(j)865 1532 y Fu(;)37 b(j)28 b Fw(=)22 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)p Fr(g)26 b Fw(the)i(canonical)e (orthonormal)g(basis)h(of)g Fv(R)2695 1502 y Ft(3)2732 1532 y Fw(\).)37 b(Th)n(us)1045 1761 y Fr(h)p Fw(\010)1137 1773 y Fs(\014)s(;`)1229 1761 y Fu(;)14 b Fw(\010)1326 1773 y Fs(\013;`)1421 1761 y Fr(i)1453 1776 y Fs(S)1497 1759 y Fm(2)1557 1761 y Fw(=)22 b Fu(\025)1692 1726 y Fs(`)1738 1648 y Fo(Z)1785 1836 y Fq(x)p Fn(2)p Fs(S)1914 1820 y Fm(2)1963 1761 y Fw(\()q Fv(x)c Fr(\017)g Fu(R)q Fw(\()p Fv(e)2264 1773 y Ft(1)2320 1761 y Fr(\000)g Fu({)p Fv(e)2474 1773 y Ft(2)2511 1761 y Fw(\)\))2575 1719 y Fs(`)p 2621 1688 247 4 v 2621 1761 a Fw(\()p Fv(x)h Fr(\017)f Fu(\013)p Fw(\))2868 1702 y Fs(`)2900 1761 y Fu(d\033)2990 1773 y Ft(2)3028 1761 y Fw(\()p Fv(x)p Fw(\))476 b(\(5.13\))648 2000 y(By)24 b(making)f(the)i(c)n(hange)e(of)h(v)-5 b(ariable)23 b Fv(y)i Fw(=)e Fu(R)2101 1969 y Fn(\000)p Ft(1)2189 2000 y Fv(x)p Fw(,)j(using)e(spherical)f(co)r(ordinates)g(for)h(the)g (v)-5 b(ariable)523 2099 y Fv(y)29 b Fw(and)f(the)g(orthogonalit)n(y)d (of)j(the)g(matrix)f(R,)g(the)h(result)g(follo)n(ws)e(b)n(y)i(straigh)n (t)e(computation.)180 b Fj(2)523 2282 y Fv(Prop)s(osition)30 b(5.4)41 b Fl(The)30 b(Bar)l(gmann)h(T)-6 b(r)l(ansform)30 b Fv(B)2251 2297 y Fs(S)2295 2280 y Fm(2)2361 2282 y Fl(of)h(the)f(state)f Fw(\010)2856 2294 y Fs(\014)s(;`)2978 2282 y Fl(is)1348 2527 y Fv(B)1416 2542 y Fs(S)1460 2525 y Fm(2)1496 2527 y Fw(\010)1556 2539 y Fs(\014)s(;`)1649 2527 y Fw(\()p Fv(z)p Fw(\))24 b(=)1876 2404 y Fr(p)p 1945 2404 220 4 v 67 x Fw(2)p Fu(`)18 b Fw(+)g(1)p 1876 2508 289 4 v 1992 2584 a Fu(`)p Fw(!)2226 2471 y Fu(\015)2269 2483 y Fs(`)p 2184 2508 158 4 v 2184 2584 a Fw(2)2226 2560 y Fs(`)p Ft(+1)2366 2527 y Fw(\()p Fu(\032)2441 2539 y Ft(2)2478 2527 y Fw(\()p Fv(z)p Fw(\))i Fr(\017)e Fu(\014)t Fw(\))2747 2485 y Fs(`)2816 2527 y Fu(:)779 b Fw(\(5.14\))523 2733 y Fl(Pr)l(o)l(of)22 b Fw(This)f(result)f(is)h (an)f(immediate)h(consequence)f(of)h(the)g(previous)e(prop)r(osition)h (and)h(the)g(expression)523 2833 y(for)27 b Fv(B)718 2848 y Fs(S)762 2831 y Fm(2)826 2833 y Fw(giv)n(en)g(in)h (equation\(5.7\).)2122 b Fj(2)648 2932 y Fw(Th)n(us)27 b(the)h(image)f(of)g(the)h(coheren)n(t)f(state)g(\010)2076 2944 y Fs(\014)s(;`)2196 2932 y Fw(under)h(the)g(transformation)e Fv(U)3214 2902 y Fn(\000)p Ft(1)3214 2953 y Fs(\033)3252 2961 y Fm(2)3331 2932 y Fw(is)1432 3182 y Fv(U)1505 3148 y Fn(\000)p Ft(1)1505 3203 y Fs(\033)1543 3211 y Fm(2)1595 3182 y Fw(\010)1655 3194 y Fs(\014)s(;`)1747 3182 y Fw(\()p Fu(\013)p Fw(\))e(=)1986 3059 y Fr(p)p 2055 3059 220 4 v 67 x Fw(2)p Fu(`)17 b Fw(+)h(1)p 1986 3163 289 4 v 2101 3239 a Fu(`)p Fw(!)2335 3126 y Fu(\015)2378 3138 y Fs(`)p 2294 3163 158 4 v 2294 3239 a Fw(2)2336 3215 y Fs(`)p Ft(+1)2475 3182 y Fw(\()q Fu(\013)g Fr(\017)g Fu(\014)t Fw(\))2723 3141 y Fs(`)3618 3182 y Fw(\(5.15\))648 3413 y(The)25 b(factor)g(\()q Fu(\013)18 b Fr(\017)g Fu(\014)t Fw(\))1300 3371 y Fs(`)1358 3413 y Fw(app)r(earing)24 b(in)i(the)g(last)g(equation)f(pro)n(vides)f(us)i(with)g(a)g(repro)r (ducing)e(k)n(ernel)523 3512 y(for)j(the)h(space)f Fr(X)1074 3524 y Fs(`)1134 3512 y Fw(as)g(the)h(follo)n(wing)e(theorem)h (establishes)g(it:)523 3695 y Fv(Theorem)j(5.5)41 b Fl(F)-6 b(or)23 b(e)l(ach)g(inte)l(ger)g(numb)l(er)f Fu(`)g Fr(\025)h Fw(1)p Fl(,)h(the)f(function)f Fu(K)2733 3707 y Fq(Q)2789 3715 y Fa(3)2825 3707 y Fs(;`)2899 3695 y Fw(:)h Fv(Q)3017 3707 y Fq(3)3062 3695 y Fr(\000)s(f)p Fw(0)p Fr(g)s(\002)s Fv(Q)3399 3707 y Fq(3)3440 3695 y Fr(\000)s(f)p Fw(0)p Fr(g)e(7!)i Fv(C)523 3795 y Fl(given)30 b(by)1589 4029 y Fu(K)1660 4041 y Fq(Q)1716 4049 y Fa(3)1752 4041 y Fs(;`)1804 4029 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))24 b Fr(\021)2176 3973 y Fw(2)2218 3943 y Fs(`)p 2130 4010 164 4 v 2130 4086 a Fw(\(2)p Fu(`)p Fw(\)!)2318 4029 y(\()p Fu(\014)f Fr(\017)18 b Fu(\013)p Fw(\))2566 3987 y Fs(`)3618 4029 y Fw(\(5.16\))523 4255 y Fl(is)30 b(a)g(r)l(epr)l(o)l (ducing)h(kernel)f(for)h(the)f(sp)l(ac)l(e)g Fr(X)1923 4267 y Fs(`)1955 4255 y Fl(,)g(that)g(is:)1174 4363 y Fo(Z)1220 4552 y Fq(Q)1276 4560 y Fa(3)1312 4552 y Fn(\000f)p Ft(0)p Fn(g)1483 4476 y Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))p Fu(K)1721 4488 y Fq(Q)1777 4496 y Fa(3)1813 4488 y Fs(;`)1865 4476 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))p Fu(d\026)2163 4488 y Ft(2)2201 4476 y Fw(\()p Fu(\013)p Fw(\))24 b(=)f Fu(f)9 b Fw(\()p Fu(\014)t Fw(\))p Fu(;)107 b(f)31 b Fr(2)24 b(X)2935 4488 y Fs(`)2990 4476 y Fu(:)605 b Fw(\(5.17\))523 4715 y Fl(Pr)l(o)l(of:)648 4815 y Fw(It)23 b(is)g(enough)g(to)g(pro)n (v)n(e)f(equation\(5.17\))g(when)h Fu(f)32 b Fw(=)22 b(\007)2395 4827 y Fs(`;m)2506 4815 y Fw(.)35 b(F)-7 b(rom)23 b(the)h(de\014nition)f(of)h Fu(d\026)3465 4827 y Ft(2)3525 4815 y Fw(w)n(e)f(ha)n(v)n(e:)792 4943 y Fo(Z)838 5132 y Fq(Q)894 5140 y Fa(3)930 5132 y Fn(\000f)p Ft(0)p Fn(g)1101 5056 y Fw(\007)1166 5068 y Fs(`;m)1276 5056 y Fw(\()p Fu(\013)p Fw(\))p Fu(K)6 b Fw(\()p Fu(\013;)14 b(\014)t Fw(\))p Fu(d\026)1768 5068 y Ft(2)1807 5056 y Fw(\()p Fu(\013)p Fw(\))25 b(=)2036 4943 y Fo(Z)2082 5132 y Fq(z)p Fn(2)p Fq(C)2214 5115 y Fm(2)2265 5056 y Fu(y)2306 5068 y Fs(`;m)2416 5056 y Fw(\()p Fv(z)p Fw(\))2578 5000 y(2)2620 4970 y Fs(`)p 2533 5037 V 2533 5113 a Fw(\(2)p Fu(`)p Fw(\)!)p 2707 4984 382 4 v 2707 5056 a(\()p Fu(\032)2782 5068 y Ft(2)2819 5056 y Fw(\()p Fv(z)p Fw(\))20 b Fr(\017)e Fu(\014)t Fw(\))3088 4998 y Fs(`)3120 5056 y Fu(d\027)3204 5068 y Ft(2)3242 5056 y Fw(\()p Fv(z)p Fw(\))24 b Fu(:)223 b Fw(\(5.18\))648 5296 y(No)n(w)27 b(let)h(us)f(de\014ne)h(the)g(function)978 5478 y(\010)1038 5490 y Fs(\032)1072 5498 y Fm(2)1109 5478 y Fw(\()p Fv(z)p Fu(;)14 b(\013)p Fw(\))84 b Fr(\021)p 1537 5406 238 4 v 83 w Fu(y)1578 5490 y Ft(1)p Fs(;)p Ft(1)1668 5478 y Fw(\()p Fv(z)p Fw(\))q(\007)1840 5490 y Ft(1)p Fs(;)p Ft(1)1929 5478 y Fw(\()p Fu(\013)p Fw(\))20 b(+)p 2149 5406 V 18 w Fu(y)2190 5490 y Ft(1)p Fs(;)p Ft(0)2280 5478 y Fw(\()p Fv(z)p Fw(\))q(\007)2452 5490 y Ft(1)p Fs(;)p Ft(0)2541 5478 y Fw(\()p Fu(\013)p Fw(\))g(+)p 2761 5406 290 4 v 18 w Fu(y)2802 5490 y Ft(1)p Fs(;)p Fn(\000)p Ft(1)2943 5478 y Fw(\()p Fv(z)p Fw(\))q(\007)3115 5490 y Ft(1)p Fs(;)p Fn(\000)p Ft(1)3257 5478 y Fw(\()p Fu(\013)p Fw(\))1389 5603 y(=)83 b Fu(\013)19 b Fr(\017)f Fu(\032)1712 5615 y Ft(2)1749 5603 y Fw(\()p Fv(z)p Fw(\))24 b Fu(:)1716 b Fw(\(5.19\))p eop %%Page: 30 30 30 29 bop 3747 324 a Fw(30)648 614 y(Since)1329 857 y(\(\010)1421 869 y Fs(\032)1455 877 y Fm(2)1492 857 y Fw(\()p Fv(z)p Fu(;)14 b(\014)t Fw(\)\))1718 822 y Fs(`)1774 857 y Fw(=)1872 800 y(\(2)p Fu(`)p Fw(\)!)p 1872 837 164 4 v 1917 914 a(2)1959 890 y Fs(`)2140 753 y(`)2094 778 y Fo(X)2060 956 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)p 2263 811 152 4 v 2263 857 a Fu(y)2304 869 y Fs(`;m)2414 857 y Fw(\()p Fv(z)p Fw(\)\007)2585 869 y Fs(`;m)2696 857 y Fw(\()p Fu(\014)t Fw(\))24 b Fu(;)760 b Fw(\(5.20\))523 1119 y(the)28 b(result)f(follo)n(ws.)2602 b Fj(2)648 1219 y Fw(W)-7 b(e)26 b(can)g(use)h(this)f(theorem,)g(in)h(particular,)e(to) i(ev)-5 b(aluate)26 b(the)g(constan)n(t)g Fu(r)r Fw(\()p Fu(`)p Fw(\))h(app)r(earing)e(in)i(equa-)523 1318 y(tion)h(\(5.9\))f (in)h(the)g(follo)n(wing)e(manner:)37 b(Let)27 b(us)h(apply)f(equation) g(\(5.9\))h(to)f(the)h(state)g(\010)3344 1330 y Fs(\014)s(;`)3436 1318 y Fw(:)740 1545 y(\010)800 1557 y Fs(\014)s(;`)892 1545 y Fw(\()p Fv(x)p Fw(\))84 b(=)f Fu(r)r Fw(\()p Fu(`)p Fw(\))1390 1432 y Fo(Z)1437 1621 y Fs(\013)p Fn(2)p Fq(Q)1581 1629 y Fm(3)1618 1545 y Fr(h)p Fw(\010)1710 1557 y Fs(\014)s(;`)1802 1545 y Fw(\010)1862 1557 y Fs(\013;`)1957 1545 y Fr(i)1989 1560 y Fs(S)2033 1543 y Fm(2)2070 1545 y Fw(\010)2130 1557 y Fs(\013;`)2225 1545 y Fw(\()p Fv(x)p Fw(\))p Fu(d\026)2432 1557 y Ft(2)2470 1545 y Fw(\()p Fu(\013)p Fw(\))1090 1778 y(=)g Fu(r)r Fw(\()p Fu(`)p Fw(\))1428 1722 y Fu(\015)1471 1734 y Fs(`)p 1387 1759 158 4 v 1387 1835 a Fw(2)1429 1811 y Fs(`)p Ft(+1)1568 1665 y Fo(Z)1614 1854 y Fs(\013)p Fn(2)p Fq(Q)1758 1862 y Fm(3)1809 1778 y Fw(\()p Fu(\013)19 b Fr(\017)f Fu(\014)t Fw(\))2057 1736 y Fs(`)2102 1778 y Fw(\010)2162 1790 y Fs(\013;`)2257 1778 y Fw(\()p Fv(x)p Fw(\))p Fu(d\026)2464 1790 y Ft(2)2503 1778 y Fw(\()p Fu(\013)p Fw(\))1090 2028 y(=)83 b Fu(r)r Fw(\()p Fu(`)p Fw(\))1386 1972 y Fu(\015)1429 1984 y Fs(`)1462 1972 y Fw(\(2)p Fu(`)p Fw(\)!)p 1386 2009 239 4 v 1411 2085 a(2)1453 2061 y Ft(2)p Fs(`)p Ft(+1)p 1636 1905 1203 4 v 1636 1915 a Fo(Z)1682 2103 y Fs(\013)p Fn(2)p Fq(Q)1826 2111 y Fm(3)1876 2028 y Fw(\()q Fu(\013)18 b Fr(\017)g Fv(x)p Fw(\))2123 1986 y Fs(`)2169 2028 y Fu(K)2240 2040 y Fq(Q)2296 2048 y Fa(3)2331 2040 y Fs(;`)2383 2028 y Fw(\()p Fu(\013;)c(\014)t Fw(\))p Fu(d\026)2681 2040 y Ft(2)2720 2028 y Fw(\()p Fu(\013)p Fw(\))24 b(=)e Fu(r)r Fw(\()p Fu(`)p Fw(\))3096 1972 y Fu(\015)3139 1984 y Fs(`)3172 1972 y Fw(\(2)p Fu(`)p Fw(\)!)p 3096 2009 239 4 v 3121 2085 a(2)3163 2061 y Ft(2)p Fs(`)p Ft(+1)3346 2028 y Fw(\010)3406 2040 y Fs(\014)s(;`)3498 2028 y Fw(\()p Fv(x)p Fw(\))6 b(\(5.21\))523 2262 y(Th)n(us)27 b(w)n(e)h(m)n(ust)f(ha) n(v)n(e)1839 2479 y Fu(r)r Fw(\()p Fu(`)p Fw(\))d(=)2123 2422 y(2)2165 2392 y Ft(2)p Fs(`)p Ft(+1)p 2099 2460 V 2099 2536 a Fu(\015)2142 2548 y Fs(`)2174 2536 y Fw(\(2)p Fu(`)p Fw(\)!)3618 2479 y(\(5.22\))648 2708 y(F)-7 b(rom)27 b(theorem)g(\(5.5\))g(w)n(e)g(also)g(obtain)g(a)g(repro)r(ducing)g(k)n (ernel)f(for)i(the)f(space)g Fr(E)3243 2720 y Ft(3)3280 2708 y Fw(:)523 2889 y Fv(Theorem)j(5.6)41 b Fl(The)31 b(function)822 3132 y Fu(K)893 3144 y Fq(Q)949 3152 y Fa(3)988 3132 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))24 b Fr(\021)1332 3029 y Fn(1)1305 3054 y Fo(X)1309 3232 y Fs(`)p Ft(=0)1494 3076 y Fw(2)1536 3046 y Fs(`)p 1449 3113 164 4 v 1449 3189 a Fw(\(2)p Fu(`)p Fw(\)!)1636 3132 y(\()q Fu(\014)e Fr(\017)c Fu(\013)p Fw(\))1884 3091 y Fs(`)1939 3132 y Fw(=)2037 3076 y(1)p 2037 3113 42 4 v 2037 3189 a(2)2102 3040 y Fo(\020)2152 3132 y Fw(exp)o(\()2310 3057 y Fo(p)p 2394 3057 225 4 v 2394 3132 a Fw(2)p Fu(\014)k Fr(\017)c Fu(\013)p Fw(\))h(+)f(exp\()p Fr(\000)2976 3057 y Fo(p)p 3059 3057 V 75 x Fw(2)p Fu(\014)k Fr(\017)c Fu(\013)q Fw(\))3316 3040 y Fo(\021)3618 3132 y Fw(\(5.23\))523 3393 y Fl(is)34 b(an)g(analytic)h(function)f(in)g (the)f(sp)l(ac)l(e)i Fr(E)1900 3405 y Ft(3)1971 3393 y Fl(\(for)f Fu(\013)g Fl(\014xe)l(d\))g(and)g(pr)l(ovides)h(a)f(r)l (epr)l(o)l(ducing)h(kernel)f(for)523 3492 y(this)c(sp)l(ac)l(e:)1203 3598 y Fo(Z)1249 3787 y Fq(Q)1305 3795 y Fa(3)1341 3787 y Fn(\000f)p Ft(0)p Fn(g)1512 3711 y Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))p Fu(K)1750 3723 y Fq(Q)1806 3731 y Fa(3)1846 3711 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))p Fu(d\026)2144 3723 y Ft(2)2183 3711 y Fw(\()p Fu(\013)p Fw(\))24 b(=)e Fu(f)9 b Fw(\()p Fu(\014)t Fw(\))p Fu(;)107 b(f)32 b Fr(2)23 b(E)2901 3723 y Ft(3)2961 3711 y Fu(:)634 b Fw(\(5.24\))523 3953 y(Note)34 b(that)h(in)g(equation)e(\(5.23\))h(w)n(e)g(can)g(tak)n (e)f(an)n(y)h(branc)n(h)f(of)h(the)h(square)e(ro)r(ot)h(function)g(and) h(w)n(e)523 4053 y(obtain)27 b(an)h(analytic)f(function)h(in)g(an)n(y)e (case.)523 4153 y Fl(Pr)l(o)l(of:)648 4264 y Fw(The)e(series)1034 4201 y Fo(P)1121 4222 y Fn(1)1121 4289 y Fs(`)p Ft(=0)1296 4231 y(2)1329 4206 y Fp(`)p 1261 4245 132 4 v 1261 4292 a Ft(\(2)p Fs(`)p Ft(\)!)1402 4264 y Fu(z)1445 4234 y Fs(`)1501 4264 y Fw(is)g(absolutely)f(con)n(v)n(ergen)n(t)e(for)j(all)g (v)-5 b(alues)23 b(of)h Fu(z)t Fr(2)p Fv(C)p Fw(,)h(so)e Fu(K)3337 4276 y Fq(Q)3393 4284 y Fa(3)3432 4264 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))26 b(is)d(an)523 4397 y(analytic)g(function) h(on)f(the)h(v)-5 b(ariable)22 b Fu(\014)28 b Fw(\()23 b Fu(\013)h Fw(\014xed\).)36 b(Moreo)n(v)n(er,)22 b(eac)n(h)g(function) i(2)3114 4367 y Fs(`)3159 4397 y Fw(\()q Fu(\014)f Fr(\017)17 b Fu(\013)p Fw(\))3407 4356 y Fs(`)3453 4397 y Fu(=)p Fw(\(2)p Fu(`)p Fw(\)!)23 b(is)g(in)523 4539 y(the)k(space)f Fr(X)945 4551 y Fs(`)1004 4539 y Fw(and)g(its)h(norm)f Fu(a)1539 4509 y Ft(2)1539 4562 y Fs(`)1599 4539 y Fr(\021)d(k)p Fw(2)1771 4509 y Fs(`)1815 4539 y Fw(\()q Fu(\014)g Fr(\017)18 b Fu(\013)p Fw(\))2063 4497 y Fs(`)2109 4539 y Fu(=)p Fw(\(2)p Fu(`)p Fw(\)!)p Fr(k)2357 4509 y Ft(2)2420 4539 y Fw(is)26 b(equal)g(to)2880 4506 y Ft(2)2913 4481 y Fm(3)p Fp(`)p Fm(+1)p 2831 4520 261 4 v 2831 4567 a Ft(\(2)p Fs(`)p Ft(+1\))p Fs(\015)3063 4576 y Fp(`)3115 4447 y Fo(\020)3217 4506 y Fs(`)p Ft(!)p 3175 4520 132 4 v 3175 4567 a(\(2)p Fs(`)p Ft(\)!)3316 4447 y Fo(\021)3366 4464 y Ft(2)3417 4539 y Fr(j)p Fu(\013)p Fr(j)3516 4509 y Ft(2)p Fs(`)3581 4539 y Fw(.)37 b(Since)523 4606 y Fo(P)611 4627 y Fn(1)611 4694 y Fs(`)p Ft(=0)741 4669 y Fu(a)785 4638 y Ft(2)785 4692 y Fs(`)856 4669 y Fw(is)d(\014nite,)j(then)d(w)n (e)g(conclude)g(that)h(the)f(k)n(ernel)g Fu(K)2521 4681 y Fq(Q)2577 4689 y Fa(3)2650 4669 y Fw(is)h(in)f Fr(E)2888 4681 y Ft(3)2959 4669 y Fw(\()p Fu(\013)h Fw(\014xed\).)58 b(The)34 b(equalit)n(y)523 4768 y(in)k(equation)g(\(5.23\))f(can)h(b)r (e)g(c)n(hec)n(k)n(ed)f(directly)g(b)n(y)h(using)g(the)g(T)-7 b(a)n(ylor)36 b(series)h(of)h(the)h(exp)r(onen)n(tial)523 4868 y(function.)2925 b Fj(2)648 4967 y Fw(The)30 b(k)n(ernel)f Fu(K)1139 4979 y Fq(Q)1195 4987 y Fa(3)1265 4967 y Fw(and)h(the)g (function)h Fu(\032)1945 4979 y Ft(2)2012 4967 y Fw(giv)n(e)e(an)h(exp) r(ected)h(repro)r(ducing)e(k)n(ernel)g(for)g(the)i(space)523 5067 y Fr(B)578 5082 y Fs(S)622 5065 y Fm(2)658 5067 y Fw(:)523 5248 y Fv(Theorem)f(5.7)41 b Fl(The)h(function)1621 5216 y Ft(1)p 1621 5230 34 4 v 1621 5277 a(2)1678 5248 y Fw(\(exp\()p Fv(z)19 b Fr(\017)f Fv(w)q Fw(\))h(+)f(exp\()p Fr(\000)p Fv(z)h Fr(\017)f Fv(w)q Fw(\)\))41 b Fl(is)h(a)f(r)l(epr)l(o) l(ducing)h(kernel)f(of)h(the)523 5348 y(sp)l(ac)l(e)30 b Fr(B)794 5363 y Fs(S)838 5346 y Fm(2)874 5348 y Fl(.)39 b(This)31 b(kernel)f(function)g(is)g(in)g(the)g(sp)l(ac)l(e)g Fr(B)2303 5363 y Fs(S)2347 5346 y Fm(2)2413 5348 y Fl(for)h Fv(w)g Fl(\014xe)l(d.)523 5529 y Fv(REMARK)f(1)648 5629 y Fw(V.)e(Bargmann)1140 5598 y Ft(20)1236 5629 y Fw(has)f(also)g (analyzed)g(the)h(space)f Fr(E)2304 5641 y Ft(3)2368 5629 y Fw(of)h(analytic)f(functions)h(on)g(the)g(quadric)f Fv(Q)3766 5641 y Fq(3)3807 5629 y Fw(.)523 5728 y(He)33 b(giv)n(es)f(to)h(suc)n(h)g(space)g(a)f(structure)h(of)g(a)g(Hilb)r (ert)h(space)e(endo)n(w)n(ed)g(with)i(a)f(inner)g(pro)r(duct)g(with)p eop %%Page: 31 31 31 30 bop 3747 324 a Fw(31)523 614 y(resp)r(ect)23 b(to)g(a)f(measure)g (lik)n(e)g(the)i(one)e(w)n(e)h(ha)n(v)n(e)f(\()p Fu(d\026)2146 626 y Ft(2)2183 614 y Fw(\).)36 b(He)23 b(also)f(considers)g(a)g(map)h Fv(A)g Fw(:)g Fr(E)3339 626 y Ft(3)3399 614 y Fr(7!)g(B)3560 626 y Ft(2)3620 614 y Fw(whic)n(h)523 714 y(is)28 b(v)n(ery)e(similar)h (to)g(the)h(transformation)e Fv(U)1945 684 y Fn(\000)p Ft(1)1945 735 y Fs(\032)1979 743 y Fm(2)2058 714 y Fw(:)d Fr(E)2148 726 y Ft(3)2208 714 y Fr(7!)g(B)2369 729 y Fs(S)2413 712 y Fm(2)2477 714 y Fw(giv)n(en)k(b)n(y:)1619 906 y Fv(U)1692 872 y Fn(\000)p Ft(1)1692 927 y Fs(\032)1726 935 y Fm(2)1782 906 y Fw(\011\()p Fv(z)p Fw(\))d(=)e(\011\()p Fu(\032)2204 918 y Ft(2)2241 906 y Fw(\()p Fv(z)p Fw(\)\))p Fu(;)85 b Fw(\011)22 b Fr(2)i(E)2697 918 y Ft(3)3618 906 y Fw(\(5.25\))523 1086 y(The)h(di\013erence)g(b)r(et)n(w)n(een)f Fv(A)h Fw(and)g Fv(U)1706 1056 y Fn(\000)p Ft(1)1706 1107 y Fs(\032)1740 1115 y Fm(2)1821 1086 y Fw(is)f(that)h(Bargmann)e (uses)i(a)f(sligh)n(tly)g(di\013eren)n(t)h(expression)f(for)523 1186 y Fu(\032)566 1198 y Ft(2)633 1186 y Fw(and)29 b(that)g Fv(A)h Fw(is)f(not)h(on)n(to.)41 b(He)30 b(sho)n(ws)e(that)i Fv(A)f Fw(preserv)n(es)e(the)j(inner)f(pro)r(duct.)43 b(In)29 b(particular)f(he)523 1285 y(also)i(\014nds)i(the)f(adjoin)n(t) h(of)f(the)g(m)n(ultiplicativ)n(e)h(op)r(erator)d(b)n(y)i(the)h(co)r (ordinate)e Fu(\013)3169 1297 y Fs(j)3204 1285 y Fu(;)73 b(j)34 b Fw(=)29 b(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3.)46 b(He)523 1385 y(do)r(es)25 b(it)g(b)n(y)g(an)f(in)n(tegration)g(b)n(y)g (parts)h(whereas)e(w)n(e)i(can)f(do)h(it)g(b)n(y)g(using)f(our)g (unitary)h(transformation)523 1485 y Fv(U)596 1497 y Fs(\032)630 1505 y Fm(2)667 1485 y Fw(,)44 b(the)c(c)n(hain)f(rule)h (and)g(taking)f(in)n(to)h(accoun)n(t)f(that)h(the)g(adjoin)n(t,)j(in)d (the)h(space)e Fr(B)3462 1500 y Fs(S)3506 1483 y Fm(2)3542 1485 y Fw(,)k(of)d(the)523 1601 y(m)n(ultiplicativ)n(e)25 b(op)r(erators)e(b)n(y)h Fu(z)1564 1571 y Ft(2)1560 1622 y(1)1601 1601 y Fu(;)14 b(z)1681 1571 y Ft(2)1677 1622 y(2)1717 1601 y Fu(;)g(z)1793 1613 y Ft(1)1830 1601 y Fu(z)1869 1613 y Ft(2)1930 1601 y Fw(are)24 b(the)h(op)r(erators)2598 1568 y Fs(@)2637 1543 y Fm(2)p 2581 1582 106 4 v 2581 1632 a Fs(@)t(z)2654 1612 y Fm(2)2652 1652 y(1)2697 1601 y Fw(,)2772 1568 y Fs(@)2811 1543 y Fm(2)p 2755 1582 V 2755 1632 a Fs(@)t(z)2828 1612 y Fm(2)2826 1652 y(2)2871 1601 y Fw(,)2997 1568 y Fs(@)3036 1543 y Fm(2)p 2929 1582 207 4 v 2929 1630 a Fs(@)t(z)3000 1638 y Fm(2)3032 1630 y Fs(@)t(z)3103 1638 y Fm(1)3170 1601 y Fw(resp)r(ectiv)n(ely)-7 b(.)35 b(Th)n(us)523 1718 y(w)n(e)27 b(\014nd)h(that)g(the)g(adjoin)n (t)g(op)r(erator:)1492 1972 y Fu(\013)1545 1938 y Fn(\003)1545 1993 y Fs(j)1607 1972 y Fw(=)1749 1916 y Fu(@)p 1705 1953 137 4 v 1705 2029 a(@)5 b(\013)1807 2041 y Fs(j)1870 1972 y Fw(+)18 b(2)2052 1869 y Ft(3)2009 1894 y Fo(X)2009 2072 y Fs(k)q Ft(=1)2143 1972 y Fu(\013)2196 1984 y Fs(k)2294 1916 y Fu(@)p 2247 1953 143 4 v 2247 2029 a(@)5 b(\013)2349 2041 y Fs(k)2453 1916 y Fu(@)p 2409 1953 137 4 v 2409 2029 a(@)g(\013)2511 2041 y Fs(j)2574 1972 y Fr(\000)18 b Fu(\013)2710 1984 y Fs(j)2746 1972 y Fw(\001)23 b Fu(;)757 b Fw(\(5.26\))523 2254 y(where)28 b(\001)c(=)946 2191 y Fo(P)1033 2212 y Ft(3)1033 2279 y Fs(k)q Ft(=1)1206 2221 y Fs(@)1245 2196 y Fm(2)p 1182 2235 119 4 v 1182 2285 a Fs(@)t(\013)1264 2265 y Fm(2)1264 2307 y Fp(k)1310 2254 y Fw(.)39 b(Our)28 b(expression)f(for)g Fu(\013)2125 2224 y Fn(\003)2125 2275 y Fs(j)2192 2254 y Fw(di\013ers)h(from)g(the)h (one)f(of)g(Bargmann)e(b)n(y)i(a)g(factor)523 2385 y(of)628 2353 y Ft(1)p 628 2367 34 4 v 628 2414 a(2)698 2385 y Fw(b)r(ecause)f(he)h(uses)f(a)h(di\013eren)n(t)f(expression)f(for)h Fu(\032)2259 2397 y Ft(2)2297 2385 y Fw(.)523 2485 y Fv(REMARK)j(2)648 2585 y Fw(V.)24 b(Guillemin)1112 2554 y Ft(21)1208 2585 y Fw(has)f(in)n(tro)r(duced)h(a)g(transformation)f Fu(p)2435 2597 y Fn(\003)2497 2585 y Fw(whic)n(h)h(is)g(related)g(to)g (our)g(unitary)f(trans-)523 2684 y(formation)35 b Fv(U)985 2696 y Fs(\033)1023 2704 y Fm(2)1097 2684 y Fw(:)i Fr(E)1201 2696 y Ft(3)1274 2684 y Fr(7!)g Fu(L)1451 2654 y Ft(2)1488 2684 y Fw(\()p Fu(S)1576 2654 y Ft(2)1613 2684 y Fw(\).)62 b(The)36 b(transformation)e Fu(p)2525 2696 y Fn(\003)2598 2684 y Fw(can)i(b)r(e)g(describ)r(ed)f(in)h(our)f(terms)h(as)523 2784 y(follo)n(ws:)42 b(The)31 b(unit)g(cotangen)n(t)f(bundle)h Fu(S)1901 2754 y Fn(\003)1939 2784 y Fw(\()p Fu(S)2027 2754 y Ft(2)2064 2784 y Fw(\))g(of)g Fu(S)2281 2754 y Ft(2)2349 2784 y Fw(can)f(b)r(e)h(iden)n(ti\014ed)g(with)h(the)f(follo) n(wing)e(sub-)523 2883 y(space)e(of)g(the)h(quadric)f Fv(Q)1352 2895 y Ft(3)1389 2883 y Fw(:)1336 3063 y Fu(X)i Fr(\021)23 b(f)p Fv(x)18 b Fw(+)g Fu({)p Fv(y)q Fr(jk)p Fv(x)p Fr(k)23 b Fw(=)g Fr(k)p Fv(y)q Fr(k)g Fw(=)f(1)p Fu(;)60 b Fv(x)19 b Fr(\017)f Fv(y)24 b Fw(=)f(0)p Fr(g)f Fu(:)767 b Fw(\(5.27\))648 3243 y(Let)19 b Fu(V)836 3255 y Fs(`)887 3243 y Fw(b)r(e)h(the)f(subspace)f(of)h Fu(L)1607 3213 y Ft(2)1644 3243 y Fw(\()p Fu(S)1732 3213 y Ft(2)1769 3243 y Fw(\))g(generated)f(b)n(y)h(the)g(spherical)f(harmonics)g(of)h (order)f Fu(`)g Fw(as)h(ab)r(o)n(v)n(e.)523 3343 y(Let)33 b Fu(f)40 b Fr(2)32 b Fu(V)893 3355 y Fs(`)925 3343 y Fw(.)53 b(Let)33 b(us)f(denote)h(b)n(y)f Fu(F)1712 3355 y Fs(`)1745 3343 y Fw(\()p Fu(x)1824 3355 y Ft(1)1862 3343 y Fu(;)14 b(x)1946 3355 y Ft(2)1983 3343 y Fu(;)g(x)2067 3355 y Ft(3)2105 3343 y Fw(\))33 b(the)g(unique)g(p)r(olynomial)f(in)h (three)g(real)e(v)-5 b(ariables)523 3456 y(\()p Fu(x)602 3468 y Ft(1)640 3456 y Fu(;)14 b(x)724 3468 y Ft(2)762 3456 y Fu(;)g(x)846 3468 y Ft(3)883 3456 y Fw(\))36 b Fr(2)g Fv(R)1114 3426 y Ft(3)1186 3456 y Fw(whic)n(h)e(is)h (homogeneous)f(of)h(degree)e Fu(`)p Fw(,)k(harmonic)d(\(\001)2972 3471 y Fq(R)3028 3455 y Fm(3)3065 3456 y Fu(F)3118 3468 y Fs(`)3186 3456 y Fr(\021)3286 3394 y Fo(P)3373 3415 y Ft(3)3373 3481 y Fs(k)q Ft(=1)3543 3424 y Fs(@)3582 3399 y Fm(2)p 3522 3438 113 4 v 3522 3487 a Fs(@)t(x)3599 3467 y Fm(2)3599 3510 y Fp(k)3645 3456 y Fu(F)3698 3468 y Fs(`)3766 3456 y Fw(=)523 3594 y(0\))41 b(and)f(whose)g(restriction)g (to)g Fu(S)1651 3564 y Ft(2)1729 3594 y Fw(is)g(equal)g(to)h Fu(f)9 b Fw(.)76 b(Let)2501 3573 y(~)2482 3594 y Fu(F)2535 3606 y Fs(`)2568 3594 y Fw(\()p Fu(w)2659 3606 y Ft(1)2697 3594 y Fu(;)14 b(w)2793 3606 y Ft(2)2830 3594 y Fu(;)g(w)2926 3606 y Ft(3)2964 3594 y Fw(\))41 b(b)r(e)g(the)g(p)r(olynomial)f(in)523 3694 y(three)d(complex)g(v)-5 b(ariables)37 b(\()p Fu(w)1526 3706 y Ft(1)1564 3694 y Fu(;)14 b(w)1660 3706 y Ft(2)1697 3694 y Fu(;)g(w)1793 3706 y Ft(3)1831 3694 y Fw(\))40 b Fr(2)g Fv(C)2067 3664 y Ft(3)2142 3694 y Fw(obtained)d(from)g Fu(F)2752 3706 y Fs(`)2822 3694 y Fw(b)n(y)g(replacing)f(\()p Fu(x)3391 3706 y Ft(1)3429 3694 y Fu(;)14 b(x)3513 3706 y Ft(2)3551 3694 y Fu(;)g(x)3635 3706 y Ft(3)3673 3694 y Fw(\))37 b(b)n(y)523 3794 y(\()p Fu(w)614 3806 y Ft(1)652 3794 y Fu(;)14 b(w)748 3806 y Ft(2)786 3794 y Fu(;)g(w)882 3806 y Ft(3)919 3794 y Fw(\).)62 b(Let)37 b(us)e(denote)h(b)n(y)g Fv(S)1761 3806 y Ft(1)1834 3794 y Fw(the)h(assignmen)n(t)d Fu(f)46 b Fr(7!)2647 3773 y Fw(~)2628 3794 y Fu(F)2681 3806 y Fs(`)2713 3794 y Fw(.)62 b(No)n(w)36 b(let)g Fv(S)3177 3806 y Ft(2)3250 3794 y Fw(b)r(e)h(the)f(op)r(erator)523 3893 y(whic)n(h)e(assigns)f(to)1179 3872 y(~)1160 3893 y Fu(F)1213 3905 y Fs(`)1280 3893 y Fw(its)h(restriction)1825 3872 y(~)1806 3893 y Fu(F)1859 3905 y Fs(`)1892 3893 y Fr(j)1915 3905 y Fs(X)2012 3893 y Fw(to)g(the)h(space)e(X.)i(Then)f (\014nally)g(let)h(us)f(in)n(tro)r(duce)g(the)523 3993 y(transformation)26 b Fu(p)1131 4005 y Fn(\003)1197 3993 y Fw(whic)n(h)h(applied)h(to)1845 3972 y(~)1826 3993 y Fu(F)1879 4005 y Fs(`)1911 3993 y Fr(j)1934 4005 y Fs(X)2025 3993 y Fw(giv)n(es)f(us)g(a)g(function)i(in)f(the)f(space)g Fu(V)3239 4005 y Fs(`)3272 3993 y Fw(:)1451 4215 y Fu(p)1493 4227 y Fn(\003)1550 4194 y Fw(~)1531 4215 y Fu(F)1584 4227 y Fs(`)1617 4215 y Fr(j)1640 4227 y Fs(X)1703 4215 y Fw(\()p Fv(x)p Fw(\))d Fr(\021)1928 4102 y Fo(Z)1974 4291 y Fq(y)q Fn(2C)2096 4299 y Fa(x)2170 4194 y Fw(~)2151 4215 y Fu(F)2204 4227 y Fs(`)2237 4215 y Fr(j)2260 4227 y Fs(X)2323 4215 y Fw(\()p Fv(x)p Fu(;)14 b Fv(y)q Fw(\))p Fu(d\026)p Fw(\()p Fv(y)q Fw(\))885 b(\(5.28\))523 4458 y(where,)28 b(for)g(\014xed)h Fv(x)p Fw(,)g(the)g(in)n(tegration)e(is)i (on)f(the)h(unit)g(circle)f Fr(C)2569 4470 y Fq(x)2641 4458 y Fw(in)h(the)g(cotangen)n(t)e(space)h(of)g Fv(x)d Fr(2)g Fu(S)3793 4428 y Ft(2)523 4558 y Fw(and)i Fu(d\026)h Fw(is)g(the)g(usual)f(normalized)g(measure)f(on)i(that)f(circle.)648 4658 y(It)j(can)g(b)r(e)h(sho)n(wn)e(that)i(the)f(op)r(erator)f Fu(p)1973 4670 y Fn(\003)2011 4658 y Fv(S)2064 4670 y Ft(2)2101 4658 y Fv(S)2154 4670 y Ft(1)2221 4658 y Fw(comm)n(utes)h (with)h(all)f(the)h(rotation)e(op)r(erators)f(in)523 4757 y(the)36 b(represen)n(tation)d(of)j Fu(S)5 b(O)r Fw(\(3\))36 b(acting)e(on)i(the)f(space)g Fu(L)2407 4727 y Ft(2)2444 4757 y Fw(\()p Fu(S)2532 4727 y Ft(2)2569 4757 y Fw(\).)60 b(Th)n(us,)38 b(b)n(y)d(Sc)n(h)n(ur's)f(lemma,)j (there)523 4857 y(exist)28 b(a)f(constan)n(t)g Fu(\025)1172 4869 y Fs(`)1232 4857 y Fw(suc)n(h)g(that)1617 5037 y Fu(p)1659 5049 y Fn(\003)1697 5037 y Fv(S)1750 5049 y Ft(2)1787 5037 y Fv(S)1840 5049 y Ft(1)1878 5037 y Fu(f)k Fw(=)23 b Fu(\025)2086 5049 y Fs(`)2118 5037 y Fu(f)t(;)106 b(f)32 b Fr(2)23 b Fu(V)2491 5049 y Fs(`)2547 5037 y Fu(:)1048 b Fw(\(5.29\))523 5217 y(The)29 b(constan)n(t)f Fu(\025)1079 5229 y Fs(`)1141 5217 y Fw(can)g(b)r(e)i(ev)-5 b(aluated)29 b(b)n(y)f(considering)g(the)h(p)r(olynomial)g(\()p Fu(x)2988 5229 y Ft(1)3045 5217 y Fw(+)19 b Fu({x)3203 5229 y Ft(2)3240 5217 y Fw(\))3272 5187 y Fs(`)3334 5217 y Fw(and)29 b(the)g(p)r(oin)n(t)523 5330 y Fv(x)23 b Fw(=)g(\(1)p Fu(;)14 b Fw(0)p Fu(;)g Fw(0\))27 b(\(see)g(reference)1490 5300 y Ft(21)1560 5330 y Fw(\).)37 b(In)28 b(this)g(w)n(a)n(y)f(w)n(e)g (ha)n(v)n(e)f Fu(\025)2448 5342 y Fs(`)2504 5330 y Fw(=)2601 5289 y Ft(\(2)p Fs(`)p Fn(\000)p Ft(1\)!!)p 2601 5311 236 4 v 2696 5358 a Fs(`)p Ft(!)2846 5330 y Fw(.)648 5429 y(Since)33 b(the)g(elemen)n(ts)g(of)g Fr(X)1522 5441 y Fs(`)1588 5429 y Fw(are)f(homogeneous)f(p)r(olynomials)h(of)h (degree)f Fu(`)h Fw(in)g(the)h(quadric)e Fv(Q)3770 5441 y Ft(3)3807 5429 y Fw(,)523 5529 y(the)c(exp)r(osition)f(ab)r(o)n(v)n (e)f(for)h(the)h(op)r(erator)e Fu(p)1945 5541 y Fn(\003)1983 5529 y Fv(S)2036 5541 y Ft(2)2074 5529 y Fv(S)2127 5541 y Ft(1)2191 5529 y Fw(suggest)h(to)g(consider)g(the)h(follo)n(wing)f (op)r(erator:)648 5629 y(Let)40 b Fu(f)52 b Fr(2)44 b Fu(V)1049 5641 y Fs(`)1122 5629 y Fw(and)39 b(consider)g(the)i (function)f Fv(U)2197 5598 y Fn(\000)p Ft(1)2197 5649 y Fs(\033)2235 5657 y Fm(2)2287 5629 y Fu(f)49 b Fw(de\014ned)40 b(on)g Fv(Q)2875 5641 y Ft(3)2912 5629 y Fw(.)74 b(Let)40 b Fu(G)3235 5641 y Fs(`)3307 5629 y Fw(b)r(e)h(the)f(unique)523 5728 y(homogeneous)23 b(p)r(olynomial)h(of)h(order)e Fu(`)i Fw(in)g(three)f(complex)h(v)-5 b(ariables)23 b(\()p Fu(z)2858 5740 y Ft(1)2895 5728 y Fu(;)14 b(z)2971 5740 y Ft(2)3008 5728 y Fu(;)g(z)3084 5740 y Ft(3)3121 5728 y Fw(\))25 b(whic)n(h)g(is)f(harmonic)p eop %%Page: 32 32 32 31 bop 3747 324 a Fw(32)523 616 y(\(\001)624 631 y Fq(C)678 614 y Fm(3)715 616 y Fu(F)768 628 y Fs(`)839 616 y Fr(\021)943 554 y Fo(P)1030 574 y Ft(3)1030 641 y Fs(k)q Ft(=1)1197 583 y Fs(@)1236 558 y Fm(2)p 1179 597 108 4 v 1179 647 a Fs(@)t(z)1252 627 y Fm(2)1250 669 y Fp(k)1296 616 y Fu(F)1349 628 y Fs(`)1420 616 y Fw(=)39 b(0\))e(and)f(whose)h(restriction)f(to)h Fv(Q)2650 628 y Ft(3)2724 616 y Fw(is)g(equal)f(to)h Fv(U)3229 586 y Fn(\000)p Ft(1)3229 636 y Fs(\033)3267 644 y Fm(2)3319 616 y Fu(f)46 b Fw(\(denote)37 b(the)523 745 y(assignmen)n(t)c Fv(U)1030 715 y Fn(\000)p Ft(1)1030 765 y Fs(\033)1068 773 y Fm(2)1119 745 y Fu(f)42 b Fr(7!)33 b Fu(G)1383 757 y Fs(`)1449 745 y Fw(b)n(y)g Fv(T)p Fw(\).)56 b(Let)34 b(us)g(no)n(w)f(consider)f(the)i(op)r(erator)e Fu(p)3055 757 y Fn(\003)3093 745 y Fv(S)3146 757 y Ft(2)3183 745 y Fv(TU)3322 715 y Fn(\000)p Ft(1)3322 765 y Fs(\033)3360 773 y Fm(2)3412 745 y Fw(.)56 b(It)34 b(can)f(b)r(e)523 844 y(sho)n(wn)f(that)h(the)g(op)r(erator)f Fu(p)1496 856 y Fn(\003)1534 844 y Fv(S)1587 856 y Ft(2)1624 844 y Fv(TU)1763 814 y Fn(\000)p Ft(1)1763 865 y Fs(\033)1801 873 y Fm(2)1886 844 y Fw(also)g(comm)n(utes)g(with)h(all)g(the)g (rotation)f(op)r(erators)f(in)i(the)523 944 y(represen)n(tation)28 b(of)h Fu(S)5 b(O)r Fw(\(3\))30 b(acting)f(on)g Fu(L)1846 914 y Ft(2)1883 944 y Fw(\()p Fu(S)1971 914 y Ft(2)2008 944 y Fw(\).)42 b(Th)n(us,)30 b(b)n(y)f(Sc)n(h)n(ur's)g(lemma,)g(there) h(exist)f(a)g(constan)n(t)523 1044 y Fu(\026)573 1056 y Fs(`)633 1044 y Fw(suc)n(h)e(that)1547 1220 y Fu(p)1589 1232 y Fn(\003)1627 1220 y Fv(S)1680 1232 y Ft(2)1717 1220 y Fv(TU)1856 1186 y Fn(\000)p Ft(1)1856 1240 y Fs(\033)1894 1248 y Fm(2)1946 1220 y Fu(f)32 b Fw(=)22 b Fu(\026)2156 1232 y Fs(`)2188 1220 y Fu(f)t(;)107 b(f)31 b Fr(2)24 b Fu(V)2562 1232 y Fs(`)2617 1220 y Fu(:)978 b Fw(\(5.30\))523 1396 y(T)-7 b(o)24 b(ev)-5 b(aluate)25 b(the)g(constan)n(t)f Fu(\026)1487 1408 y Fs(`)1518 1396 y Fw(,)i(w)n(e)e(also)g(consider)f (the)i(p)r(olynomial)f(\()p Fu(x)2815 1408 y Ft(1)2866 1396 y Fw(+)13 b Fu({x)3018 1408 y Ft(2)3055 1396 y Fw(\))3087 1366 y Fs(`)3143 1396 y Fw(in)25 b Fu(V)3285 1408 y Fs(`)3342 1396 y Fw(and)g(the)g(p)r(oin)n(t)523 1496 y Fv(x)e Fw(=)g(\(1)p Fu(;)14 b Fw(0)p Fu(;)g Fw(0\),)27 b(to)g(obtain)h Fu(\026)1408 1508 y Fs(`)1463 1496 y Fw(=)1667 1463 y Ft(1)p 1560 1477 247 4 v 1560 1532 a Fs(`)p Ft(!)1607 1485 y Fn(p)p 1661 1485 145 3 v 1661 1532 a Ft(2)p Fs(`)p Ft(+1)1816 1496 y Fw(.)648 1607 y(Th)n(us)f(w)n(e)g(\014nally)g(ha)n(v)n(e,)g(for) g Fu(F)35 b Fr(2)23 b(X)1798 1619 y Fs(`)1831 1607 y Fw(:)1575 1817 y Fu(p)1617 1829 y Fn(\003)1655 1817 y Fv(S)1708 1829 y Ft(2)1745 1817 y Fv(T)p Fu(F)35 b Fw(=)2149 1761 y(1)p 1997 1798 346 4 v 1997 1882 a Fu(`)p Fw(!)2055 1815 y Fr(p)p 2124 1815 220 4 v 67 x Fw(2)p Fu(`)17 b Fw(+)h(1)2353 1817 y Fv(U)2426 1829 y Fs(\033)2464 1837 y Fm(2)2501 1817 y Fu(F)35 b(:)1006 b Fw(\(5.31\))648 2037 y(Therefore,)26 b(w)n(e)g(can)h(think)h(the)f(op)r(erator)e Fv(U)2088 2049 y Fs(\033)2126 2057 y Fm(2)2191 2037 y Fw(as)h(the)i(unitary-ization)e(of)h(the)g(op)r(erator)e Fu(p)3612 2049 y Fn(\003)3650 2037 y Fv(S)3703 2049 y Ft(2)3741 2037 y Fv(T)p Fw(.)523 2208 y(V.3)j Fv(B)746 2223 y Fs(S)790 2206 y Fm(3)858 2208 y Fv(as)k(a)g(coheren)m(t)g(state) g(transform.)648 2354 y Fw(This)27 b(section)g(is)h(the)g(analog)e(of)h (the)h(previous)f(section)g(IV.2)g(concerning)g(the)h(Bargmann)e(T)-7 b(rans-)523 2454 y(form)29 b Fv(B)789 2469 y Fs(S)833 2452 y Fm(3)870 2454 y Fw(.)42 b(Namely)-7 b(,)30 b(this)g(transform)f (can)g(also)f(b)r(e)i(written)g(as)f(a)g(coheren)n(t)g(state)g (transform)f(\(with)523 2554 y(the)c(coheren)n(t)e(states)i(de\014ned)f (in)h(the)g(same)f(w)n(a)n(y)f(as)h(ab)r(o)n(v)n(e\))g(and)g(w)n(e)g (also)f(ha)n(v)n(e)h(a)g(repro)r(ducing)f(k)n(ernel)523 2653 y(for)27 b(the)h(spaces)f Fr(Z)1108 2665 y Fs(n)1153 2653 y Fw(,)h Fr(E)1248 2665 y Ft(4)1313 2653 y Fw(and)f Fr(F)1534 2665 y Ft(4)1571 2653 y Fw(.)648 2753 y(F)-7 b(rom)27 b(the)h(de\014nition)g(of)f Fv(B)1540 2768 y Fs(S)1584 2751 y Fm(3)1648 2753 y Fw(\(2.47\))g(w)n(e)g(obtain:)523 2929 y Fv(Theorem)j(5.8)1064 3151 y(B)1132 3166 y Fs(S)1176 3149 y Fm(3)1213 3151 y Fw(\011\()p Fv(z)p Fw(\))23 b(=)1525 3047 y Fn(1)1498 3072 y Fo(X)1495 3248 y Fs(n)p Ft(=1)1644 3035 y Fr(p)p 1713 3035 50 4 v 60 x Fu(n)p Fw(2)1805 3065 y Fs(n)p Fn(\000)p Ft(1)p 1644 3132 291 4 v 1649 3208 a Fw(\()p Fu(n)c Fr(\000)f Fw(1\)!)1945 3151 y Fr(h)p Fw(\011)p Fu(;)c Fw(\010)2139 3166 y Fs(\032)2173 3174 y Fm(3)2205 3166 y Ft(\()p Fq(z)p Ft(\))p Fs(;n)p Fn(\000)p Ft(1)2440 3151 y Fr(i)2472 3166 y Fs(S)2516 3149 y Fm(3)2553 3151 y Fu(;)83 b Fw(\011)23 b Fr(2)g Fu(L)2882 3117 y Ft(2)2919 3151 y Fw(\()p Fu(S)3007 3117 y Ft(3)3044 3151 y Fw(\))h Fu(;)495 b Fw(\(5.32\))523 3409 y Fl(wher)l(e)30 b(the)g(c)l(oher)l(ent)g(states)g(in)g Fu(L)1618 3379 y Ft(2)1654 3409 y Fw(\()p Fu(S)1742 3379 y Ft(3)1779 3409 y Fw(\))g Fl(ar)l(e)h(de\014ne)l(d)f(in)f(the)h(same)h(way)f(as)g (b)l(efor)l(e)1214 3593 y Fw(\010)1274 3605 y Fs(\013;n)p Fn(\000)p Ft(1)1467 3593 y Fw(\()p Fv(x)p Fw(\))24 b(=)e(\()q Fv(x)d Fr(\017)e Fu(\013)p Fw(\))1939 3552 y Fs(n)p Fn(\000)p Ft(1)2083 3593 y Fu(;)83 b(\013)24 b Fr(2)f Fv(Q)2416 3605 y Ft(4)2472 3593 y Fr(\000)18 b(f)p Fw(0)p Fr(g)44 b Fu(n)23 b Fr(\025)g Fw(1)f Fu(:)645 b Fw(\(5.33\))648 3770 y(These)27 b(states)g(pro)n(vide)f(a)i(resolution)e(of)i(the)g (iden)n(tit)n(y:)648 3869 y(F)-7 b(or)27 b(\011)22 b Fr(2)i(M)1063 3881 y Fs(n)1107 3869 y Fw(,)k(the)g(follo)n(wing)f (equation)g(holds:)1013 4083 y(\011\()p Fv(x)p Fw(\))d(=)1480 4027 y Fu(n)p 1314 4064 383 4 v 1314 4140 a Fw(\(\()p Fu(n)18 b Fr(\000)g Fw(1\)!\))1658 4116 y Ft(2)1720 3970 y Fo(Z)1766 4159 y Fq(Q)1822 4167 y Fm(4)1854 4159 y Fn(\000f)p Ft(0)p Fn(g)2012 4083 y Fr(h)p Fw(\011)p Fu(;)c Fw(\010)2206 4095 y Fs(\013;n)p Fn(\000)p Ft(1)2398 4083 y Fr(i)2430 4098 y Fs(S)2474 4081 y Fm(3)2511 4083 y Fw(\010)2571 4095 y Fs(\013;n)p Fn(\000)p Ft(1)2764 4083 y Fw(\()p Fv(x)p Fw(\))p Fu(d\026)2971 4095 y Ft(4)3010 4083 y Fw(\()p Fu(\013)p Fw(\))24 b Fu(;)444 b Fw(\(5.34\))648 4316 y(This)27 b(equation)g(is)h(a)f(consequence)f(of)i(the)g(follo)n (wing)f(four)g(facts:)523 4416 y(\(a\))523 4576 y Fv(Prop)s(osition)j (5.9)41 b Fl(The)30 b(inner)g(pr)l(o)l(duct)g(of)h(two)f(c)l(oher)l (ent)f(states)h(in)g Fu(L)2863 4546 y Ft(2)2899 4576 y Fw(\()p Fu(S)2987 4546 y Ft(3)3025 4576 y Fw(\))g Fl(is:)1216 4792 y Fr(h)p Fw(\010)1308 4804 y Fs(\014)s(;k)1409 4792 y Fu(;)14 b Fw(\010)1506 4804 y Fs(\013;`)1601 4792 y Fr(i)1633 4807 y Fs(S)1677 4790 y Fm(3)1737 4792 y Fw(=)1981 4735 y(1)p 1834 4773 336 4 v 1834 4849 a(\()p Fu(k)22 b Fw(+)c(1\)2)2130 4825 y Fs(k)2194 4792 y Fw(\()p Fu(\013)h Fr(\017)f Fu(\014)t Fw(\))2441 4750 y Fs(k)2496 4792 y Fu(\016)2533 4804 y Fs(`;k)2621 4792 y Fu(;)83 b(k)26 b Fr(\025)d Fw(0)g Fu(:)646 b Fw(\(5.35\))523 5016 y(\(b\))28 b(the)g(in)n(v)-5 b(ariance)27 b(of)g(the)h(measure)f Fu(d\026)1851 5028 y Ft(4)1916 5016 y Fw(under)g(rotations)g Fu(R)c Fr(2)h Fu(S)5 b(O)r Fw(\(4\):)523 5177 y Fv(Prop)s(osition)30 b(5.10)40 b Fl(L)l(et)g Fu(R)i Fr(2)f Fu(S)5 b(O)r Fw(\(4\))q Fl(.)68 b(for)41 b(any)f Fu(d\026)2331 5189 y Ft(4)2369 5177 y Fr(\000)p Fl(inte)l(gr)l(able)g(function)f Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))41 b Fl(the)f(fol)t(lowing)523 5277 y(e)l(quation)30 b(holds:)1258 5378 y Fo(Z)1304 5567 y Fq(Q)1360 5575 y Fa(4)1396 5567 y Fn(\000f)p Ft(0)p Fn(g)1567 5491 y Fu(f)9 b Fw(\()p Fu(R)q(\013)p Fw(\))p Fu(d\026)1891 5503 y Ft(4)1929 5491 y Fw(\()p Fu(\013)p Fw(\))24 b(=)2158 5378 y Fo(Z)2204 5567 y Fq(Q)2260 5575 y Fa(4)2296 5567 y Fn(\000f)p Ft(0)p Fn(g)2467 5491 y Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))p Fu(d\026)2727 5503 y Ft(4)2765 5491 y Fw(\()p Fu(\013)p Fw(\))24 b Fu(:)689 b Fw(\(5.36\))523 5728 y(\(c\))p eop %%Page: 33 33 33 32 bop 3747 324 a Fw(33)523 614 y Fv(Prop)s(osition)30 b(5.11)40 b Fl(The)31 b(Bar)l(gmann)f(tr)l(ansform)g Fv(B)2273 629 y Fs(S)2317 613 y Fm(3)2383 614 y Fl(maps)h(the)f(c)l (oher)l(ent)g(states)f(as)h(fol)t(lows:)1263 848 y Fv(B)1331 863 y Fs(S)1375 846 y Fm(3)1412 848 y Fw(\010)1472 860 y Fs(\014)s(;k)1573 848 y Fw(\()p Fv(z)p Fw(\))24 b(=)1851 724 y Fr(p)p 1920 724 189 4 v 68 x Fu(k)e Fw(+)c(1)p 1801 829 359 4 v 1801 905 a(\()p Fu(k)j Fw(+)d(1\)!2)2119 881 y Fs(k)2183 848 y Fw(\()q Fu(\032)2259 860 y Ft(3)2296 848 y Fw(\()p Fv(z)p Fw(\))h Fr(\017)f Fu(\014)t Fw(\))2565 806 y Fs(k)2620 848 y Fu(;)83 b(k)25 b Fr(\025)e Fw(0)694 b(\(5.37\))523 1080 y Fl(and)30 b(then)g(the)g(image)h(of)f(the)g(c)l (oher)l(ent)g(states)f Fw(\010)2107 1092 y Fs(\014)s(;k)2238 1080 y Fl(under)h(the)g(unitary)g(tr)l(ansformation)g Fv(U)3544 1050 y Fn(\000)p Ft(1)3544 1101 y Fs(\033)3582 1109 y Fm(3)3663 1080 y Fl(is)1295 1326 y Fv(U)1368 1292 y Fn(\000)p Ft(1)1368 1347 y Fs(\033)1406 1355 y Fm(3)1457 1326 y Fw(\010)1517 1338 y Fs(\014)s(;k)1618 1326 y Fw(\()p Fu(\013)p Fw(\))25 b(=)1908 1203 y Fr(p)p 1977 1203 189 4 v 67 x Fu(k)c Fw(+)d(1)p 1857 1307 359 4 v 1857 1383 a(\()p Fu(k)k Fw(+)c(1\)!2)2176 1359 y Fs(k)2240 1326 y Fw(\()p Fu(\013)h Fr(\017)f Fu(\014)t Fw(\))2487 1285 y Fs(k)2542 1326 y Fu(;)83 b(k)26 b Fr(\025)d Fw(0)f Fu(:)726 b Fw(\(5.38\))523 1554 y(\(d\))523 1716 y Fv(Theorem)30 b(5.12)41 b Fl(F)-6 b(or)30 b(e)l(ach)i(inte)l(ger)f(numb)l(er)e Fu(n)c Fr(\025)f Fw(1)p Fl(,)31 b(the)f(function)h Fu(K)2854 1728 y Fq(Q)2910 1736 y Fa(4)2945 1728 y Fs(;n)3035 1716 y Fw(:)24 b Fv(Q)3154 1728 y Fq(4)3215 1716 y Fr(\002)18 b Fv(Q)3370 1728 y Fq(4)3436 1716 y Fr(7!)25 b Fv(C)30 b Fl(given)523 1816 y(by)1296 2029 y Fu(K)1367 2041 y Fq(Q)1423 2049 y Fa(4)1459 2041 y Fs(;n)p Fn(\000)p Ft(1)1608 2029 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))25 b Fr(\021)2192 1973 y Fw(1)p 1935 2010 554 4 v 1935 2086 a(\(\()p Fu(n)19 b Fr(\000)f Fw(1\)!\))2280 2062 y Ft(2)2318 2086 y Fw(2)2360 2062 y Fs(n)p Fn(\000)p Ft(1)2513 2029 y Fw(\()p Fu(\014)23 b Fr(\017)18 b Fu(\013)p Fw(\))2761 1987 y Fs(n)p Fn(\000)p Ft(1)3618 2029 y Fw(\(5.39\))523 2252 y Fl(is)30 b(a)g(r)l(epr)l(o)l (ducing)h(kernel)f(for)h(the)f(sp)l(ac)l(e)g Fr(Z)1924 2264 y Fs(n)1969 2252 y Fl(,)g(that)g(is:)1118 2355 y Fo(Z)1164 2544 y Fq(Q)1220 2552 y Fa(4)1256 2544 y Fn(\000f)p Ft(0)p Fn(g)1427 2468 y Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))p Fu(K)1665 2480 y Fq(Q)1721 2488 y Fa(4)1757 2480 y Fs(;n)p Fn(\000)p Ft(1)1907 2468 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))p Fu(d\026)2205 2480 y Ft(4)2243 2468 y Fw(\()p Fu(\013)p Fw(\))24 b(=)f Fu(f)9 b Fw(\()p Fu(\014)t Fw(\))p Fu(;)107 b(f)31 b Fr(2)24 b(Z)2978 2480 y Fs(n)3046 2468 y Fu(:)549 b Fw(\(5.40\))648 2703 y(The)31 b(pro)r(ofs)h(of)f(these)h (facts)g(are)f(similar)g(to)h(those)f(giv)n(en)g(in)h(section)g(IV.2.) 49 b(W)-7 b(e)33 b(just)f(p)r(oin)n(t)g(out)523 2803 y(that)c(to)f(pro)n(v)n(e)e(equation\(5.36\))h(w)n(e)h(need)h(to)f(use) g(the)h(fact)f(that)h(for)f(an)n(y)f(rotation)g Fu(R)e Fr(2)g Fu(S)5 b(O)r Fw(\(4\))27 b(there)523 2902 y(exist)g(a)f(couple)h (of)g(matrices)f(U)i(and)e(V)i(in)f Fu(S)5 b(U)k Fw(\(2\))27 b(suc)n(h)f(that)i Fu(R)q(\032)2635 2914 y Ft(3)2671 2902 y Fw(\()p Fv(z)p Fw(\))c(=)f Fu(\032)2932 2914 y Ft(2)2969 2902 y Fw(\()p Fu(M)3082 2914 y Fs(U)o(;V)3204 2902 y Fv(z)p Fw(\))28 b(where)e Fu(M)3626 2914 y Fs(U)o(;V)3774 2902 y Fw(is)523 3002 y(a)h(4)18 b Fr(\002)g Fw(4)27 b(matrix)g(with)i(the)f(2)17 b Fr(\002)i Fw(2)27 b(matrices)f(U)i(and)g (V)g(on)f(the)h(diagonal)e(and)i(zeros)e(elsewhere.)648 3101 y(Theorem)g(5.12)h(pro)n(vides)f(us)h(with)h(a)g(repro)r(ducing)e (k)n(ernel)h(for)g(the)h(space)f Fr(E)3107 3113 y Ft(4)3144 3101 y Fw(:)523 3280 y Fv(Theorem)j(5.13)41 b Fl(The)30 b(function)1300 3520 y Fu(K)1371 3532 y Fq(Q)1427 3540 y Fa(4)1466 3520 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))24 b Fr(\021)1812 3417 y Fn(1)1786 3442 y Fo(X)1783 3617 y Fs(n)p Ft(=1)2188 3464 y Fw(1)p 1932 3501 V 1932 3577 a(\(\()p Fu(n)19 b Fr(\000)f Fw(1\)!\))2277 3553 y Ft(2)2314 3577 y Fw(2)2356 3553 y Fs(n)p Fn(\000)p Ft(1)2510 3520 y Fw(\()p Fu(\014)23 b Fr(\017)18 b Fu(\013)p Fw(\))2757 3479 y Fs(n)p Fn(\000)p Ft(1)3618 3520 y Fw(\(5.41\))523 3775 y Fl(b)l(elongs)30 b(to)g(the)g(sp)l(ac)l(e)g Fr(E)1305 3787 y Ft(4)1372 3775 y Fl(\(for)h Fu(\013)f Fl(\014xe)l(d\))f(and)h (it)g(is)g(a)g(r)l(epr)l(o)l(ducing)h(kernel)f(of)h(this)f(sp)l(ac)l (e:)1203 3883 y Fo(Z)1249 4072 y Fq(Q)1305 4080 y Fa(4)1341 4072 y Fn(\000f)p Ft(0)p Fn(g)1512 3996 y Fu(f)9 b Fw(\()p Fu(\013)p Fw(\))p Fu(K)1750 4008 y Fq(Q)1806 4016 y Fa(4)1846 3996 y Fw(\()p Fu(\013;)14 b(\014)t Fw(\))p Fu(d\026)2144 4008 y Ft(4)2183 3996 y Fw(\()p Fu(\013)p Fw(\))24 b(=)e Fu(f)9 b Fw(\()p Fu(\014)t Fw(\))p Fu(;)107 b(f)32 b Fr(2)23 b(E)2901 4008 y Ft(4)2961 3996 y Fu(:)634 b Fw(\(5.42\))648 4231 y(F)-7 b(rom)27 b(this)h(last)f(theorem)g(and)g(the)h(map)g Fu(\032)2040 4243 y Ft(3)2105 4231 y Fw(w)n(e)f(obtain)g(the)h(follo)n (wing)523 4394 y Fv(Theorem)i(5.14)41 b Fl(The)30 b(function)984 4613 y Fw(~)962 4634 y Fu(K)1033 4646 y Fq(Q)1089 4654 y Fa(4)1128 4634 y Fw(\()p Fv(z)p Fu(;)14 b Fv(w)q Fw(\))25 b Fr(\021)1483 4531 y Fn(1)1456 4555 y Fo(X)1453 4731 y Fs(n)p Ft(=1)1773 4578 y Fw(1)p 1602 4615 383 4 v 1602 4691 a(\(\()p Fu(n)19 b Fr(\000)f Fw(1\)!\))1947 4667 y Ft(2)2008 4634 y Fw(\()q(\()p 2073 4589 97 4 v Fu(w)2132 4646 y Ft(1)2170 4634 y Fu(z)2209 4646 y Ft(1)2264 4634 y Fw(+)p 2347 4589 V 18 w Fu(w)2406 4646 y Ft(2)2444 4634 y Fu(z)2483 4646 y Ft(2)2519 4634 y Fw(\)\()p 2583 4589 V Fu(w)2642 4646 y Ft(3)2681 4634 y Fu(z)2720 4646 y Ft(3)2775 4634 y Fw(+)p 2858 4589 V 18 w Fu(w)2917 4646 y Ft(4)2955 4634 y Fu(z)2994 4646 y Ft(4)3031 4634 y Fw(\)\))3095 4593 y Fs(n)p Fn(\000)p Ft(1)3618 4634 y Fw(\(5.43\))523 4889 y Fl(b)l(elongs)30 b(to)g(the)g(sp)l(ac)l(e)g Fr(F)1321 4901 y Ft(4)1388 4889 y Fl(\(for)h Fv(w)g Fl(\014xe)l(d\))e (and)h(it)g(is)g(a)g(r)l(epr)l(o)l(ducing)h(kernel)f(of)g(this)g(sp)l (ac)l(e:)1272 4997 y Fo(Z)1318 5186 y Fq(C)1372 5194 y Fa(4)1426 5110 y Fu(f)9 b Fw(\()p Fv(w)q Fw(\))1633 5089 y(~)1610 5110 y Fu(K)1681 5122 y Fq(Q)1737 5130 y Fa(4)1777 5110 y Fw(\()p Fv(z)p Fu(;)14 b Fv(w)q Fw(\))p Fu(d\027)2074 5122 y Ft(4)2113 5110 y Fw(\()p Fv(w)q Fw(\))24 b(=)e Fu(f)9 b Fw(\()p Fv(z)p Fw(\))p Fu(;)107 b(f)32 b Fr(2)23 b(F)2855 5122 y Ft(4)2892 5110 y Fu(:)703 b Fw(\(5.44\))523 5338 y Fv(REMARK)30 b(3)648 5438 y Fw(V.)c(Bargmann)1138 5408 y Ft(20)1232 5438 y Fw(also)f(considers)f(a) h(transformation)f Fu(A)g Fw(:)f Fr(E)2558 5450 y Ft(4)2618 5438 y Fr(7!)g(B)2779 5450 y Ft(4)2842 5438 y Fw(whic)n(h)i(is)h(v)n (ery)e(similar)h(to)h(our)523 5537 y(transformation)g Fv(U)1162 5507 y Fn(\000)p Ft(1)1162 5558 y Fs(\032)1196 5566 y Fm(3)1275 5537 y Fw(:)d Fr(E)1365 5549 y Ft(4)1425 5537 y Fr(7!)g(F)1591 5549 y Ft(4)1656 5537 y Fw(giv)n(en)j(b)n(y)1536 5728 y Fv(U)1609 5694 y Fn(\000)p Ft(1)1609 5749 y Fs(\032)1643 5757 y Fm(3)1699 5728 y Fw(\011\()p Fv(z)p Fw(\))e(=)e(\011\()p Fu(\032)2121 5740 y Ft(3)2158 5728 y Fw(\()p Fv(z)p Fw(\)\))p Fu(;)85 b Fw(\011)22 b Fr(2)i(E)2614 5740 y Ft(4)3618 5728 y Fw(\(5.45\))p eop %%Page: 34 34 34 33 bop 3747 324 a Fw(34)523 614 y(V.)24 b(Bargmann)f(uses)g(a)h (sligh)n(tly)f(di\013eren)n(t)h(expression)f(for)g(the)i(map)f Fu(\032)2767 626 y Ft(3)2804 614 y Fw(.)35 b(He)25 b(pro)n(v)n(es)d (that)i Fu(A)g Fw(preserv)n(es)523 714 y(the)36 b(inner)g(pro)r(duct)g (and)g(that)g(the)g(image)f(of)h Fr(E)2119 726 y Ft(4)2192 714 y Fw(under)g(the)h(map)e Fu(A)i Fw(is)e(the)i(set:)53 b Fu(A)p Fw(\()p Fr(E)3438 726 y Ft(4)3476 714 y Fw(\))37 b(=)g Fr(f)p Fu(f)45 b Fr(2)523 814 y(B)578 826 y Ft(4)615 814 y Fr(j)p Fu(M)9 b(f)31 b Fw(=)23 b Fu(f)9 b Fr(g)27 b Fw(where)g Fu(M)9 b(f)35 b Fw(is)28 b(the)g(mean)f(v)-5 b(alue)28 b(of)f Fu(f)9 b Fw(:)691 1043 y Fu(M)g(f)g Fw(\()p Fu(z)902 1055 y Ft(1)939 1043 y Fu(;)14 b(z)1015 1055 y Ft(2)1051 1043 y Fu(;)g(z)1127 1055 y Ft(3)1164 1043 y Fu(;)g(z)1240 1055 y Ft(4)1277 1043 y Fw(\))23 b Fr(\021)1420 930 y Fo(Z)1503 951 y Fs(\031)1466 1119 y Ft(0)1562 1043 y Fu(f)9 b Fw(\()p Fu(z)1683 1055 y Ft(1)1733 1043 y Fw(exp\()p Fu({ )s Fw(\))p Fu(;)14 b(z)2084 1055 y Ft(2)2135 1043 y Fw(exp\()p Fu({ )s Fw(\))p Fu(;)g(z)2486 1055 y Ft(3)2537 1043 y Fw(exp\()p Fr(\000)p Fu({ )s Fw(\))p Fu(;)g(z)2953 1055 y Ft(4)3003 1043 y Fw(exp\()p Fr(\000)p Fu({ )s Fw(\)\))3385 987 y Fu(d )p 3385 1024 101 4 v 3389 1100 a Fw(2)p Fu(\031)3618 1043 y Fw(\(5.46\))648 1268 y(This)27 b(motiv)-5 b(ates)28 b(to)f(establish)g(the)h(follo)n (wing)523 1434 y Fv(Prop)s(osition)i(5.15)40 b Fl(The)31 b(sp)l(ac)l(es)g Fu(A)p Fw(\()p Fr(E)1809 1446 y Ft(4)1846 1434 y Fw(\))f Fl(and)g Fr(F)2129 1446 y Ft(4)2196 1434 y Fl(ar)l(e)g(the)g(same.)523 1600 y Fw(One)d(can)h(pro)n(v)n(e)d(this) j(prop)r(osition)f(b)n(y)g(follo)n(wing)g(the)h(pro)r(of)f(of)g(prop)r (osition)g(\(2.1\).)648 1700 y(As)22 b(noted)h(b)n(y)f(V.)h(Bargmann) 1588 1669 y Ft(20)1657 1700 y Fw(,)g(the)g(function)2184 1679 y(~)2161 1700 y Fu(K)2232 1712 y Fq(Q)2288 1720 y Fa(4)2351 1700 y Fw(can)f(b)r(e)h(expressed)e(in)i(terms)f(of)h(the)g (mo)r(di\014ed)523 1819 y(Bessel)k(function)h(of)g(the)g(\014rst)f (kind)h Fu(I)1729 1831 y Ft(0)1766 1819 y Fw(\()p Fu(z)t Fw(\))23 b Fr(\021)1984 1757 y Fo(P)2072 1777 y Fn(1)2072 1844 y Fs(m)p Ft(=0)2307 1786 y(1)p 2243 1800 162 4 v 2243 1848 a(\()p Fs(m)p Ft(!\))2373 1831 y Fm(2)2428 1752 y Fo(\000)2476 1786 y Fs(z)p 2476 1800 35 4 v 2477 1848 a Ft(2)2520 1752 y Fo(\001)2558 1769 y Ft(2)p Fs(m)2655 1819 y Fw(:)1185 2028 y(~)1163 2049 y Fu(K)1234 2061 y Fq(Q)1290 2069 y Fa(4)1329 2049 y Fw(\()p Fv(z)p Fu(;)14 b Fv(w)q Fw(\))24 b(=)f Fu(I)1690 2061 y Ft(0)1742 1957 y Fo(\020)1791 2049 y Fw(2)1833 1974 y Fo(p)p 1916 1974 1023 4 v 75 x Fw(\()p 1948 2003 97 4 v Fu(w)2007 2061 y Ft(1)2045 2049 y Fu(z)2084 2061 y Ft(1)2139 2049 y Fw(+)p 2222 2003 V 18 w Fu(w)2281 2061 y Ft(2)2319 2049 y Fu(z)2358 2061 y Ft(2)2395 2049 y Fw(\)\()p 2459 2003 V Fu(w)2518 2061 y Ft(3)2556 2049 y Fu(z)2595 2061 y Ft(3)2650 2049 y Fw(+)p 2733 2003 V 18 w Fu(w)2792 2061 y Ft(4)2830 2049 y Fu(z)2869 2061 y Ft(4)2906 2049 y Fw(\))2938 1957 y Fo(\021)3001 2049 y Fu(:)594 b Fw(\(5.47\))523 2268 y Fv(REMARK)33 b(4)d Fw(W)-7 b(e)31 b(can)f(also)f(relate)g(our)h (transformation)e Fv(U)2574 2280 y Fs(\033)2612 2288 y Fm(3)2677 2268 y Fw(:)g Fr(E)2772 2280 y Ft(4)2836 2268 y Fr(7!)g Fu(L)3004 2238 y Ft(\(2\))3092 2268 y Fw(\()p Fu(S)3180 2238 y Ft(3)3217 2268 y Fw(\))j(with)g(the)f(trans-) 523 2368 y(formation)d Fu(p)946 2380 y Fn(\003)1011 2368 y Fw(of)h(V.)g(Guillemin)1574 2338 y Ft(21)1645 2368 y Fw(:)37 b(F)-7 b(or)27 b Fu(F)35 b Fr(2)23 b(Z)2080 2380 y Fs(n)1663 2585 y Fu(p)1705 2597 y Fn(\003)1743 2585 y Fv(S)1796 2597 y Ft(2)1834 2585 y Fv(T)p Fu(F)35 b Fw(=)2172 2529 y(1)p 2086 2566 215 4 v 2086 2642 a Fu(n)p Fw(!)2181 2582 y Fr(p)p 2251 2582 50 4 v 2251 2642 a Fu(n)2310 2585 y Fv(U)2383 2597 y Fs(\033)2421 2605 y Fm(3)2459 2585 y Fu(F)1106 b Fw(\(5.48\))523 2811 y(where)28 b(the)g(op)r(erators)e Fv(S)1328 2823 y Ft(2)1393 2811 y Fw(and)i Fv(T)h Fw(are)e(de\014ned)h(in)g(a)g(similar)f(w)n(a)n (y)g(as)g(ab)r(o)n(v)n(e.)37 b(The)28 b(pro)r(of)f(of)h(this)h(last)523 2911 y(equation)d(follo)n(ws)f(the)i(same)f(steps)h(as)e(the)i(one)f (to)h(pro)n(v)n(e)e(equation)h(\(5.31\).)35 b(Th)n(us)27 b(w)n(e)f(can)g(think)h(the)523 3010 y(op)r(erator)f Fv(U)931 3022 y Fs(\033)969 3030 y Fm(3)1034 3010 y Fw(as)h(the)h (unitary-ization)e(of)i Fu(p)1981 3022 y Fn(\003)2019 3010 y Fv(S)2072 3022 y Ft(2)2109 3010 y Fv(T)p Fw(.)523 3346 y Fv(A)m(CKNO)m(WLEDGEMENTS)648 3446 y Fw(W)-7 b(e)31 b(thank)g(the)g(follo)n(wing)f(p)r(eople)i(for)e(v)n(ery)g (illuminating)h(discussions)f(and)h(imp)r(ortan)n(t)g(sugges-)523 3545 y(tions:)40 b(Stephan)30 b(De)g(Bievre,)f(Amine)h(El)f(Gradec)n (hi,)g(Brian)g(C.)h(Hall,)f(Stephen)i(B.)e(Son)n(tz,)h(La)n(wrence)523 3645 y(E.)d(Thomas,)g(Alejandro)g(Urib)r(e-Ah)n(umada,)g(Alb)r(erto)h (V)-7 b(erjo)n(vsky-Sola.)648 3745 y(This)21 b(article)f(w)n(as)g (partially)g(supp)r(orted)h(b)n(y)g(the)g(gran)n(ts)f(P)-7 b(API)r(IT-UNAM)21 b(IN102998)d(and)j(CONA-)523 3844 y(CYT)28 b(32146-E.)p eop %%Page: 35 35 35 34 bop 3747 324 a Fw(35)1921 614 y Fv(App)s(endix)31 b(A)1198 774 y(The)h(Kernel)f(of)h(the)g(Bargmann)f(transform)g(for)h Fu(S)3091 744 y Ft(2)3128 774 y Fv(.)648 1190 y Fw(The)d(purp)r(ose)f (of)h(this)h(app)r(endix)f(is)g(to)g(pro)n(v)n(e)e(that)j(the)f(k)n (ernel)f Fu(K)2810 1205 y Fs(S)2854 1188 y Fm(2)2890 1190 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))31 b(can)d(b)r(e)i (expressed)e(as)523 1290 y(a)f(p)r(o)n(w)n(er)g(series)f(of)i(the)g (function)g(\010)1680 1305 y Fs(S)1724 1288 y Fm(2)1760 1290 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\).)39 b(That)27 b(is)1258 1535 y Fu(K)1329 1550 y Fs(S)1373 1533 y Fm(2)1409 1535 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)1741 1431 y Fn(1)1714 1456 y Fo(X)1714 1635 y Fs(k)q Ft(=0)1858 1412 y Fr(p)p 1928 1412 231 4 v 1928 1479 a Fw(2)p Fu(k)c Fw(+)e(1)p 1858 1516 300 4 v 1974 1592 a Fu(k)s Fw(!)2182 1418 y Fo(\022)2287 1479 y Fw(1)p 2253 1516 111 4 v 2253 1533 a Fr(p)p 2322 1533 42 4 v 68 x Fw(3)2373 1418 y Fo(\023)2434 1435 y Fs(k)2489 1535 y Fw(\(\010)2581 1550 y Fs(S)2625 1533 y Fm(2)2662 1535 y Fw(\()p Fv(x)p Fu(;)c Fv(z)p Fw(\)\))2888 1493 y Fs(k)523 1786 y Fw(W)-7 b(e)28 b(w)n(an)n(t)f(to)g(do)h(it)g(b)n(y)f(sho)n(wing)f(that)i(the)g(follo)n (wing)f(equation)g(holds)g(for)g(all)h Fu(k)d Fr(\025)e Fw(0:)1342 2027 y Fr(K)1405 2039 y Fs(k)1446 2027 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)1761 1903 y Fr(p)p 1830 1903 231 4 v 68 x Fw(2)p Fu(k)d Fw(+)d(1)p 1761 2008 300 4 v 1876 2084 a Fu(k)s Fw(!)2084 1910 y Fo(\022)2190 1971 y Fw(1)p 2155 2008 111 4 v 2155 2024 a Fr(p)p 2224 2024 42 4 v 69 x Fw(3)2276 1910 y Fo(\023)2337 1927 y Fs(k)2392 2027 y Fw(\(\010)2484 2042 y Fs(S)2528 2025 y Fm(2)2564 2027 y Fw(\))2597 1985 y Fs(k)2651 2027 y Fw(\()p Fv(x)p Fu(;)c Fv(z)p Fw(\))523 2248 y(with)1528 2485 y Fr(K)1591 2497 y Fs(k)1632 2485 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))24 b(=)2018 2381 y Fs(k)1976 2406 y Fo(X)1937 2585 y Fs(m)p Ft(=)p Fn(\000)p Fs(k)2149 2485 y Fu(Y)2216 2451 y Fn(\003)2197 2505 y Fs(k)q(m)2297 2485 y Fw(\()p Fv(x)p Fw(\))p Fu(y)2452 2497 y Fs(k)q(m)2552 2485 y Fw(\()p Fv(z)p Fw(\))648 2747 y(The)30 b(main)h(idea)f(of)h(the) g(pro)r(of)f(is)h(based)f(on)g(the)h(fact)g(that)g(the)g(follo)n(wing)f (function)h Fu(G)3502 2759 y Fs(k)3543 2747 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))30 b Fr(\021)523 2858 y(K)586 2870 y Fs(k)627 2858 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))p Fu(=c)p Fw(\()p Fu(k)s Fw(\))g(\()q(\010)1115 2873 y Fs(S)1159 2856 y Fm(2)1196 2858 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\)\))1422 2816 y Fs(k)1502 2858 y Fw(\(with)39 b Fu(c)p Fw(\()p Fu(k)s Fw(\))g(some)f(constan)n(t\))g(satis\014es)f (an)h(homogeneous)f(system)h(of)523 2958 y(\014rst)58 b(order)28 b(di\013eren)n(tial)h(equations)f(with)i(initial)f (condition)g Fu(G)2597 2970 y Fs(k)2638 2958 y Fw(\()p Fv(x)2720 2970 y Ft(0)2758 2958 y Fu(;)14 b Fv(z)2837 2970 y Ft(0)2875 2958 y Fw(\))25 b(=)h(1.)41 b(Then)29 b(w)n(e)g(sho)n(w)f(that)523 3057 y(the)36 b(gradien)n(t)f(of)g Fu(G)1175 3069 y Fs(k)1252 3057 y Fw(m)n(ust)h(b)r(e)h(zero,)f(whic)n (h)g(implies)g(that)g(the)g(solution)f(of)h(the)g(system)g(m)n(ust)g(b) r(e)523 3157 y Fu(G)588 3169 y Fs(k)629 3157 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))25 b Fr(\021)d Fw(1,)27 b(i.e.)37 b Fr(K)1232 3169 y Fs(k)1273 3157 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))25 b(=)d Fu(c)p Fw(\()p Fu(k)s Fw(\))14 b(\()q(\010)1831 3172 y Fs(S)1875 3155 y Fm(2)1911 3157 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\)\))2138 3115 y Fs(k)2179 3157 y Fw(.)37 b(Next,)28 b(w)n(e)f(ev)-5 b(aluate)27 b(the)h(constan)n(t)f Fu(c)p Fw(\()p Fu(k)s Fw(\).)648 3257 y(W)-7 b(e)33 b(kno)n(w)f(that)h(the)g(Bargmann)e (transform)h Fv(B)2216 3272 y Fs(S)2260 3255 y Fm(2)2329 3257 y Fw(in)n(tert)n(wines)g(the)h(op)r(erators)e Fu(L)3337 3269 y Fs(i)3397 3257 y Fw(and)i Fu(L)3621 3226 y Fn(0)3621 3278 y Fs(i)3648 3257 y Fw(,)h Fu(i)e Fw(=)523 3356 y(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3.)35 b(Th)n(us)28 b(for)f(an)n(y)g(smo)r (oth)g(function)h(\011\()p Fv(x)p Fw(\))1743 3532 y Fu(L)1800 3498 y Fn(0)1800 3553 y Fs(i)1828 3532 y Fv(B)1896 3547 y Fs(S)1940 3530 y Fm(2)1976 3532 y Fw(\011)23 b(=)f Fv(B)2219 3547 y Fs(S)2263 3530 y Fm(2)2303 3532 y Fw( )-27 b(L)2352 3544 y Fs(i)2379 3532 y Fw(\011)523 3708 y(No)n(w)27 b(w)n(e)g(w)n(an)n(t)g(to)h(sho)n(w)e(that)i(for)f(an)n(y)g Fu(k)f Fr(\025)d Fw(0)1486 3884 y(\()p Fu(L)1575 3896 y Fs(i)1621 3884 y Fw(+)18 b Fu(L)1761 3850 y Fn(0)1761 3905 y Fs(i)1788 3884 y Fw(\))p Fr(K)1883 3896 y Fs(k)1925 3884 y Fw(\()p Fv(x)p Fu(;)c Fv(z)p Fw(\))24 b(=)f(0)91 b Fu(i)23 b Fw(=)g(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)648 4060 y(Let)40 b Fu(P)862 4072 y Fs(k)943 4060 y Fw(b)r(e)g(the)h(pro)5 b(jector)38 b(on)i(the)g(span)g(of)g Fr(f)p Fu(y)2274 4072 y Fs(k)q(m)2373 4060 y Fw(\()p Fv(z)p Fw(\))p Fu(=)k(m)g Fw(=)f Fr(\000)p Fu(k)s(;)14 b(:)g(:)g(:)f(;)h(k)s Fr(g)p Fw(.)73 b Fu(P)3322 4072 y Fs(k)3404 4060 y Fw(acts)39 b(on)h(the)523 4160 y(Bargmann)26 b(Space)h Fu(B)1226 4172 y Ft(2)1263 4160 y Fw(.)37 b(Th)n(us)28 b(w)n(e)f(ha)n(v)n(e)f (for)h(an)n(y)g(smo)r(oth)h(function)g(\011\()p Fv(x)p Fw(\))1371 4378 y Fu(P)1424 4390 y Fs(k)1465 4378 y Fv(B)1533 4393 y Fs(S)1577 4376 y Fm(2)1613 4378 y Fw(\011\()p Fv(z)p Fw(\))c(=)1896 4265 y Fo(Z)1942 4454 y Fq(x)p Fn(2)p Fs(S)2071 4437 y Fm(2)2120 4378 y Fr(K)2183 4390 y Fs(k)2224 4378 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\)\011\()p Fv(x)p Fw(\))p Fu(d)p Fw(\012\()p Fv(x)p Fw(\))523 4600 y(whic)n(h)28 b(implies)1287 4814 y Fu(P)1340 4826 y Fs(k)1381 4814 y Fv(B)1449 4829 y Fs(S)1493 4812 y Fm(2)1529 4814 y Fu(L)1586 4826 y Fs(i)1613 4814 y Fw(\011\()p Fv(z)p Fw(\))c(=)1896 4701 y Fo(Z)1942 4890 y Fq(x)p Fn(2)p Fs(S)2071 4873 y Fm(2)2120 4814 y Fr(K)2183 4826 y Fs(k)2224 4814 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))p Fu(L)2474 4826 y Fs(i)2503 4814 y Fw(\011\()p Fv(x)p Fw(\))p Fu(d)p Fw(\012\()p Fv(x)p Fw(\))1679 5041 y(=)22 b Fr(\000)1845 4928 y Fo(Z)1891 5116 y Fq(x)p Fn(2)p Fs(S)2020 5100 y Fm(2)2056 5041 y Fw(\()p Fu(L)2145 5053 y Fs(i)2172 5041 y Fr(K)2235 5053 y Fs(k)2276 5041 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\)\)\011\()p Fv(x)p Fw(\))p Fu(d)p Fw(\012\()p Fv(x)p Fw(\))523 5263 y(where)27 b(w)n(e)g(ha)n(v)n(e)g(used)g(the)h(selfadjoin)n(tness)f(of)h Fu(L)2109 5275 y Fs(i)2136 5263 y Fw(.)648 5362 y(In)g(the)g(other)f (hand,)h(if)g(w)n(e)g(apply)f Fu(P)1823 5374 y Fs(k)1892 5362 y Fw(to)h(b)r(oth)g(sides)g(of)g(the)g(equation)f Fu(L)3024 5332 y Fn(0)3024 5384 y Fs(i)3051 5362 y Fv(B)3119 5377 y Fs(S)3163 5361 y Fm(2)3200 5362 y Fw(\011)c(=)g Fv(B)3444 5377 y Fs(S)3488 5361 y Fm(2)3528 5362 y Fw( )-27 b(L)3576 5374 y Fs(i)3604 5362 y Fw(\011)27 b(and)523 5462 y(since)g Fu(P)779 5474 y Fs(k)848 5462 y Fw(comm)n(utes)g(with)i Fu(L)1485 5432 y Fn(0)1485 5484 y Fs(i)1512 5462 y Fu(;)60 b(i)22 b Fw(=)h(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3,)26 b(w)n(e)h(get)1287 5681 y Fu(P)1340 5693 y Fs(k)1381 5681 y Fv(B)1449 5696 y Fs(S)1493 5680 y Fm(2)1529 5681 y Fu(L)1586 5693 y Fs(i)1613 5681 y Fw(\011\()p Fv(z)p Fw(\))d(=)1896 5568 y Fo(Z)1942 5757 y Fq(x)p Fn(2)p Fs(S)2071 5740 y Fm(2)2120 5681 y Fu(L)2177 5647 y Fn(0)2177 5702 y Fs(i)2205 5681 y Fr(K)2268 5693 y Fs(k)2309 5681 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\)\011\()p Fv(x)p Fw(\))p Fu(d)p Fw(\012\()p Fv(x)p Fw(\))p eop %%Page: 36 36 36 35 bop 3747 324 a Fw(36)648 614 y(Since)37 b(the)h(last)f(t)n(w)n(o) g(equations)g(hold)g(for)g(a)h(dense)f(set)h(in)f Fu(L)2693 584 y Ft(2)2730 614 y Fw(\()p Fu(S)2818 584 y Ft(2)2855 614 y Fw(\),)k(w)n(e)c(conclude)g(that)h(\()p Fu(L)3713 626 y Fs(i)3766 614 y Fw(+)523 714 y Fu(L)580 684 y Fn(0)580 736 y Fs(i)607 714 y Fw(\))p Fr(K)702 726 y Fs(k)744 714 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)f(0)91 b Fu(i)23 b Fw(=)g(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3.)648 814 y(The)31 b(function)h Fr(K)1214 826 y Fs(k)1287 814 y Fw(satis\014es)f(another)f(di\013eren)n(tial)i(equation)f(whic)n(h)g (is)h(a)f(consequence)f(that)i Fr(K)3789 826 y Fs(k)523 913 y Fw(is)26 b(an)g(homogeneous)e(p)r(olynomial)h(of)h(degree)f(2)p Fu(k)j Fw(in)f(the)f(v)-5 b(ariables)25 b(\()p Fu(z)2766 925 y Ft(1)2803 913 y Fu(;)14 b(z)2879 925 y Ft(2)2916 913 y Fw(\).)36 b(Th)n(us)26 b(w)n(e)g(m)n(ust)g(ha)n(v)n(e)f(for)523 1013 y(all)i Fu(k)f Fr(\025)d Fw(0:)1786 1196 y(\()p Fu(H)1887 1208 y Ft(0)1943 1196 y Fr(\000)18 b Fw(2)p Fu(k)s Fw(\))p Fr(K)2209 1208 y Fs(k)2272 1196 y Fw(=)23 b(0)523 1378 y(with)1777 1561 y Fu(H)1846 1573 y Ft(0)1906 1561 y Fw(=)g Fu(z)2033 1573 y Ft(1)2070 1561 y Fu(@)2114 1573 y Ft(1)2170 1561 y Fw(+)18 b Fu(z)2292 1573 y Ft(2)2329 1561 y Fu(@)2373 1573 y Ft(2)648 1743 y Fw(The)27 b(analysis)f(ab)r(o)n (v)n(e)h(m)n(ust)g(hold,)h(in)g(particular,)e(for)h Fu(k)f Fw(=)d(1.)36 b(Th)n(us)28 b(w)n(e)f(ha)n(v)n(e:)1468 1926 y(\()p Fu(L)1557 1938 y Fs(i)1603 1926 y Fw(+)18 b Fu(L)1743 1892 y Fn(0)1743 1947 y Fs(i)1770 1926 y Fw(\)\010)1862 1941 y Fs(S)1906 1924 y Fm(2)1943 1926 y Fw(\()p Fv(x)p Fu(;)c Fv(z)p Fw(\))24 b(=)f(0)92 b Fu(i)22 b Fw(=)h(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)523 2109 y(and)1767 2291 y(\()p Fu(H)1868 2303 y Ft(0)1924 2291 y Fr(\000)k Fw(2)p Fu(k)s Fw(\)\010)2187 2306 y Fs(S)2231 2290 y Fm(2)2291 2291 y Fw(=)k(0)648 2474 y(Since)27 b Fu(L)921 2486 y Fs(i)948 2474 y Fu(;)37 b(L)1065 2444 y Fn(0)1065 2496 y Fs(i)1120 2474 y Fw(and)28 b Fu(H)1351 2486 y Ft(0)1416 2474 y Fw(are)e(\014rst)i(order)e(linear)h(di\013eren) n(tial)g(op)r(erators,)f(w)n(e)h(ha)n(v)n(e)f(for)h(all)h Fu(k)e Fr(\025)c Fw(0)1787 2657 y(\()p Fu(L)1876 2622 y Fn(0)1876 2677 y Fs(i)1922 2657 y Fw(+)c Fu(L)2062 2669 y Fs(i)2089 2657 y Fw(\)\010)2181 2622 y Fs(`)2181 2679 y(S)2225 2662 y Fm(2)2285 2657 y Fw(=)23 b(0)1773 2781 y(\()p Fu(H)1874 2793 y Ft(0)1930 2781 y Fr(\000)18 b Fw(2)p Fu(`)p Fw(\)\010)2182 2796 y Fs(S)2226 2779 y Fm(2)2285 2781 y Fw(=)23 b(0)648 2964 y(No)n(w)k(consider)f(a)h (particular)g(p)r(oin)n(t)g(\()p Fv(x)1911 2976 y Ft(0)1949 2964 y Fu(;)14 b Fv(z)2028 2976 y Ft(0)2066 2964 y Fw(\))28 b(where)f(\010)2426 2979 y Fs(S)2470 2962 y Fm(2)2506 2964 y Fw(\()p Fv(x)2588 2976 y Ft(0)2626 2964 y Fu(;)14 b Fv(z)2705 2976 y Ft(0)2743 2964 y Fw(\))23 b Fr(6)p Fw(=)g(0.)36 b(Let)1750 3194 y Fu(c)p Fw(\()p Fu(k)s Fw(\))24 b(=)2036 3138 y Fr(K)2099 3150 y Fs(k)2140 3138 y Fw(\()p Fv(x)2222 3150 y Ft(0)2260 3138 y Fu(;)14 b Fv(z)2339 3150 y Ft(0)2376 3138 y Fw(\))p 2017 3175 410 4 v 2017 3254 a(\010)2077 3226 y Fs(k)2077 3282 y(S)2121 3266 y Fm(2)2158 3254 y Fw(\()p Fv(x)2240 3266 y Ft(0)2278 3254 y Fu(;)g Fv(z)2357 3266 y Ft(0)2394 3254 y Fw(\))648 3431 y(De\014ne)28 b(the)g(function)1637 3641 y Fu(G)1702 3653 y Fs(k)1744 3641 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)2150 3585 y Fr(K)2213 3597 y Fs(k)2254 3585 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))p 2059 3622 482 4 v 2059 3701 a Fu(c)p Fw(\()p Fu(k)s Fw(\)\010)2265 3672 y Fs(k)2265 3729 y(S)2309 3712 y Fm(2)2345 3701 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))523 3878 y(This)28 b(function)g(satis\014es)f(the)h(follo)n(wing)e(system)h(of)h (homogeneous)e(di\013eren)n(tial)h(equations:)1508 4060 y(\()p Fu(L)1597 4026 y Fn(0)1597 4081 y Fs(i)1643 4060 y Fw(+)18 b Fu(L)1783 4072 y Fs(i)1810 4060 y Fw(\))p Fu(G)1907 4072 y Fs(k)1949 4060 y Fw(\()p Fv(x)p Fu(;)c Fv(z)p Fw(\))84 b(=)f(0)91 b Fu(i)23 b Fw(=)g(1)p Fu(;)14 b Fw(2)p Fu(;)g Fw(3)1736 4185 y Fu(H)1805 4197 y Ft(0)1843 4185 y Fu(G)1908 4197 y Fs(k)1949 4185 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))84 b(=)f(0)523 4368 y(with)28 b(initial)g (condition)1830 4550 y Fu(G)1895 4562 y Fs(k)1936 4550 y Fw(\()p Fv(x)2018 4562 y Ft(0)2056 4550 y Fu(;)14 b Fv(z)2135 4562 y Ft(0)2173 4550 y Fw(\))23 b(=)g(1)648 4733 y(The)k(system)h(of)f(equations)g(can)g(also)g(b)r(e)h(written)f (as)1884 4916 y Fv(M)p Fr(r)p Fu(G)2109 4928 y Fs(k)2173 4916 y Fw(=)c(0)523 5115 y(with)28 b Fr(r)23 b Fw(=)892 5023 y Fo(\020)959 5082 y Fs(@)p 952 5096 73 4 v 952 5143 a(@)t(\022)1034 5115 y Fu(;)1091 5082 y Fs(@)p 1081 5096 79 4 v 1081 5143 a(@)t(\036)1170 5115 y Fu(;)1239 5082 y Fs(@)p 1217 5096 104 4 v 1217 5143 a(@)t(z)1288 5151 y Fm(1)1330 5115 y Fu(;)1399 5082 y Fs(@)p 1377 5096 V 1377 5143 a(@)t(z)1448 5151 y Fm(2)1490 5023 y Fo(\021)1567 5115 y Fw(and)k Fv(M)h Fw(a)f(4)18 b Fr(\002)g Fw(4)27 b(matrix)g(giv)n(en)g(b)n(y)1160 5526 y Fv(M)22 b Fw(=)1361 5310 y Fo(2)1361 5456 y(6)1361 5506 y(6)1361 5559 y(4)1495 5376 y Fu({)14 b Fw(sin)o(\()p Fu(\036)p Fw(\))121 b Fu({)14 b Fw(cot)o(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))2504 5343 y Ft(1)p 2504 5357 34 4 v 2504 5405 a(2)2547 5376 y Fu(z)2586 5388 y Ft(2)2760 5343 y Fn(\000)p Ft(1)p 2760 5357 85 4 v 2786 5405 a(2)2855 5376 y Fu(z)2894 5388 y Ft(1)1458 5476 y Fr(\000)p Fu({)g Fw(cos)n(\()p Fu(\036)p Fw(\))89 b Fu({)14 b Fw(cot)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))88 b Fr(\000)2536 5443 y Ft(1)p 2525 5457 57 4 v 2525 5505 a(2)p Fs({)2591 5476 y Fu(z)2630 5488 y Ft(2)2786 5443 y(1)p 2774 5457 V 2774 5505 a(2)p Fs({)2841 5476 y Fu(z)2880 5488 y Ft(1)1602 5576 y Fw(0)430 b Fr(\000)p Fu({)2504 5543 y Ft(1)p 2504 5557 34 4 v 2504 5605 a(2)2547 5576 y Fu(z)2586 5588 y Ft(1)2786 5543 y(1)p 2786 5557 V 2786 5605 a(2)2829 5576 y Fu(z)2868 5588 y Ft(2)1602 5676 y Fw(0)454 b(0)348 b Fr(\000)p Fu(z)2592 5688 y Ft(1)2802 5676 y Fu(z)2841 5688 y Ft(2)2972 5310 y Fo(3)2972 5456 y(7)2972 5506 y(7)2972 5559 y(5)p eop %%Page: 37 37 37 36 bop 3747 324 a Fw(37)648 614 y(The)27 b(determinan)n(t)h(of)f (the)h(matrix)f Fv(M)h Fw(is:)1600 815 y Fu(det)p Fv(M)23 b Fw(=)2062 759 y(1)p 1924 796 319 4 v 1924 813 a Fr(p)p 1993 813 42 4 v 69 x Fw(3)o(sin\()p Fu(\022)r Fw(\))2252 815 y(\010)2312 830 y Fs(S)2356 814 y Fm(2)2392 815 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))648 1056 y(Since)31 b(\010)928 1071 y Fs(S)972 1054 y Fm(2)1008 1056 y Fw(\()p Fv(x)1090 1068 y Ft(0)1128 1056 y Fu(;)14 b Fv(z)1207 1068 y Ft(0)1245 1056 y Fw(\))29 b Fr(6)p Fw(=)g(0,)i(then)h Fv(M)f Fw(is)g(in)n(v)n(ertible)f(on)h(a)g(neigh)n(b)r(orho)r(o)r(d)f Fr(V)37 b Fw(of)31 b(the)h(p)r(oin)n(t)f(\()p Fv(x)3620 1068 y Ft(0)3658 1056 y Fu(;)14 b Fv(z)3737 1068 y Ft(0)3775 1056 y Fw(\).)523 1156 y(Therefore,)26 b(w)n(e)i(m)n(ust)f(ha)n(v)n(e) 1549 1338 y Fu(G)1614 1350 y Fs(k)1655 1338 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)f(1)45 b(for)27 b(all)c(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b Fr(2)g(V)523 1521 y Fw(and)j(then)1427 1703 y Fr(K)1490 1715 y Fs(k)1554 1703 y Fw(=)c Fu(c)p Fw(\()p Fu(k)s Fw(\)\010)1848 1669 y Fs(k)1848 1726 y(S)1892 1709 y Fm(2)1929 1703 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))47 b(for)27 b(all)c(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b Fr(2)f(V)648 1886 y Fw(Since)i Fr(K)925 1898 y Fs(k)991 1886 y Fw(and)g(\010)1210 1856 y Fs(k)1210 1913 y(S)1254 1896 y Fm(2)1316 1886 y Fw(are)f(analytical)g(functions)h(of)g(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\),)27 b(then)f(the)f(last)g(equation)g(m)n(ust)g (actually)523 1986 y(hold)j(for)f(all)g(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\).)648 2085 y(Finally)-7 b(,)28 b(w)n(e)f(w)n(an)n(t)h(to) g(ev)-5 b(aluate)27 b(the)i(constan)n(t)e Fu(c)p Fw(\()p Fu(k)s Fw(\))h(app)r(earing)f(in)h(the)h(last)e(equation.)38 b(By)28 b(using)523 2185 y(the)g(equations)1128 2430 y Fu(Y)1176 2442 y Ft(11)1247 2430 y Fw(\()p Fv(x)p Fw(\))c(=)e Fr(\000)1537 2299 y Fo(r)p 1620 2299 62 4 v 1630 2373 a Fw(3)p 1630 2410 42 4 v 1630 2486 a(2)1695 2430 y(sin\()p Fu(\022)r Fw(\))14 b(exp\()p Fu({\036)p Fw(\))p Fu(;)107 b(Y)2361 2442 y Ft(10)2432 2430 y Fw(\()p Fv(x)p Fw(\))24 b(=)2658 2357 y Fr(p)p 2727 2357 V 73 x Fw(3)13 b(cos)o(\()p Fu(\022)r Fw(\))p Fu(;)1351 2665 y(Y)1399 2677 y Ft(1)p Fn(\000)p Ft(1)1521 2665 y Fw(\()p Fv(x)p Fw(\))24 b(=)1747 2535 y Fo(r)p 1830 2535 62 4 v 1840 2609 a Fw(3)p 1840 2646 42 4 v 1840 2722 a(2)1905 2665 y(sin\()p Fu(\022)r Fw(\))14 b(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))1281 2888 y Fu(y)1322 2900 y Ft(11)1392 2888 y Fw(\()p Fv(z)p Fw(\))24 b(=)1635 2832 y Fu(z)1678 2801 y Ft(2)1674 2852 y(1)p 1619 2869 111 4 v 1619 2885 a Fr(p)p 1688 2885 42 4 v 69 x Fw(2)1740 2888 y Fu(;)106 b(y)1910 2900 y Ft(10)1980 2888 y Fw(\()p Fv(z)p Fw(\))24 b(=)f Fu(z)2237 2900 y Ft(1)2274 2888 y Fu(z)2313 2900 y Ft(2)2350 2888 y Fu(;)106 b(y)2520 2900 y Ft(1)p Fn(\000)p Ft(1)2642 2888 y Fw(\()p Fv(z)p Fw(\))24 b(=)2885 2832 y Fu(z)2928 2801 y Ft(2)2924 2852 y(2)p 2869 2869 111 4 v 2869 2885 a Fr(p)p 2938 2885 42 4 v 69 x Fw(2)2990 2888 y Fu(;)523 3114 y Fw(w)n(e)j(obtain)807 3337 y(\010)867 3352 y Fs(S)911 3335 y Fm(2)947 3337 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))25 b(=)1262 3212 y Fr(p)p 1331 3212 V 69 x Fw(3)p 1262 3318 111 4 v 1297 3394 a(2)1397 3270 y Fo(\000)1435 3337 y Fr(\000)14 b Fw(sin)o(\()p Fu(\022)r Fw(\))g(exp)q(\()p Fr(\000)p Fu({\036)p Fw(\))p Fu(z)2110 3302 y Ft(2)2106 3357 y(1)2165 3337 y Fw(+)k(2)c(cos)n(\()p Fu(\022)r Fw(\))p Fu(z)2558 3349 y Ft(1)2596 3337 y Fu(z)2635 3349 y Ft(2)2690 3337 y Fw(+)19 b(sin)o(\()p Fu(\022)r Fw(\))14 b(exp)q(\()p Fu({\036)p Fw(\))p Fu(z)3305 3302 y Ft(2)3301 3357 y(2)3342 3270 y Fo(\001)648 3543 y Fw(No)n(w)27 b(consider)f(the)i(relation)1172 3790 y Fu(Y)1220 3802 y Fs(k)q(m)1320 3790 y Fw(\()p Fv(x)p Fw(\))c(=)1546 3641 y Fo(s)p 1629 3641 623 4 v 1639 3734 a Fw(\(2)p Fu(k)d Fw(+)d(1\)\()p Fu(k)j Fr(\000)d Fu(m)p Fw(\)!)p 1639 3771 603 4 v 1786 3847 a(\()p Fu(k)k Fw(+)c Fu(m)p Fw(\)!)2251 3790 y Fu(P)2316 3756 y Fs(m)2304 3811 y(k)2379 3790 y Fw(\(cos\()p Fu(\022)r Fw(\)\))c(exp\()p Fu({m\036)p Fw(\))523 4040 y(with)28 b Fu(P)777 4010 y Fs(m)765 4064 y(k)868 4040 y Fw(the)g(asso)r(ciated)e(Legendre)h(p)r(olynomials.)35 b(By)28 b(taking)f Fu(\022)e Fw(=)e(0)k(and)g(since)h Fu(P)3294 4010 y Fs(m)3282 4064 y(k)3357 4040 y Fw(\(1\))23 b(=)f(1)p Fu(\016)3652 4052 y Fs(m)p Ft(0)3748 4040 y Fw(,)1475 4206 y Fo(p)p 1558 4206 295 4 v 75 x Fw(\(2)p Fu(k)f Fw(+)d(1\))1863 4225 y Fu(z)1906 4195 y Fs(k)1902 4246 y Ft(1)1946 4225 y Fu(z)1989 4195 y Fs(k)1985 4246 y Ft(2)p 1863 4262 167 4 v 1911 4338 a Fu(k)s Fw(!)2062 4281 y(=)23 b Fu(c)p Fw(\()p Fu(k)s Fw(\))2310 4189 y Fo(\020)2359 4209 y Fr(p)p 2429 4209 42 4 v 2429 4281 a Fw(3)o Fu(z)2509 4293 y Ft(1)2546 4281 y Fu(z)2585 4293 y Ft(2)2622 4189 y Fo(\021)2671 4207 y Fs(k)523 4487 y Fw(whic)n(h)28 b(in)f(turn)h(implies)1652 4735 y Fu(c)p Fw(\()p Fu(k)s Fw(\))23 b(=)1919 4612 y Fr(p)p 1988 4612 231 4 v 67 x Fw(2)p Fu(k)e Fw(+)d(1)p 1919 4716 300 4 v 2034 4792 a Fu(k)s Fw(!)2242 4618 y Fo(\022)2348 4679 y Fw(1)p 2313 4716 111 4 v 2313 4733 a Fr(p)p 2382 4733 42 4 v 68 x Fw(3)2434 4618 y Fo(\023)2495 4635 y Fs(k)p eop %%Page: 38 38 38 37 bop 3747 324 a Fw(38)1923 614 y Fv(App)s(endix)31 b(B)1200 774 y(The)h(Kernel)f(of)h(the)g(Bargmann)f(transform)g(for)h Fu(S)3093 744 y Ft(3)3130 774 y Fw(.)648 1197 y(The)27 b(purp)r(ose)g(of)h(this)g(app)r(endix)f(is)h(to)f(pro)n(v)n(e)f(the)i (equation)1482 1441 y Fu(K)1553 1456 y Fs(S)1597 1439 y Fm(3)1656 1441 y Fw(=)1773 1337 y Fn(1)1746 1362 y Fo(X)1743 1538 y Fs(n)p Ft(=1)1952 1316 y Fr(p)p 2021 1316 92 4 v 69 x Fw(2)p Fu(n)p 1892 1422 281 4 v 1892 1498 a Fw(\()p Fu(n)19 b Fr(\000)f Fw(1\)!)2249 1385 y(1)p 2193 1422 154 4 v 2193 1500 a(2)2235 1476 y Fs(n=)p Ft(2)2370 1441 y Fw(\()q(\010)2463 1456 y Fs(S)2507 1439 y Fm(3)2543 1441 y Fw(\))2575 1399 y Fs(n)p Fn(\000)p Ft(1)648 1695 y Fw(W)-7 b(e)28 b(w)n(an)n(t)e(to)i(do)f(it)h(b)n(y)g (sho)n(wing)e(that)i(for)f(all)g Fu(n)c Fr(\025)g Fw(1:)1387 1913 y(~)1371 1934 y Fr(K)1434 1946 y Fs(n)1480 1934 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)1855 1810 y Fr(p)p 1924 1810 92 4 v 68 x Fw(2)p Fu(n)p 1795 1915 281 4 v 1795 1991 a Fw(\()p Fu(n)18 b Fr(\000)g Fw(1\)!)2151 1878 y(1)p 2095 1915 154 4 v 2095 1993 a(2)2137 1969 y Fs(n=)p Ft(2)2273 1934 y Fw(\(\010)2365 1949 y Fs(S)2409 1933 y Fm(3)2445 1934 y Fw(\))2478 1893 y Fs(n)p Fn(\000)p Ft(1)2622 1934 y Fw(\()p Fv(x)p Fu(;)c Fv(z)p Fw(\))523 2161 y(with)1444 2384 y(~)1428 2405 y Fr(K)1491 2417 y Fs(n)1536 2405 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))25 b(=)1841 2301 y Fs(n)p Fn(\000)p Ft(1)1844 2326 y Fo(X)1848 2505 y Fs(`)p Ft(=0)2007 2301 y Fs(m)p Ft(=)p Fs(`)2016 2326 y Fo(X)1981 2505 y Fs(m)p Ft(=)p Fn(\000)p Fs(`)2185 2405 y Fu(Y)2251 2371 y Fn(\003)2233 2425 y Fs(n`m)2365 2405 y Fw(\()p Fv(x)p Fw(\))p Fu(y)2520 2417 y Fs(n`m)2652 2405 y Fw(\()p Fv(z)p Fw(\))648 2669 y(The)31 b(idea)h(of)f(the)h(pro)r (of)f(is)h(exactly)f(the)h(same)f(as)g(the)h(one)g(w)n(e)f(used)g(in)h (app)r(endix)g(A.)g(Th)n(us)g(for)523 2769 y(eac)n(h)27 b Fu(n)c Fr(\025)f Fw(1)p Fu(;)28 b Fw(let)g(us)f(de\014ne)h(the)g (function)1309 2971 y Fu(F)1362 2983 y Fs(n)1408 2971 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))24 b(=)1729 2950 y(~)1713 2971 y Fr(K)1776 2983 y Fs(n)1821 2971 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))p Fu(=)2070 2879 y Fo(\020)2121 2971 y Fu(c)p Fw(\()p Fu(n)p Fw(\))g(\(\010)2377 2986 y Fs(S)2421 2970 y Fm(3)2458 2971 y Fw(\))2490 2930 y Fs(n)p Fn(\000)p Ft(1)2634 2971 y Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))2827 2879 y Fo(\021)523 3179 y Fw(\(in)25 b(a)f(neigh)n(b)r(orho)r(o)r(d)f Fr(V)32 b Fw(of)24 b(a)g(p)r(oin)n(t)h (\()p Fv(x)1769 3191 y Ft(0)1806 3179 y Fu(;)14 b Fv(z)1885 3191 y Ft(0)1923 3179 y Fw(\))25 b(where)f(\(\010)2309 3194 y Fs(S)2353 3177 y Fm(3)2390 3179 y Fw(\))14 b(\()p Fv(x)2518 3191 y Ft(0)2556 3179 y Fu(;)g Fv(z)2635 3191 y Ft(0)2672 3179 y Fw(\))23 b Fr(6)p Fw(=)g(0\))h(suc)n(h)h(that)f Fu(F)3327 3191 y Fs(n)3397 3179 y Fw(is)h(a)f(solution)523 3279 y(of)k(the)g(system)f(of)g(di\013eren)n(tial)h(equations:)1888 3461 y Fv(M)p Fr(r)p Fu(F)2101 3473 y Fs(n)2170 3461 y Fw(=)22 b(0)523 3644 y(with)28 b Fr(r)23 b Fw(=)g(\()942 3611 y Fs(@)p 934 3625 73 4 v 934 3673 a(@)t(\022)1017 3644 y Fu(;)1074 3611 y Fs(@)p 1064 3625 79 4 v 1064 3673 a(@)t(\036)1152 3644 y Fu(;)1210 3611 y Fs(@)p 1199 3625 V 1199 3673 a(@)t(\025)1288 3644 y Fu(;)1357 3611 y Fs(@)p 1335 3625 104 4 v 1335 3673 a(@)t(z)1406 3681 y Fm(1)1448 3644 y Fu(;)1517 3611 y Fs(@)p 1495 3625 V 1495 3673 a(@)t(z)1566 3681 y Fm(2)1608 3644 y Fu(;)1677 3611 y Fs(@)p 1654 3625 V 1654 3673 a(@)t(z)1725 3681 y Fm(3)1768 3644 y Fu(;)1837 3611 y Fs(@)p 1814 3625 V 1814 3673 a(@)t(z)1885 3681 y Fm(4)1928 3644 y Fw(\))28 b(and)f Fv(M)g Fw(the)h(7)18 b Fr(\002)g Fw(7)27 b(matrix)683 3781 y Fo(2)683 3927 y(6)683 3977 y(6)683 4027 y(6)683 4077 y(6)683 4126 y(6)683 4176 y(6)683 4226 y(6)683 4276 y(6)683 4326 y(6)683 4379 y(4)895 3845 y Fu({)14 b Fw(sin)o(\()p Fu(\036)p Fw(\))199 b Fu({)14 b Fw(cot)o(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))340 b(0)2606 3811 y Fs(z)2638 3819 y Fm(2)p 2606 3826 65 4 v 2621 3874 a Ft(2)2837 3811 y Fs(z)2869 3819 y Fm(1)p 2837 3826 V 2853 3874 a Ft(2)3069 3811 y Fs(z)3101 3819 y Fm(4)p 3069 3826 V 3084 3874 a Ft(2)3301 3811 y Fs(z)3333 3819 y Fm(3)p 3301 3826 V 3316 3874 a Ft(2)858 3945 y Fr(\000)p Fu({)14 b Fw(cos)n(\()p Fu(\036)p Fw(\))167 b Fu({)14 b Fw(cot)o(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\036)p Fw(\))344 b(0)337 b Fr(\000)2638 3911 y Fs(z)2670 3919 y Fm(2)p 2638 3926 V 2642 3973 a Ft(2)p Fs({)2837 3911 y(z)2869 3919 y Fm(1)p 2837 3926 V 2841 3973 a Ft(2)p Fs({)3027 3945 y Fr(\000)3102 3911 y Fs(z)3134 3919 y Fm(4)p 3101 3926 V 3105 3973 a Ft(2)p Fs({)3301 3911 y(z)3333 3919 y Fm(3)p 3301 3926 V 3305 3973 a Ft(2)p Fs({)1002 4044 y Fw(0)508 b Fr(\000)p Fu({)540 b Fw(0)2606 4010 y Fs(z)2638 4018 y Fm(1)p 2606 4025 V 2621 4073 a Ft(2)2795 4044 y Fr(\000)2870 4010 y Fs(z)2902 4018 y Fm(2)p 2870 4025 V 2885 4073 a Ft(2)3069 4010 y Fs(z)3101 4018 y Fm(3)p 3069 4025 V 3084 4073 a Ft(2)3258 4044 y Fr(\000)3333 4010 y Fs(z)3365 4018 y Fm(4)p 3333 4025 V 3348 4073 a Ft(2)899 4158 y Fu(f)9 b Fw(\()p Fu(\036;)14 b(\025)p Fw(\))1383 4118 y Fn(\000)d Ft(cot\()p Fs(\025)p Ft(\))h(sin)o(\()p Fs(\036)p Ft(\))p 1383 4139 428 4 v 1496 4187 a Fs({)g Ft(sin\()p Fs(\022)r Ft(\))1928 4158 y Fr(\000)p Fu({)i Fw(sin)o(\()p Fu(\022)r Fw(\))g(cos\()p Fu(\036)p Fw(\))2606 4124 y Fs(z)2638 4132 y Fm(2)p 2606 4139 65 4 v 2621 4187 a Ft(2)2837 4124 y Fs(z)2869 4132 y Fm(1)p 2837 4139 V 2853 4187 a Ft(2)3027 4158 y 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Fw(\()p Fv(x)p Fu(;)g Fv(z)p Fw(\))p eop %%Page: 39 39 39 38 bop 3747 324 a Fw(39)648 614 y(It)28 b(is)f(just)h(left)g(to)g (ev)-5 b(aluate)27 b(the)h(constan)n(t)f Fu(c)p Fw(\()p Fu(n)p Fw(\).)37 b(Since)1069 806 y Fu(Y)1117 818 y Ft(100)1304 806 y Fw(=)82 b(1)p Fu(;)h(Y)1647 818 y Ft(200)1774 806 y Fw(=)22 b(2)14 b(cos)o(\()p Fu(\025)p Fw(\))p Fu(;)84 b(Y)2295 818 y Ft(211)2421 806 y Fw(=)23 b Fr(\000)p Fu({)2601 733 y Fr(p)p 2670 733 42 4 v 73 x Fw(2)13 b(sin\()p Fu(\022)r Fw(\))h(sin)q(\()p Fu(\025)p Fw(\))p Fu(e)3200 772 y Fs({\036)3267 806 y Fu(;)1069 939 y(Y)1117 951 y Ft(210)1304 939 y Fw(=)82 b(2)p Fu({)14 b Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin\()p Fu(\025)p Fw(\))p Fu(;)84 b(Y)2133 951 y Ft(21)p Fn(\000)p Ft(1)2312 939 y Fw(=)22 b Fu({)2426 866 y Fr(p)p 2495 866 V 73 x Fw(2)14 b(sin\()p Fu(\022)r Fw(\))g(sin\()p Fu(\025)p Fw(\))p Fu(e)3025 905 y Fn(\000)p Fs({\036)1077 1103 y Fu(y)1118 1115 y Ft(100)1304 1103 y Fw(=)82 b(1)p Fu(;)h(y)1640 1115 y Ft(200)1766 1103 y Fw(=)1898 1047 y(1)p 1864 1084 111 4 v 1864 1100 a Fr(p)p 1933 1100 42 4 v 69 x Fw(2)1984 1103 y([)p Fu(z)2046 1115 y Ft(1)2083 1103 y Fu(z)2122 1115 y Ft(4)2178 1103 y Fr(\000)18 b Fu(z)2300 1115 y Ft(2)2336 1103 y Fu(z)2375 1115 y Ft(3)2412 1103 y Fw(])p Fu(;)83 b(y)2582 1115 y Ft(211)2709 1103 y Fw(=)22 b Fu(z)2835 1115 y Ft(1)2872 1103 y Fu(z)2911 1115 y Ft(3)2948 1103 y Fu(;)1077 1323 y(y)1118 1335 y Ft(210)1304 1323 y Fw(=)1496 1267 y(1)p 1461 1304 111 4 v 1461 1320 a Fr(p)p 1530 1320 42 4 v 69 x Fw(2)1582 1323 y([)p Fu(z)1644 1335 y Ft(1)1681 1323 y Fu(z)1720 1335 y Ft(4)1775 1323 y Fw(+)c Fu(z)1897 1335 y Ft(2)1934 1323 y Fu(z)1973 1335 y Ft(3)2010 1323 y Fw(])p Fu(;)83 b(y)2180 1335 y Ft(21)p Fn(\000)p Ft(1)2358 1323 y Fw(=)23 b Fu(z)2485 1335 y Ft(2)2522 1323 y Fu(z)2561 1335 y Ft(4)523 1549 y Fw(w)n(e)k(\014nd)729 1732 y(\010)789 1747 y Fs(S)833 1730 y Fm(3)870 1732 y Fw(\()p Fv(x)p Fu(;)14 b Fv(z)p Fw(\))84 b(=)1295 1659 y Fr(p)p 1364 1659 V 73 x Fw(2)13 b([)p Fu(z)1481 1744 y Ft(1)1518 1732 y Fu(z)1557 1744 y Ft(4)1594 1732 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))20 b Fr(\000)e Fu({)c Fw(cos)o(\()p Fu(\022)r Fw(\))g(sin\()p Fu(\025)p Fw(\)\))20 b Fr(\000)e Fu(z)2611 1744 y Ft(2)2648 1732 y Fu(z)2687 1744 y Ft(3)2724 1732 y Fw(\(cos)o(\()p Fu(\025)p Fw(\))i(+)e Fu({)c Fw(cos)n(\()p Fu(\022)r Fw(\))g(sin)q(\()p Fu(\025)p Fw(\)\))1295 1856 y(+)23 b Fu({z)1449 1868 y Ft(1)1486 1856 y Fu(z)1525 1868 y Ft(3)1575 1856 y Fw(sin\()p Fu(\022)r Fw(\))14 b(sin)q(\()p Fu(\025)p Fw(\))g(exp\()p Fr(\000)p Fu({\036)p Fw(\))19 b Fr(\000)f Fu({z)2525 1868 y Ft(2)2562 1856 y Fu(z)2601 1868 y Ft(4)2652 1856 y Fw(sin\()p Fu(\022)r Fw(\))c(sin\()p Fu(\025)p Fw(\))g(exp)q(\()p Fu({\036)p Fw(\)])648 2039 y(No)n(w)27 b(consider)f Fu(\025)e Fw(=)e(0.)37 b(Since)28 b Fu(Y)1686 2051 y Fs(n`m)1818 2039 y Fw(\()p Fu(\022)r(;)14 b(\036;)g(\025)24 b Fw(=)f(0\))g(=)f Fu(n\016)2445 2051 y Fs(`)p Ft(0)2538 2039 y Fw(w)n(e)27 b(get)1397 2263 y Fu(ny)1488 2275 y Fs(n)p Ft(00)1599 2263 y Fw(\()p Fv(z)p Fw(\))d(=)f Fu(c)p Fw(\()p Fu(n)p Fw(\))1981 2171 y Fo(\020)2030 2191 y Fr(p)p 2099 2191 V 72 x Fw(2\()p Fu(z)2212 2275 y Ft(1)2249 2263 y Fu(z)2288 2275 y Ft(4)2343 2263 y Fr(\000)18 b Fu(z)2465 2275 y Ft(2)2502 2263 y Fu(z)2541 2275 y Ft(3)2578 2263 y Fw(\))2610 2171 y Fo(\021)2660 2188 y Fs(n)p Fn(\000)p Ft(1)648 2466 y Fw(By)27 b(taking)g(the)h(norm) f(in)h(b)r(oth)g(sides)f(of)h(last)f(equation,)g(w)n(e)g(ha)n(v)n(e) 1593 2649 y Fu(n)c Fw(=)g Fu(c)p Fw(\()p Fu(n)p Fw(\)2)1946 2615 y Ft(\()p Fs(n)p Fn(\000)p Ft(1\))p Fs(=)p Ft(2)2195 2585 y Fr(p)p 2264 2585 50 4 v 64 x Fu(n)p Fw(\()p Fu(n)18 b Fr(\000)g Fw(1\)!)523 2832 y(where)27 b(w)n(e)g(used)h Fr(k)p Fw(\()p Fu(z)1188 2844 y Ft(1)1225 2832 y Fu(z)1264 2844 y Ft(4)1319 2832 y Fr(\000)18 b Fu(z)1441 2844 y Ft(2)1478 2832 y Fu(z)1517 2844 y Ft(3)1554 2832 y Fw(\))1586 2801 y Fs(n)p Fn(\000)p Ft(1)1716 2832 y Fr(k)23 b Fw(=)1868 2772 y Fr(p)p 1937 2772 V 60 x Fu(n)p Fw(\()p Fu(n)c Fr(\000)f Fw(1\)!.)37 b(Th)n(us)27 b(w)n(e)g(conclude)1726 3075 y Fu(c)p Fw(\()p Fu(n)p Fw(\))c(=)2057 2950 y Fr(p)p 2126 2950 92 4 v 69 x Fw(2)p Fu(n)p 1997 3056 281 4 v 1997 3132 a Fw(\()p Fu(n)c Fr(\000)f Fw(1\)!)2354 3019 y(1)p 2297 3056 154 4 v 2297 3134 a(2)2339 3110 y Fs(n=)p Ft(2)p eop %%Page: 40 40 40 39 bop 3747 324 a Fw(40)565 584 y Ft(1)602 614 y Fw(V.Bargmann.)34 b Fl(On)25 b(a)h(Hilb)l(ert)g(Sp)l(ac)l(e)g(of)g(A)n(nalytic)g(F)-6 b(unctions)25 b(and)i(an)e(Asso)l(ciate)l(d)i(Inte)l(gr)l(al)e(T)-6 b(r)l(ans-)602 714 y(form.)39 b(Part)30 b(I)p Fw(,)e(Comm.)37 b(on)27 b(Pure)g(and)g(Appl.)38 b(Math.,)28 b Fv(14)f Fw(\(1961\),)f(187-214.)565 780 y Ft(2)602 811 y Fw(B.)32 b(Hall.)49 b Fl(Holomorphic)36 b(metho)l(ds)f(in)f(analysis)h(and)f (mathematic)l(al)h(physics)p Fw(,)g(In)d(\\First)f(Summer)602 910 y(Sc)n(ho)r(ol)26 b(in)h(Analysis)g(and)g(Mathematical)f(Ph)n (ysics.)36 b(Quan)n(tization,)26 b(the)h(Segal-Bargmann)d(T)-7 b(rans-)602 1010 y(form)28 b(and)g(Semiclassical)f(Analysis".)38 b(Con)n(temp)r(orary)26 b(Mathematics,)i(v)n(ol.)38 b Fv(260)p Fw(,)28 b(AMS-SMM,)h(S.)602 1110 y(P)n(\023)-39 b(erez-Estev)-5 b(a)24 b(and)j(C.)h(Villegas-Blas,)e(editors,)h (\(2000\))565 1176 y Ft(3)602 1206 y Fw(I.)33 b(Segal.)53 b Fl(Mathematic)l(al)37 b(pr)l(oblems)f(of)f(r)l(elativistic)i(Physics) p Fw(,)f(In)d(\\Pro)r(ceedings)e(of)i(the)h(Summer)602 1306 y(Seminar,)d(Boulder,)h(Colorado,)d(1960,)i(V)-7 b(ol)31 b(I)r(I".)g(Lectures)f(in)i(Applied)g(Mathematics,)f(American) 602 1405 y(Math.)37 b(So)r(c.,)27 b(Pro)n(vidence,)f(Rho)r(de)i (Island,)f(M.)h(Kac,)f(editor,)g(\(1963\))565 1472 y Ft(4)602 1502 y Fw(Cohen-T)-7 b(annoudji,)31 b(B.)f(Diu,)i(F.)f(Lalo)r (e,)g(B.Dui.)47 b Fl(Quantum)31 b(Me)l(chanics)p Fw(,)i(John)d(Wiley)h (and)f(Sons,)602 1602 y(\(1992\).)565 1668 y Ft(5)602 1698 y Fw(S.)e(Gra\016)g(and)g(A.)g(P)n(armeggiani.)36 b Fl(Quantum)28 b(Evolution)j(and)g(Classic)l(al)h(Flow)f(in)f(Complex) i(Phase)602 1798 y(Sp)l(ac)l(e)p Fw(.)37 b(Comm.)g(Math.)g(Ph)n(ysics,) 26 b Fv(128)h Fw(\(1990\),)g(393-409.)565 1864 y Ft(6)602 1895 y Fw(L.)k(Thomas)f(and)g(S.)i(W)-7 b(assel.)46 b Fl(Semiclassic)l(al)35 b(Appr)l(oximation)f(for)f(Schr\177)-42 b(odinger)35 b(Op)l(er)l(ators)e(on)f(a)602 1994 y(2-spher)l(e)f(at)e (High)i(Ener)l(gy)p Fw(,)d(J.)f(Math.)37 b(Ph)n(ysics,)27 b Fv(36)g Fw(\(1995\),)f(5480-5505.)565 2061 y Ft(7)602 2091 y Fw(C.)j(Villegas-Blas.)40 b Fl(The)33 b(L)l(aplacian)g(on)f(the) f(n-spher)l(e,)i(the)e(Hydr)l(o)l(gen)h(A)n(tom)f(and)h(the)f(Bar)l (gmann)602 2190 y(Sp)l(ac)l(e)k(R)l(epr)l(esentation)p Fw(,)f(Ph.)52 b(D.)34 b(Thesis,)f(Mathematics)g(Departmen)n(t,)h(Univ)n (ersit)n(y)e(of)h(Virginia,)602 2290 y(\(1996\).)565 2357 y Ft(8)602 2387 y Fw(A.)j(Urib)r(e)h Fl(Symb)l(ol)h(Calculus)g (for)h(a)f(Class)h(of)f(Pseudidi\013er)l(ential)i(Op)l(er)l(ators)e(on) g Fv(S)3389 2357 y Fs(n)3472 2387 y Fl(and)g(Band)602 2486 y(Asymptotics)p Fw(.)f(J.)28 b(F)-7 b(unct.)37 b(Anal.)g Fv(59)27 b Fw(\(1984\),)g(535-556.)565 2553 y Ft(9)602 2583 y Fw(S.)h(Stern)n(b)r(erg)e Fl(Gr)l(oup)k(The)l(ory)i(and)e (Physics)p Fw(,)f(Cam)n(bridge)e(Univ)n(ersit)n(y)f(Press,)g(\(1995\).) 565 2652 y Ft(10)635 2683 y Fw(J.)32 b(Moser.)48 b Fl(R)l(e)l (gularization)35 b(of)f(Kepler's)h(pr)l(oblem)g(and)f(the)g(aver)l (aging)i(metho)l(d)e(on)g(a)g(manifold)p Fw(,)635 2782 y(Comm.)j(Pure)26 b(Appl.)38 b(Math.)f Fv(23)p Fw(,)27 b(609-636,)e(\(1970\).)565 2849 y Ft(11)635 2879 y Fw(B.)37 b(Hall.)66 b Fl(The)39 b(Se)l(gal-Bar)l(gmann)h(\\c)l(oher)l(ent")e (state)g(tr)l(ansform)h(for)h(c)l(omp)l(act)f(Lie)g(Gr)l(oups)p Fw(,)h(J.)635 2978 y(F)-7 b(unct.)38 b(Anal.)e Fv(122)p Fw(,)27 b(103-151,)e(\(1994\).)565 3045 y Ft(12)635 3075 y Fw(B.)e(Hall.)35 b Fl(Harmonic)26 b(analysis)h(with)f(r)l(esp)l(e)l (ct)f(to)g(he)l(at)h(kernel)g(me)l(asur)l(e)c Fw(,)i(Bull.)35 b(Amer.)h(Math.)f(So)r(c.)635 3175 y Fv(38)p Fw(,)27 b(43-78,)f(\(2001\).)565 3241 y Ft(13)635 3271 y Fw(M.)35 b(Stenzel.)58 b Fl(The)37 b(Se)l(gal-Bar)l(gmann)g(tr)l(ansform)f(on)g (a)h(symmetric)f(sp)l(ac)l(e)h(of)g(c)l(omp)l(act)g(typ)l(e)p Fw(,)f(J.)635 3371 y(F)-7 b(unct.)38 b(Anal.)e Fv(165)p Fw(,)27 b(44-58,)f(\(1999\).)565 3438 y Ft(14)635 3468 y Fw(K.)39 b(Ii.)71 b Fl(On)39 b(a)i(Bar)l(gmann-typ)l(e)f(tr)l (ansform)h(and)g(a)f(Hilb)l(ert)g(sp)l(ac)l(e)h(of)g(holomorphic)j (functions)p Fw(,)635 3567 y(T)-7 b(ohoku)27 b(Mat.)37 b(J.)27 b(\(2\))h(38,)e(no.1,)h(57-69,)f(\(1986\).)565 3634 y Ft(15)635 3664 y Fw(R.)31 b(W)-7 b(ada.)47 b Fl(On)32 b(the)h(F)-6 b(ourier-Bor)l(el)34 b(tr)l(ansformations)g(of)g(analytic) g(functionals)f(on)g(the)g(c)l(omplex)635 3764 y(spher)l(e)p Fw(,)c(T)-7 b(ohoku)26 b(Mat.)37 b(J.)28 b(\(2\))f(38,)g(no.3,)g (417-432,)e(\(1986\).)565 3830 y Ft(16)635 3860 y Fw(J.)h(Ra)n(wnsley) -7 b(.)35 b Fl(A)29 b(nonunitary)f(p)l(airing)i(of)f(p)l(olarizations)i (for)e(the)g(kepler)h(pr)l(oblem)p Fw(,)e(T)-7 b(rans.)35 b(Amer.)635 3960 y(Math.)i(So)r(c.)g(.)g Fv(250)p Fw(,)27 b(167-180,)d(\(1979\).)565 4026 y Ft(17)635 4056 y Fw(B.)j(Kursunoglu,) g Fl(Mo)l(dern)j(Quantum)e(The)l(ory)p Fw(,)i(W.)e(H.)g(F)-7 b(reeman)27 b(and)g(Compan)n(y)g(\(1962\).)565 4126 y Ft(18)635 4156 y Fw(M.E.)19 b(Rose,)i Fl(Elementary)i(The)l(ory)g(of)h (A)n(ngular)d(Momentum)p Fw(,)g(John)e(Wiley)h(and)f(Sons,)i(Inc.)34 b(\(1957\).)565 4226 y Ft(19)635 4256 y Fw(P)-7 b(.)31 b(Kramer,)g(M.Moshinsky)g(and)h(T)f(Seligman,)h Fl(Complex)j (Extensions)e(of)h(Canonic)l(al)h(T)-6 b(r)l(ansfor-)635 4355 y(mations)25 b(and)g(Quantum)e(Me)l(chanics)p Fw(.)37 b(In)22 b(Group)f(Theory)g(and)h(its)g(Applications,)h(edited)g(b)n(y)e (E.M.)635 4455 y(Lo)r(eblm,)28 b(Academic)f(Press)f(New)i(Y)-7 b(ork)27 b(\(1975\),)f(V)-7 b(ol)28 b(I)r(I)r(I.)565 4521 y Ft(20)635 4552 y Fw(V.)40 b(Bargmann)e(and)i(I.)g(T.)g(T)-7 b(o)r(doro)n(v,)41 b Fl(Sp)l(ac)l(es)g(of)h(analytic)g(functions)f(on)g (a)h(c)l(omplex)f(c)l(one)h(as)635 4651 y(c)l(arriers)35 b(for)f(the)g(symmetric)g(tensor)g(r)l(epr)l(esentations)g(of)g Fu(S)5 b(O)r Fw(\()p Fu(n)p Fw(\).)51 b(J.)31 b(Math.)50 b(Ph)n(ysics,)32 b(V)-7 b(ol)32 b(18,)635 4751 y(No.6,)27 b(June)g(\(1977\),)g(1141-1148.)565 4817 y Ft(21)635 4847 y Fw(V.)42 b(Guillemin,)j Fl(T)-6 b(o)l(eplitz)44 b(Op)l(er)l(ators)e(in)h(n-dimensions)p Fw(.)78 b(In)n(tegral)40 b(Equations)h(and)g(Op)r(erator)635 4947 y(Theory)-7 b(,)27 b(V)-7 b(ol.)36 b(7,)28 b(\(1984\),)e(145-205.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF