Content-Type: multipart/mixed; boundary="-------------0311130735488" This is a multi-part message in MIME format. ---------------0311130735488 Content-Type: text/plain; name="03-498.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-498.keywords" Magnetic field, gauge invariance, pseudodifferential operator, Weyl calculus, canonical commutation relations, quantization, Moyal product ---------------0311130735488 Content-Type: application/postscript; name="nou.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nou.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: nou.dvi %%Pages: 21 1 %%BoundingBox: 0 0 596 843 %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution 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b(pla)n(y)g(a)h(decissiv)n(e)-118 5356 y(role)i(in)h(determining)g(the)g(explicit)g(form)n(ula)f(ab)r(o)n(v)n (e.)43 b(They)30 b(are)f(th)n(us)h(also)e(basic)i(in)g(deducing)g(the)g (explicit)g(pro)r(duct)g(rule)-118 5456 y Fs(\()p Fu(f)t(;)14 b(g)s Fs(\))23 b Fl(7!)g Fu(f)k Fl(\016)18 b Fu(g)30 b Fz(and)e(in)n(v)n (olution)e Fu(f)32 b Fl(7!)23 b Fu(f)1178 5426 y Fm(\016)1243 5456 y Fz(leading)k(to)h Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))p Fr(Op)o Fs(\()p Fu(g)s Fs(\))24 b(=)e Fr(Op)o Fs(\()p Fu(f)28 b Fl(\016)18 b Fu(g)s Fs(\))27 b Fz(and)h Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))2942 5426 y Fm(\003)3003 5456 y Fs(=)23 b Fr(Op)o Fs(\()p Fu(f)3283 5426 y Fm(\016)3321 5456 y Fs(\))p Fz(.)p -118 5530 1607 4 v -27 5583 a Fk(\003)9 5606 y Fj(Institute)18 b(of)e(Mathematics)h (\020Simion)e(Stoilo)n(w\021)22 b(of)16 b(the)i(Romanian)d(A)n(cadem)n(y)-6 b(,)18 b(P)-6 b(.O.)16 b(Bo)n(x)h(1-764,)h(Bucarest,)h(R)n(O-70700,)f (Romania,)f(Electronic)-118 5685 y(mail:)29 b(man)n(toiu@imar.ro,)22 b(purice@imar.ro)1869 5989 y Fz(1)p eop %%Page: 2 2 bop 6 154 a Fz(When)23 b(a)e(magnetic)h(\034eld)g Fu(B)k Fz(is)c(turned)g (on,)g(w)n(e)g(are)f(faced)h(with)g(the)g(problem)f(of)h(mo)r(difying)g(the)g (form)n(ula)f(for)h Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))22 b Fz(in)g(a)-118 254 y(w)n(a)n(y)c(taking)g(in)n(to)h(accoun)n(t)f(the)h(presence)g(of)g(the)g (magnetic)g(\034eld)g(in)g(a)g(correct,)g(ph)n(ysical)f(w)n(a)n(y)-7 b(.)33 b(A)19 b(mistak)n(en)g(pro)r(cedure)f(whic)n(h)-118 353 y(app)r(ears)31 b(from)h(time)g(to)g(time)h(in)g(the)f(literature)f(is)i (the)f(follo)n(wing:)45 b(One)32 b(c)n(ho)r(oses)f(a)g(v)n(ector)g(p)r(oten)n (tial)h Fu(A)h Fz(corresp)r(onding)-118 453 y(to)27 b(the)h(magnetic)f (\034eld)h(\()p Fu(B)g Fs(=)22 b Fu(dA)p Fz(\))29 b(and,)e(b)n(y)h(an)f (\(unjusti\034ed\))i(application)e(of)g(the)h(minimal)g(coupling)f (principle,)h(one)f(sets)-118 552 y Fr(Op)-8 572 y Fv(A)46 552 y Fs(\()p Fu(f)9 b Fs(\))23 b(:=)g Fr(Op)o Fs(\()p Fu(f)477 564 y Fv(A)531 552 y Fs(\))p 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y Fu(dy)i(dk)g(e)1626 1015 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)p Fv(k)1887 1050 y Fu(f)1950 933 y Fq(\022)2021 993 y Fu(x)19 b Fs(+)f Fu(y)p 2021 1030 193 4 v 2097 1107 a Fs(2)2224 1050 y Fu(;)c(k)21 b Fl(\000)d Fu(A)2484 933 y Fq(\022)2555 993 y Fu(x)h Fs(+)f Fu(y)p 2555 1030 V 2631 1107 a Fs(2)2758 933 y Fq(\023\023)2894 1050 y Fu(u)p Fs(\()p Fu(y)s Fs(\))23 b(=)589 b Fz(\(0.2\))1237 1342 y Fs(=)1345 1229 y Fq(Z)1307 1472 y Fp(R)1354 1455 y Fo(2)p Fn(N)1462 1342 y Fu(dy)26 b(dp)d(e)1719 1307 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)p Fv(p)1977 1342 y Fu(e)2016 1306 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)p Fv(A)2287 1312 y Fs(\()2329 1279 y Fn(x)p Fo(+)p Fn(y)p 2329 1293 108 3 v 2368 1326 a Fo(2)2446 1312 y Fs(\))2482 1342 y Fu(f)2546 1225 y Fq(\022)2617 1286 y Fu(x)c Fs(+)f Fu(y)p 2617 1323 193 4 v 2692 1399 a Fs(2)2819 1342 y Fu(;)c(p)2898 1225 y Fq(\023)2973 1342 y Fu(u)p Fs(\()p Fu(y)s Fs(\))598 b Fz(\(0.3\))-118 1621 y(cannot)29 b(b)r(e)h(the)g(righ)n(t) f(one,)h(since)f(it)h(lac)n(ks)e(gauge)h(co)n(v)-5 b(ariance:)38 b(If)30 b(one)f(c)n(ho)r(oses)g(another)f(v)n(ector)g(p)r(oten)n(tial)i Fu(A)3477 1590 y Fm(0)3530 1621 y Fz(asso)r(ciated)-118 1720 y(to)25 b Fu(B)t Fz(,)i(di\033ering)e(from)g(the)i(initial)e(one)h(b)n(y)f (the)h(gradien)n(t)e(of)i(a)f(scalar)f(function,)j Fu(A)2525 1690 y Fm(0)2572 1720 y Fs(=)22 b Fu(A)15 b Fs(+)f Fl(r)p Fu(\032)p Fz(,)27 b(then)f Fx(the)i(exp)l(e)l(cte)l(d)g(formula)-118 1820 y Fu(e)-79 1790 y Fv(i\032)-17 1820 y Fr(Op)93 1840 y Fv(A)147 1820 y Fs(\()p Fu(f)9 b Fs(\))p Fu(e)300 1790 y Fm(\000)p Fv(i\032)455 1820 y Fs(=)41 b Fr(Op)671 1840 y Fv(A)721 1823 y Fi(0)747 1820 y Fs(\()p Fu(f)9 b Fs(\))39 b Fx(do)l(es)h(not)g(hold)p Fz(.)71 b(In)38 b(fact,)k(this)c(form)n(ula)g(is)g(restored)f(if)i(one)f (replaces)f(the)i(phase)f(factor)-118 1941 y Fu(e)-79 1905 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)p Fv(A)192 1911 y Fs(\()233 1878 y Fn(x)p Fo(+)p Fn(y)p 233 1892 108 3 v 273 1925 a Fo(2)350 1911 y Fs(\))411 1941 y Fz(b)n(y)24 b Fu(e)562 1910 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)783 1863 y Fh(R)827 1884 y Fo(1)815 1932 y(0)871 1910 y Fv(dsA)p Ft(\(\(1)p Fm(\000)p Fv(s)p Ft(\))p Fv(x)p Ft(+)p Fv(sy)r Ft(\))1367 1941 y Fz(.)36 b(It)24 b(is)h(reassuring)d(to)i(note)g (that)h Fl(\000)p Fs(\()p Fu(x)12 b Fl(\000)g Fu(y)s Fs(\))g Fl(\001)2795 1874 y Fq(R)2851 1895 y Ft(1)2834 1970 y(0)2902 1941 y Fu(dsA)p Fs(\(\(1)g Fl(\000)g Fu(s)p Fs(\))p Fu(x)g Fs(+)g Fu(sy)s Fs(\))25 b Fz(is)f(in)h(fact)-118 2041 y(the)i(circulation)g Fs(\000)487 2010 y Fv(A)541 2041 y Fs(\([)p Fu(x;)14 b(y)s Fs(]\))28 b Fz(of)f(the)g(v)n(ector)f(p)r(oten)n(tial)h Fu(A)h Fz(through)e(the)h(segmen)n(t)g(leading)f(from)h Fu(x)h Fz(to)f Fu(y)s Fz(.)36 b(Th)n(us)27 b(the)h(form)n(ula)-118 2140 y(w)n(e)f(prop)r (ose)f(instead)i(of)f(\(0.2\))h(is)700 2438 y Fs([)p Fr(Op)833 2401 y Fv(A)887 2438 y Fs(\()p Fu(f)9 b Fs(\))p Fu(u)p Fs(]\()p Fu(x)p Fs(\))24 b(:=)1318 2325 y Fq(Z)1364 2514 y Fp(R)1411 2497 y Fo(2)p Fn(N)1506 2438 y Fu(dy)i(dp)d(e)1763 2404 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)p Fv(p)2021 2438 y Fu(e)2060 2404 y Fm(\000)p Fv(i)p Ft(\000)2176 2379 y Fn(A)2223 2404 y Ft(\([)p Fv(x;y)r Ft(]\))2410 2438 y Fu(f)2473 2321 y Fq(\022)2544 2382 y Fu(x)c Fs(+)f Fu(y)p 2544 2419 193 4 v 2620 2495 a Fs(2)2747 2438 y Fu(;)c(p)2826 2321 y Fq(\023)2901 2438 y Fu(u)p Fs(\()p Fu(y)s Fs(\))p Fu(:)647 b Fz(\(0.4\))6 2652 y(The)38 b(main)g(purp)r(ose)e(of)i(our)f(article)g(is)g(to)h(giv)n(e)e (an)h(explanation)g(of)h(these)f(facts.)67 b(As)37 b(a)h(prologue)d(for)i (doing)g(this,)-118 2751 y(in)32 b(the)g(\034rst)g(Section)f(w)n(e)h(review)f (some)g(facts)h(related)f(to)g(canonical)g(comm)n(utation)g(relations)g(and)g (W)-7 b(eyl)32 b(calculus)g(when)-118 2851 y(no)j(magnetic)h(\034eld)g(is)g (presen)n(t.)61 b(The)36 b(main)f(topic)h(will)g(b)r(e)g(the)h (justi\034cation)f(of)f(the)i(form)n(ula)e(for)g Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))36 b Fz(as)f(a)h(sort)f(of)-118 2951 y(in)n(tegrated)30 b(form)i(of)f Fx(the)j(W)-6 b(eyl)33 b(system)p Fz(,)g(whic)n(h)e(is)h(a)f (family)h(of)f(unitary)g(op)r(erators)f Fl(f)p Fu(W)12 b Fs(\()p Fu(\030)t Fs(\))p Fl(g)2933 2963 y Fv(\030)3000 2951 y Fz(indexed)32 b(b)n(y)f(the)h(p)r(oin)n(ts)g(of)-118 3050 y(the)f(phase)f(space)g(and)h (con)n(taining)f(the)h(relev)-5 b(an)n(t)30 b(information)g(on)g(the)h(comm)n (utation)f(relations)g(b)r(et)n(w)n(een)h(the)g(op)r(erators)-118 3150 y Fu(Q)j Fz(and)g Fu(P)12 b Fz(.)58 b(W)-7 b(e)35 b(claim)f(no)g (originalit)n(y)f(\(see)i(for)f(example)g([Hor1)n(]\);)39 b(w)n(e)34 b(include)h(this)g(here)f(b)r(ecause)g(it)h(seems)f(to)g(b)r(e)h(an)-118 3250 y(argumen)n(t)26 b(largely)f(ignored,)g(whic)n(h)i(is)g(basic)f(to)h (our)f(approac)n(h.)34 b(The)27 b(sym)n(b)r(olic)f(calculus)g(b)r(ey)n(ond)h (the)g(W)-7 b(eyl)27 b(prescription)-118 3349 y(is)j(the)g(famous)f(Mo)n(y)n (al)g(pro)r(duct.)44 b(Other)29 b(references)g(emphasizing)g(the)h (connection)g(b)r(et)n(w)n(een)f(Quan)n(tum)h(Mec)n(hanics)f(and)-118 3449 y(pseudo)r(di\033eren)n(tial)e(theory)g(are)f([GLS])i(and)g([F)-7 b(o)o(].)6 3548 y(When)27 b(a)f(magnetic)g(\034eld)h(is)f(presen)n(t,)g(the)h (W)-7 b(eyl)26 b(system)g(has)g(to)g(b)r(e)h(mo)r(di\034ed.)37 b(Instead)26 b(of)g(the)h(group)e(of)h(translations,)-118 3648 y(app)r(earing)32 b(naturally)h(in)h(the)g(form)n(ula)f(giving)f Fu(W)12 b Fs(\()p Fu(\030)t Fs(\))35 b Fz(\(see)e(\(1.8\)\),)j(one)d(has)g (to)h(w)n(ork)e(with)i(the)g(magnetic)f(translations,)-118 3748 y(forming)27 b(a)g(sort)g(of)g(generalized)f(pro)5 b(jectiv)n(e)26 b(represen)n(tation)g(of)i Fw(R)2025 3718 y Fv(N)2094 3748 y Fz(.)37 b(This)27 b(is)h(presen)n(ted)f(in)g(the)h(second)f(Section.)6 3847 y(In)32 b(the)f(third)g(Section,)h(the)f(form)n(ula)f Fr(Op)1353 3810 y Fv(A)1438 3847 y Fz(for)h(the)g(functional)g(calculus)f (with)i(magnetic)e(\034eld)h(is)g(deduced.)47 b(F)-7 b(or)30 b(this)-118 3947 y(w)n(e)i(apply)g(the)h(same)e(strategy)g(as)h(in)h(Section) f(1,)h(but)g(using)f(no)n(w)g(the)g(magnetic)g(W)-7 b(eyl)33 b(system,)g(in)n(tro)r(duced)f(in)h(Section)-118 4047 y(2.)65 b(The)38 b(setting)f(relies)f(on)h(the)h(c)n(hoice)e(of)h(a)g(v)n(ector)f(p)r (oten)n(tial,)k(but)e(no)n(w)e(gauge)g(co)n(v)-5 b(ariance)35 b(is)i(a)n(v)-5 b(ailable;)41 b(equiv)-5 b(alen)n(t)-118 4146 y(v)n(ector)24 b(p)r(oten)n(tials)h(lead)h(to)f(unitarily)g(equiv)-5 b(alen)n(t)25 b(op)r(erators.)35 b(The)25 b(basic)g(expression)f(for)h Fr(Op)2919 4109 y Fv(A)2974 4146 y Fs(\()p Fu(f)9 b Fs(\))25 b Fz(requires)g(rather)f(strong)-118 4246 y(conditions)j(on)g(the)g(function) h Fu(f)9 b Fz(.)36 b(But)28 b(for)e(large)g(classes)g(of)h(magnetic)g (\034elds)g(one)g(can)g(extend)g(it,)h(as)f(in)g(the)h(non-magnetic)-118 4345 y(case,)e(to)g(all)g(temp)r(ered)h(distributions,)f(b)n(y)g(a)g (suitable)g(in)n(terpretation)g(of)g Fr(Op)2388 4309 y Fv(A)2442 4345 y Fs(\()p Fu(f)9 b Fs(\))27 b Fz(as)f(a)g(linear)f(con)n(tin)n(uous)h (op)r(erator)e(from)-118 4445 y(the)35 b(Sc)n(h)n(w)n(artz)e(space)i Fl(S)6 b Fs(\()p Fw(R)764 4415 y Fv(N)833 4445 y Fs(\))36 b Fz(to)e(its)h(dual.)59 b(This)35 b(is)g(based)f(on)h(a)f(study)h(of)g(the)h (distribution)e(k)n(ernel)g(of)h(this)g(op)r(erator.)-118 4545 y(These)27 b(and)h(some)e(other)h(fundamen)n(tal)h(facts)f(are)g(also)f (considered)h(in)h(Section)f(3.)6 4644 y(Bey)n(ond)j(these)h(op)r(erators)e (lies)i(a)f(sym)n(b)r(olic)g(calculus)h(whic)n(h)f(is)h Fx(manifestly)j (gauge)f(invariant)p Fz(,)g(b)r(eing)e(de\034ned)g(only)f(in)-118 4744 y(terms)21 b(of)h(the)g(magnetic)g(\034eld.)35 b(The)22 b(comp)r(osition)f(is)h(a)f(magnetic)g(correction)f(of)i(the)g(Mo)n(y)n(al)f (pro)r(duct,)h(while)g(the)h(in)n(v)n(olution)-118 4844 y(is)i(just)h(the)f (usual)g(complex)g(conjugation)f(of)i(functions.)36 b(In)25 b(Section)g(4)g(w)n(e)g(study)h(this)f(sym)n(b)r(olic)g(calculus.)35 b(Once)25 b(again)f(an)-118 4943 y(imp)r(ortan)n(t)29 b(problem)f(is)h(to)g (extend)g(the)h(form)n(ulae)e(when)h(ob)n(vious)f(in)n(tegrabilit)n(y)g (conditions)g(are)g(not)h(satis\034ed.)41 b(This)29 b(is)g(a)-118 5043 y(more)23 b(complicated)h(task)f(for)h(the)g(pro)r(duct)g(than)h(it)f(w) n(as)f(for)h(the)g(quan)n(tization)g(itself,)h(esp)r(ecially)e(if)i(one)f (aims)f(at)h(obtaining)-118 5112 y Fm(\003)-80 5142 y Fz(-algebras.)34 b(W)-7 b(e)28 b(p)r(ostp)r(one)f(the)g(application)g(of)g(the)g(mac)n(hinery) f(of)h(oscillatory)e(in)n(tegrals)h(and)h(classical)f(sym)n(b)r(ol)g (function)-118 5242 y(spaces)h(to)h(a)f(future)h(article.)38 b(F)-7 b(or)27 b(our)g(presen)n(t)g(purp)r(oses)g(the)i(strategy)d(of)i (extension)g(b)n(y)f(dualit)n(y)h(metho)r(ds)g(\(see)g([GBV1],)-118 5342 y([GBV2])e(for)g(the)g(non-magnetic)f(case\))h(is)g(more)g(fruitful.)37 b(It)27 b(will)f(lead)g(to)g(a)g(large,)f(in)n(teresting)2992 5312 y Fm(\003)3030 5342 y Fz(-algebra)f(of)i(distributions)-118 5441 y(whic)n(h)h(will)h(b)r(e)g(called)f Fx(the)j(magnetic)g(Moyal)i(algebr) l(a)p Fz(.)6 5541 y(Of)j(course,)f(for)g(certain)g(problems,)h(a)f(w)n (ell-justi\034ed)g(norm)g(on)g(\(restricted\))2575 5511 y Fm(\003)2613 5541 y Fz(-algebras)e(of)i(sym)n(b)r(ols)f(could)h(b)r(e)h(v)n(ery)-118 5641 y(useful.)40 b(It)29 b(happ)r(ens)g(that)f(this)h(is)g(easier)e(to)h(ac) n(hiev)n(e)g(after)g(p)r(erforming)f(a)i(partial)e(F)-7 b(ourier)28 b(transform.)38 b(Surprisingly)-7 b(,)28 b(one)-118 5740 y(naturally)h (encoun)n(ters)g(certain)g Fu(C)1002 5710 y Fm(\003)1040 5740 y Fz(-algebras)f(whic)n(h)i(w)n(ere)f(studied)h(in)g(pure)g(mathematics,)g (with)g(little)h(connection)e(with)1869 5989 y(2)p eop %%Page: 3 3 bop -118 154 a Fz(ph)n(ysics.)49 b(These)32 b(are)f(sp)r(ecial)h(t)n(yp)r(es) g(of)g(t)n(wisted)g(crossed)e(pro)r(ducts,)j(asso)r(ciated)e(to)h(t)n(wisted) g(actions)f(of)h Fw(R)3395 124 y Fv(N)3496 154 y Fz(on)g(suitable)-118 254 y(ab)r(elian)25 b Fu(C)233 223 y Fm(\003)271 254 y Fz(-algebras)e(of)i(p) r(osition)f(observ)-5 b(ables.)35 b(They)25 b(w)n(ere)f(already)f(related)i (to)g(quan)n(tum)f(magnetic)h(\034elds)g(in)g([MP)q(];)h(see)-118 353 y(also)f([GI)q(],)i([Be1)o(])f(and)h([Be2)o(])g(for)f(related)g(w)n (orks.)35 b(In)26 b(a)h(future)f(publication)h(w)n(e)f(shall)g(extend)h (their)g(study)f(and)h(outline)f(the)-118 453 y(connection)h(with)h(our)f (pseudo)r(di\033eren)n(tial)g(calculus.)6 552 y(Concerning)i(the)h (di\033erence)f(b)r(et)n(w)n(een)g(the)h(expressions)e(\(0.2\))h(\(or)g (\(0.3\)\))g(and)h(\(0.4\))f(some)g(commen)n(ts)g(are)f(necessary)-7 b(.)-118 652 y(In)25 b(Subsection)g(3.4)f(w)n(e)g(shall)h(outline)g(some)f (situations)g(when)h(they)g(giv)n(e)f(the)h(same)f(result.)36 b(By)25 b(admitting)g(the)g(con)n(v)n(enien)n(t)-118 752 y(assumption)f(that) g(the)h(comp)r(onen)n(ts)f(of)g Fu(A)h Fz(are)e(smo)r(oth)h(functions)h(with) f(temp)r(erate)h(gro)n(wth,)e(b)r(oth)i(\(0.2\))f(and)g(\(0.4\))g(can)g(b)r (e)-118 851 y(extended)29 b(to)h(an)n(y)e(temp)r(erate)h(distribution)h Fu(f)9 b Fz(.)41 b(Then)30 b Fx(we)h(shal)t(l)i(have)g Fr(Op)2306 871 y Fv(A)2360 851 y Fs(\()p Fu(f)9 b Fs(\))26 b(=)f Fr(Op)2701 814 y Fv(A)2755 851 y Fs(\()p Fu(f)9 b Fs(\))31 b Fx(for)h(al)t(l)h Fu(f)40 b Fx(if)32 b(and)g(only)g(if)g Fu(A)f Fx(is)-118 951 y(line)l(ar)c Fz(\(this)g(is)g(one)f(of)g(the)h(most)f(imp)r(ortan)n(t)h (cases)e(app)r(earing)g(usually)h(in)h(the)g(literature\).)36 b(Remark)25 b(that)i(this)g(condition)-118 1051 y(corresp)r(onds)21 b(to)h(a)g(constan)n(t)g(magnetic)g(\034eld,)i(but)g Fx(it)h(is)g(not)g (gauge)h(invariant)p Fz(:)35 b(for)22 b(some)g(other)g Fu(A)3037 1020 y Fm(0)3084 1051 y Fz(with)h Fu(dA)3373 1020 y Fm(0)3420 1051 y Fs(=)f Fu(B)27 b Fs(=)c Fz(const,)-118 1160 y Fr(Op)-8 1180 y Fv(A)42 1163 y Fi(0)98 1160 y Fz(and)29 b Fr(Op)371 1123 y Fv(A)421 1098 y Fi(0)476 1160 y Fz(will)h(b)r(e)f(di\033eren)n(t!)41 b(\(The)29 b(pseudo)r(di\033eren)n(tial)g(calculus)f(corresp)r(onding)f(to)i (a)g(linear)f Fu(A)h Fz(w)n(as)f(dev)n(elopp)r(ed)-118 1260 y(in)j([BMGH)q(])g(in)h(connection)e(with)i(some)f(problems)f(in)i(pure)e (PDE)j(theory;)f(in)g(fact)f(the)h(term)f(\020magnetic)g(\034eld\021)38 b(is)31 b(nev)n(er)-118 1359 y(explicitly)e(men)n(tioned.\))43 b(One)29 b(will)h(also)e(ha)n(v)n(e)h Fr(Op)1525 1379 y Fv(A)1579 1359 y Fs(\()p Fu(f)9 b Fs(\))26 b(=)g Fr(Op)1920 1322 y Fv(A)1974 1359 y Fs(\()p Fu(f)9 b Fs(\))30 b Fx(for)j(any)d Fu(A)f Fz(if)h Fx(f)48 b Fz(is)29 b(a)g(p)r(olynomial)g(of)h(order)e Fl(\024)e Fs(2)p Fz(.)42 b(The)-118 1459 y(most)33 b(studied)h(magnetic)e(op)r(erators) f(are)i(the)g(Sc)n(hr\366)r(dinger)f(magnetic)h(Hamiltonians)f Fs(\()p Fu(P)j Fl(\000)21 b Fu(A)p Fs(\()p Fu(Q)p Fs(\)\))3183 1429 y Ft(2)3244 1459 y Fs(+)g Fu(V)e Fs(\()p Fu(Q)p Fs(\))p Fz(,)35 b(cf)f([AHS)q(])-118 1559 y(and)26 b([MR])h(for)f(instance.)36 b(They)26 b(are)g(obtained)g(b)n(y)g(quan)n(tizing)g(the)h(sum)f(b)r(et)n(w)n (een)g(a)h(quadratic)e(function)i(dep)r(endind)g(only)-118 1658 y(on)33 b Fu(p)g Fz(and)g(a)g(function)h(dep)r(ending)g(only)f(on)g Fu(x)p Fz(,)i(so)e(no)g(care)f(is)h(needed)h(in)f(this)h(case.)53 b(Ho)n(w)n(ev)n(er,)33 b(ev)n(en)g(their)g(study)g(ma)n(y)-118 1758 y(in)n(v)n(olv)n(e)25 b(applying)i(the)g(W)-7 b(eyl)27 b(calculus)g(to)g(more)f(complicated)g(sym)n(b)r(ols.)36 b(An)n(yho)n(w,)27 b(for)f(p)r(olynomials)g(of)h(order)f(three)h(in)g Fu(p)p Fz(,)-118 1857 y(\(0.2\))d(and)g(\(0.4\))g(already)f(giv)n(e)h(di\033eren)n(t)g (results.)35 b(W)-7 b(e)25 b(remark)e(that)i(using)f(the)g(W)-7 b(eyl)25 b(calculus)f(coupled)g(with)h(the)g(minimal)-118 1957 y(coupling)g(principle)h(\(as)f(in)i(\(0.2\)\))e(for)h(a)f(non-constan)n(t)g (magnetic)g(\034eld)h(and)g(a)f(complicated)h(sym)n(b)r(ol)f(is)h(still)g (legitimate)f(as)-118 2057 y(long)j(as)h(this)h(is)f(a)g(tec)n(hnical)g(to)r (ol)g(and)g(not)g(the)h(quan)n(tization)f(of)g(the)h(classical)d(observ)-5 b(able)28 b(represen)n(ted)g(b)n(y)h(the)h(sym)n(b)r(ol.)-118 2156 y(This)e(is)f(often)i(the)f(case)f(in)h(solid)f(state)h(ph)n(ysics,)f (in)h(argumen)n(ts)f(concerning)f(the)i(P)n(eierls)g(substitution,)g(as)f(in) i([GMS])f(and)-118 2256 y([PST].)6 2356 y(Our)e(feeling)g(is)g(that)h(the)f (magnetic)g(W)-7 b(eyl)26 b(calculus)g(elab)r(orated)f(in)h(this)h(pap)r(er)e (is)h(b)r(oth)h(an)f(in)n(teresting)f(mathematical)-118 2455 y(ob)5 b(ject)37 b(and)g(a)g(signi\034can)n(t)f(formalism)g(for)h (theoretical)f(ph)n(ysics.)65 b(W)-7 b(e)38 b(in)n(tended)f(to)g(b)r(e)h (accessible)e(to)h(p)r(eople)g(that)h(are)-118 2555 y(only)31 b(v)-5 b(aguely)31 b(familiar)h(with)g(pseudo)r(di\033eren)n(tial)f(theory;)j (more)d(tec)n(hnical)g(dev)n(elopmen)n(ts)g(or)g(applications)g(are)g (defered)-118 2654 y(to)c(future)h(w)n(orks.)35 b(W)-7 b(e)28 b(w)n(ere)f(encouraged)f(in)h(this)h(attitude)g(b)n(y)g(a)f(discussion)g (with)h(Joseph)e(A)-9 b(vron)27 b(and)g(Omri)g(Gat.)-118 2929 y Fy(1)131 b(Canonical)64 b(comm)l(utation)f(relations)h(and)e(pseudo)t 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Fl(2)d Fu(X)r(;)42 b(p)23 b Fl(2)g Fu(X)2652 3511 y Fv(?)3727 3545 y Fz(\(1.7\))-118 3728 y(holds.)46 b(No)n(w)30 b(there)g(is)h(no)f(am)n(biguit)n(y)g(in)h(addressing)e(the)i (abstract)f(problem)g(of)h(the)g(classi\034cation)e(of)h(triples)h Fs(\()p Fl(H)q Fu(;)14 b(U;)g(V)k Fs(\))p Fz(,)-118 3827 y(where)23 b Fl(H)i Fz(is)f(a)f(Hilb)r(ert)h(space)f(and)h Fu(U)32 b Fs(:)23 b Fu(X)29 b Fl(!)23 b(U)8 b Fs(\()p Fl(H)q Fs(\))p Fz(,)26 b Fu(V)41 b Fs(:)24 b Fu(X)1811 3797 y Fv(?)1871 3827 y Fl(!)f(U)8 b Fs(\()p Fl(H)q Fs(\))25 b Fz(are)e(strongly)f(con)n(tin)n(uous)h(unitary)g (represen)n(tations)-118 3927 y(satisfying)28 b(\(1.7\).)40 b(And)29 b(there)f(is)h(a)f(simple)h(answ)n(er,)f(giv)n(en)f(b)n(y)i(the)g (Stone-v)n(on)e(Neumann)i(Theorem)f(\(for)g(a)h(more)e(explicit)-118 4027 y(statemen)n(t)i(and)f(for)g(the)h(pro)r(of)f(w)n(e)h(send)f(to)h([F)-7 b(o]\):)39 b(If)29 b(one)g(also)e(assumes)h(irreducibilit)n(y)g(of)h(the)g (family)f Fl(f)p Fu(U)9 b Fs(\()p Fu(q)s Fs(\))p Fu(;)14 b(V)19 b Fs(\()p Fu(p)p Fs(\))25 b Fl(j)g Fu(q)j Fl(2)-118 4126 y Fu(X)r(;)14 b(p)26 b Fl(2)h Fu(X)216 4096 y Fv(?)253 4126 y Fl(g)p Fz(,)j(then)h(an)n(y)e(solution)g(is)h(unitarily)f(equiv)-5 b(alen)n(t)29 b(to)h(\(1.6\))f(\(whic)n(h)h(is)g(called)f Fx(the)j(Schr)l (\366)l(dinger)h(r)l(epr)l(esentation)p Fz(\).)-118 4226 y(And)28 b(a)f(non-irreducible)f(triple)i(is)f(just)i(a)e(m)n(ultiple)h(of)f(this)h (Sc)n(hr\366)r(dinger)e(represen)n(tation.)6 4325 y(A)39 b(con)n(v)n(enien)n (t)e(w)n(a)n(y)f(to)i(condense)f(the)i(t)n(w)n(o)e(ob)5 b(jects)37 b Fu(U)47 b Fz(and)38 b Fu(V)57 b Fz(in)n(to)38 b(a)f(single)g(one)h(is)g(to) g(de\034ne)g Fx(the)h(W)-6 b(eyl)40 b(system)-118 4425 y Fl(f)p Fu(W)12 b Fs(\()p Fu(\030)t Fs(\))23 b Fl(j)g Fu(\030)k Fl(2)d Fs(\004)p Fl(g)f(\032)f(U)8 b Fs(\()p Fl(H)q Fs(\))29 b Fz(b)n(y)672 4707 y Fu(W)12 b Fs(\()p Fu(q)s(;)i(p)p Fs(\))23 b(:=)g Fu(e)1131 4651 y Fn(i)p 1128 4660 29 3 v 1128 4693 a Fo(2)1166 4673 y Fv(q)r Fm(\001)p Fv(p)1279 4707 y Fu(U)9 b Fs(\()p Fl(\000)p Fu(q)s Fs(\))p Fu(V)19 b Fs(\()p Fu(p)p Fs(\))24 b(=)e Fu(e)1837 4673 y Fm(\000)1902 4651 y Fn(i)p 1899 4660 V 1899 4693 a Fo(2)1937 4673 y Fv(q)r Fm(\001)p Fv(p)2051 4707 y Fu(V)d Fs(\()p Fu(p)p Fs(\))p Fu(U)9 b Fs(\()p Fl(\000)p Fu(q)s Fs(\))p Fu(;)69 b(q)26 b Fl(2)e Fu(X)r(;)41 b(p)23 b Fl(2)g Fu(X)3047 4673 y Fv(?)3085 4707 y Fu(:)619 b Fz(\(1.8\))-118 4890 y(A)28 b(short)f(calculation)f(sho)n (ws)h(that)g Fu(W)40 b Fz(satis\034es)1103 5172 y Fu(W)12 b Fs(\()p Fu(\030)t Fs(\))p Fu(W)g Fs(\()p Fu(\021)s Fs(\))24 b(=)f Fu(e)1659 5116 y Fn(i)p 1655 5125 V 1655 5158 a Fo(2)1694 5138 y Fv(\033)r Ft(\()p Fv(\030)r(;\021)r Ft(\))1902 5172 y Fu(W)12 b Fs(\()p Fu(\030)22 b Fs(+)c Fu(\021)s Fs(\))p Fu(;)98 b(\030)t(;)14 b(\021)26 b Fl(2)d Fs(\004)p Fu(;)1065 b Fz(\(1.9\))-118 5374 y(i.e.)37 b Fu(W)i Fz(is)28 b(a)f(pro)5 b(jectiv)n(e)26 b(represen)n(tation)g(of)i(the)g(group)e Fs(\004)i Fz(with)g Fs(2)p Fz(-co)r(cycle)e(\(phase)h(factor\))g Fu(e)2908 5321 y Fn(i)p 2905 5330 V 2905 5363 a Fo(2)2943 5343 y Fv(\033)2988 5374 y Fz(.)6 5473 y(Of)37 b(course,)h Fu(W)49 b Fz(can)37 b(b)r(e)g(de\034ned)g(for)g(an)n(y)f(abstract)f(triple)i Fs(\()p Fl(H)q Fu(;)14 b(U;)g(V)19 b Fs(\))p Fz(.)65 b(But,)39 b(as)e(a)f (consequence)g(of)h(the)g(Stone-v)n(on)-118 5573 y(Neumann)32 b(Theorem,)h(it)g(is)f(enough)g(to)g(w)n(ork)f(with)h(the)h(Sc)n(hr\366)r (dinger)e(represen)n(tation.)49 b(The)32 b(corresp)r(onding)e Fu(W)45 b Fz(will)32 b(b)r(e)-118 5672 y(called)27 b Fx(the)j(Schr)l(\366)l (dinger)h(W)-6 b(eyl)30 b(system)d Fz(and)h(is)f(explicitely)h(giv)n(en)e(on) i Fu(L)2213 5642 y Ft(2)2249 5672 y Fs(\()p Fu(X)7 b Fs(\))28 b Fz(b)n(y)1869 5989 y(4)p eop %%Page: 5 5 bop 1223 326 a Fs([)p Fu(W)12 b Fs(\()p Fu(q)s(;)i(p)p Fs(\))p Fu(u)p Fs(]\()p Fu(y)s Fs(\))23 b(=)g Fu(e)1848 290 y Fm(\000)p Fv(i)1923 296 y Fs(\()1965 268 y Fo(1)p 1965 277 29 3 v 1965 310 a(2)2003 290 y Fv(q)r Ft(+)p Fv(y)2122 296 y Fs(\))2155 290 y Fm(\001)p Fv(p)2236 326 y Fu(u)p Fs(\()p Fu(y)e Fs(+)d Fu(q)s 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(w)n(een)h(the)g(basic)f(op)r(erators)f Fu(Q)i Fz(and)-118 897 y Fu(P)12 b Fz(.)49 b(In)32 b(the)g(next)g(paragraph,)e(the)i(quan)n (tization)e(b)n(y)i(pseudo)r(di\033eren)n(tial)f(op)r(erators)e(will)j(b)r(e) g(obtained)f(as)g(an)h(in)n(tegrated)-118 997 y(form)27 b(of)h(this)f(W)-7 b(eyl)28 b(system.)-118 1227 y Fg(1.3)112 b(Pseudo)s(di\033eren)m(tial)36 b(op)s(erators)-118 1381 y Fz(If)27 b(a)e(family)i(of)f(self-adjoin)n(t)f(op) r(erators)f Fu(S)1219 1393 y Ft(1)1257 1381 y Fu(;)14 b(:)g(:)g(:)f(;)h(S) 1492 1393 y Fv(m)1581 1381 y Fz(is)26 b(giv)n(en)g(suc)n(h)g(that)g(for)g(an) n(y)f Fu(i;)14 b(j)5 b Fz(,)26 b Fu(S)2729 1393 y Fv(i)2783 1381 y Fz(and)g Fu(S)2994 1393 y Fv(j)3055 1381 y Fz(comm)n(ute,)h(then)f (one)g(can)-118 1480 y(de\034ne)i(a)f(functional)g(calculus)h(for)f(this)g (family)h(b)n(y)f(one)g(of)h(the)g(t)n(w)n(o)f(form)n(ulae)1084 1701 y Fu(f)9 b Fs(\()p Fu(S)c Fs(\))23 b(=)1364 1588 y Fq(Z)1410 1777 y Fp(R)1457 1760 y Fn(m)1525 1701 y Fu(f)9 b Fs(\()p Fu(\025)p Fs(\))p Fu(dE)1791 1713 y Fv(S)1841 1701 y Fs(\()p Fu(\025)p Fs(\))24 b(=)2064 1588 y Fq(Z)2110 1777 y Fp(R)2157 1760 y Fn(m)2225 1701 y Fu(dt)2340 1679 y Fs(\024)2322 1701 y Fu(f)8 b Fs(\()p Fu(t)p Fs(\))p Fu(e)2504 1667 y Fm(\000)p Fv(it)p Fm(\001)p Fv(S)2673 1701 y Fu(:)-118 1891 y Fz(Here)41 b Fu(E)153 1903 y Fv(S)242 1891 y Fz(is)g(the)h(sp)r(ectral)e(measure)g(\(on)i Fw(R)1374 1861 y Fv(m)1443 1891 y Fz(\))f(of)g(the)h(family)f Fu(S)2099 1903 y Ft(1)2136 1891 y Fu(;)14 b(:)g(:)g(:)g(;)g(S)2372 1903 y Fv(m)2434 1891 y Fz(,)45 b(under)c(suitable)g(assumptions)82 b Fu(t)27 b Fl(\001)h Fu(S)50 b Fs(:=)-118 1991 y Fu(t)-88 2003 y Ft(1)-51 1991 y Fu(S)0 2003 y Ft(1)50 1991 y Fs(+)13 b Fl(\001)h(\001)g(\001)e Fs(+)h Fu(t)345 2003 y Fv(m)408 1991 y Fu(S)459 2003 y Fv(m)547 1991 y Fz(is)24 b(a)h(w)n(ell-de\034ned)f (self-adjoin)n(t)g(op)r(erator)f(and)2078 1969 y Fs(\024)2060 1991 y Fu(f)34 b Fz(is)24 b(the)h(in)n(v)n(erse)f(F)-7 b(ourier)23 b(tranform)h(of)h Fu(f)9 b Fz(,)25 b(con)n(v)n(enien)n(tly)-118 2090 y(normalized.)6 2190 y(If,)e(once)e(again,)g Fu(S)580 2202 y Ft(1)617 2190 y Fu(;)14 b(:)g(:)g(:)g(;)g(S)853 2202 y Fv(m)937 2190 y Fz(are)20 b(self-adjoin)n(t,)i(but)g(they)f(no)g(longer)f (comm)n(ute,)i(there)f(is)g(usualy)f(no)h(reasonable)e(sp)r(ectral)-118 2290 y(measure)32 b Fu(E)272 2302 y Fv(S)320 2290 y Fz(.)52 b(One)33 b(can)f(try)h(to)f(use)h(the)g(op)r(erator)e(v)n(ersion)g(of)i(the)g (F)-7 b(ourier)32 b(in)n(v)n(ersion)f(form)n(ula)h(to)g(de\034ne)h(a)g (functional)-118 2389 y(calculus.)j(The)27 b(k)n(ey)g(p)r(oin)n(t)g(w)n(ould) g(b)r(e)g(the)h(abilit)n(y)e(of)i(de\034ning)f(a)f(suitable)h(analogue)f(of)h Fu(e)2782 2359 y Fm(\000)p Fv(it)p Fm(\001)p Fv(S)2950 2389 y Fz(.)37 b(This)27 b(strategy)e(is)j(outlined)-118 2489 y(in)g([An1])g (\(see)g(also)f([An2]\))h(for)g(v)n(ery)e(general)h(situations.)37 b(But)28 b(the)h(prop)r(erties)e(of)h(the)g(resulting)f(functional)h (calculus)g(are)-118 2588 y(quite)g(mo)r(dest)f(if)h(the)g(comm)n(utation)f (relations)g(of)g(the)h(op)r(erators)e Fu(S)2098 2600 y Fv(j)2160 2588 y Fz(ha)n(v)n(e)h(no)g(in)n(teresting)g(p)r(eculiarities.)6 2688 y(W)-7 b(e)33 b(shall)e(sho)n(w)g(ho)n(w)g(this)i(program)c(can)j(b)r(e) g(implemen)n(ted)h(for)e(the)h(case)f Fu(m)g Fs(=)e(2)p Fu(N)9 b Fz(,)33 b Fu(S)2893 2700 y Fv(j)2958 2688 y Fs(=)d Fu(Q)3119 2700 y Fv(j)3186 2688 y Fz(if)i Fu(j)j Fs(=)30 b(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(N)41 b Fz(and)-118 2788 y Fu(S)-67 2800 y Fv(j)-9 2788 y Fs(=)22 b Fu(P)131 2800 y Fv(j)192 2788 y Fz(for)j Fu(j)j Fs(=)23 b Fu(N)g Fs(+)14 b(1)p Fu(;)g(:)g(:)g(:)f(;)h Fs(2)p Fu(N)9 b Fz(.)35 b(In)26 b(an)f(analoguous,)f(but)i(more)e (complicated)h(w)n(a)n(y)-7 b(,)25 b(in)h(Section)f(3)g(w)n(e)g(shall)g(do)h (the)f(same)-118 2887 y(for)i Fu(S)60 2899 y Fv(j)118 2887 y Fs(=)22 b Fu(Q)271 2899 y Fv(j)333 2887 y Fz(if)28 b Fu(j)g Fs(=)23 b(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(N)36 b Fz(and)27 b Fu(S)1100 2899 y Fv(j)1158 2887 y Fs(=)c(\005)1308 2857 y Fv(A)1308 2909 y(j)1390 2887 y Fz(for)k Fu(j)h Fs(=)22 b Fu(N)27 b Fs(+)18 b(1)p Fu(;)c(:)g(:)g(:)f(;)h Fs(2)p Fu(N)9 b Fz(,)26 b(with)i Fs(\005)2487 2857 y Fv(A)2487 2909 y(j)2565 2887 y Fs(=)22 b Fu(P)2705 2899 y Fv(j)2759 2887 y Fl(\000)17 b Fu(A)2903 2899 y Fv(j)2938 2887 y Fs(\()p Fu(Q)p Fs(\))28 b Fz(the)g Fu(j)5 b Fz('th)28 b(comp)r(onen)n(t)f(of)-118 2987 y(the)k(magnetic)f(momen) n(tum)g(de\034ned)h(b)n(y)f(a)h(v)n(ector)e(p)r(oten)n(tial)h Fu(A)p Fz(.)46 b(But)31 b(w)n(e)f(stop)g(for)g(a)g(momen)n(t)h(to)f(\034x)h (some)f(con)n(v)n(en)n(tions)-118 3087 y(on)d(F)-7 b(ourier)27 b(transforms)f(that)i(will)f(also)g(b)r(e)h(useful)g(later)f(on.)6 3186 y(In)f(fact)g(w)n(e)f(are)f(faced)i(with)f(t)n(w)n(o)g(problems:)35 b(normalization)24 b(and)h(the)h(c)n(hoice)f(of)g(a)g(go)r(o)r(d)g (de\034nition)h(on)f(the)h(symplectic)-118 3286 y(space.)55 b(The)34 b(Leb)r(egue)f(measures)f(on)i Fu(X)7 b Fz(,)35 b Fu(X)1358 3256 y Fv(?)1429 3286 y Fz(and)f Fs(\004)g Fz(are)f(not)h(the)g (most)f(con)n(v)n(enien)n(t)g(Haar)g(\(=)g(p)r(ositiv)n(e,)i(translational) -118 3385 y(in)n(v)-5 b(arian)n(t)37 b(Borel\))g(measures,)j(since)e(they)g (lead)g(to)g(the)h(app)r(earance)e(of)h(spurious)f(constan)n(ts.)68 b(Let)38 b(us)g(start)g(with)h(t)n(w)n(o)-118 3485 y(arbitrary)25 b(Haar)i(measures)f Fu(dx)i Fz(on)g Fu(X)34 b Fz(and)27 b Fu(dp)h Fz(on)f Fu(X)1599 3455 y Fm(\003)1636 3485 y Fz(.)37 b(One)28 b(de\034nes)f(at)g(the)h(lev)n(el)f(of)h(temp)r(ered)g(distributions)817 3684 y Fl(F)877 3696 y Fv(X)940 3684 y Fu(;)p 977 3618 68 4 v 14 w Fl(F)1045 3696 y Fv(X)1131 3684 y Fs(:)23 b Fl(S)1233 3650 y Fm(0)1257 3684 y Fs(\()p Fu(X)7 b Fs(\))23 b Fl(!)g(S)1582 3650 y Fm(0)1606 3684 y Fs(\()p Fu(X)1714 3650 y Fm(\003)1751 3684 y Fs(\))p Fu(;)97 b Fl(F)1963 3696 y Fv(X)2022 3680 y Fn(?)2061 3684 y Fu(;)p 2098 3618 V 14 w Fl(F)2166 3696 y Fv(X)2225 3680 y Fn(?)2287 3684 y Fs(:)23 b Fl(S)2389 3650 y Fm(0)2413 3684 y Fs(\()p Fu(X)2521 3650 y Fv(?)2558 3684 y Fs(\))h Fl(!)f(S)2776 3650 y Fm(0)2800 3684 y Fs(\()p Fu(X)7 b Fs(\))p Fu(;)-118 3828 y Fz(uniquely)28 b(determined)f(b)n(y)g(the)h(follo)n(wing)f(actions)g (on)g(in)n(tegrable)f(functions:)782 4049 y Fs(\()p Fl(F)874 4061 y Fv(X)937 4049 y Fu(u)p Fs(\)\()p Fu(p)p Fs(\))d(=)1234 3935 y Fq(Z)1280 4124 y Fv(X)1357 4049 y Fu(dx)h(e)1510 4014 y Fm(\000)p Fv(ix)p Fm(\001)p Fv(p)1681 4049 y Fu(u)p Fs(\()p Fu(x)p Fs(\))p Fu(;)97 b Fs(\()p 1992 3982 V Fl(F)2060 4061 y Fv(X)2123 4049 y Fu(u)p Fs(\)\()p Fu(p)p Fs(\))24 b(=)2420 3935 y Fq(Z)2466 4124 y Fv(X)2543 4049 y Fu(dx)g(e)2696 4014 y Fv(ix)p Fm(\001)p Fv(p)2815 4049 y Fu(u)p Fs(\()p Fu(x)p Fs(\))p Fu(;)730 4313 y Fs(\()p Fl(F)822 4325 y Fv(X)881 4309 y Fn(?)920 4313 y Fu(v)s Fs(\)\()p Fu(x)p Fs(\))g(=)1218 4200 y Fq(Z)1264 4389 y Fv(X)1323 4372 y Fn(?)1376 4313 y Fu(dp)f(e)1523 4279 y Fm(\000)p Fv(ix)p Fm(\001)p Fv(p)1693 4313 y Fu(v)s Fs(\()p Fu(p)p Fs(\))p Fu(;)98 b Fs(\()p 1995 4246 V Fl(F)2063 4325 y Fv(X)2122 4309 y Fn(?)2161 4313 y Fu(v)s Fs(\)\()p Fu(p)p Fs(\))24 b(=)2453 4200 y Fq(Z)2500 4389 y Fv(X)2559 4372 y Fn(?)2611 4313 y Fu(dp)f(e)2758 4279 y Fv(ix)p Fm(\001)p Fv(p)2877 4313 y Fu(v)s Fs(\()p Fu(p)p Fs(\))p Fu(:)-118 4498 y Fz(It)28 b(is)f(easily)g(sho)n(wn)g(that)h(there)f(exists)g Fu(c)c(>)g Fs(0)k Fz(suc)n(h)g(that)p 923 4630 V 923 4697 a Fl(F)991 4709 y Fv(X)1050 4693 y Fi(\003)1107 4697 y Fl(\016)18 b(F)1227 4709 y Fv(X)1313 4697 y Fs(=)k Fu(c)h Fz(id)1529 4712 y Fm(S)1574 4695 y Fi(0)1596 4712 y Ft(\()p Fv(X)5 b Ft(\))1794 4697 y Fz(and)83 b Fl(F)2071 4709 y Fv(X)2152 4697 y Fl(\016)p 2212 4630 V 18 w(F)2280 4709 y Fv(X)2339 4693 y Fn(?)2401 4697 y Fs(=)22 b Fu(c)h Fz(id)2617 4712 y Fm(S)2662 4695 y Fi(0)2684 4712 y Ft(\()p Fv(X)2769 4695 y Fn(?)2804 4712 y Ft(\))2834 4697 y Fu(:)-118 4852 y Fz(Th)n(us,)29 b(b)n(y)f(rede\034ning)g Fu(dx)i Fz(and)e Fu(dp)p Fz(,)h(one)g(gets)f Fl(F)1430 4816 y Fm(\000)p Ft(1)1422 4876 y Fv(X)1544 4852 y Fs(=)p 1633 4785 V 24 w Fl(F)1701 4864 y Fv(X)1760 4848 y Fn(?)1828 4852 y Fz(and)g Fl(F)2058 4816 y Fm(\000)p Ft(1)2050 4876 y Fv(X)2109 4860 y Fn(?)2173 4852 y Fs(=)p 2262 4785 V 24 w Fl(F)2330 4864 y Fv(X)2393 4852 y Fz(.)41 b(W)-7 b(e)29 b(\034x)f(suc)n(h)h(a)f(c)n(hoice)g (for)g Fu(dx)h Fz(and)g Fu(dp)p Fz(,)g(but)-118 4951 y(ob)n(viously)d Fu(d\030)h Fs(:=)c Fu(dx)c Fl(\012)f Fu(dp)28 b Fz(do)r(es)f(not)h(dep)r(end) g(on)f(this)h(c)n(hoice.)36 b(W)-7 b(e)28 b(also)e(set)i(the)g(symplectic)f (F)-7 b(ourier)27 b(transforms)1429 5151 y Fl(F)1489 5163 y Ft(\004)1536 5151 y Fu(;)14 b Fl(F)1641 5115 y Fm(\000)p Ft(1)1633 5175 y(\004)1753 5151 y Fs(:)23 b Fl(S)1855 5116 y Fm(0)1879 5151 y Fs(\(\004\))h Fl(!)f(S)2184 5116 y Fm(0)2208 5151 y Fs(\(\004\))p Fu(;)695 5340 y Fs(\()p Fl(F)787 5352 y Ft(\004)835 5340 y Fu(f)9 b Fs(\)\()p Fu(\030)t Fs(\))24 b(:=)1155 5227 y Fq(Z)1202 5416 y Ft(\004)1263 5340 y Fu(d\021)j(e)1413 5306 y Fm(\000)p Fv(i\033)r Ft(\()p Fv(\030)r(;\021)r Ft(\))1673 5340 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))p Fu(;)97 b Fs(\()p Fl(F)2051 5305 y Fm(\000)p Ft(1)2043 5364 y(\004)2140 5340 y Fu(f)9 b Fs(\)\()p Fu(\030)t Fs(\))24 b(:=)2461 5227 y Fq(Z)2507 5416 y Ft(\004)2569 5340 y Fu(d\021)i(e)2718 5306 y Fv(i\033)r Ft(\()p Fv(\030)r(;\021)r Ft(\))2926 5340 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))-118 5541 y Fz(and)27 b(note)h(that)g Fl(F)468 5553 y Ft(\004)538 5541 y Fs(=)23 b Fl(I)i(\016)18 b Fs(\()p Fl(F)848 5553 y Fv(X)929 5541 y Fl(\012)p 1012 5474 V 18 w(F)1080 5553 y Fv(X)1139 5537 y Fn(?)1178 5541 y Fs(\))p Fz(,)28 b(where)f Fl(I)i Fs(:)23 b Fl(S)1677 5511 y Fm(0)1701 5541 y Fs(\()p Fu(X)1809 5511 y Fv(?)1865 5541 y Fl(\002)18 b Fu(X)7 b Fs(\))23 b Fl(!)g(S)2241 5511 y Fm(0)2265 5541 y Fs(\()p Fu(X)i Fl(\002)18 b Fu(X)2550 5511 y Fv(?)2588 5541 y Fs(\))p Fz(,)28 b Fs(\()p Fl(I)6 b Fu(g)s Fs(\)\()p Fu(x;)14 b(p)p Fs(\))24 b(:=)e Fu(g)s Fs(\()p Fu(p;)14 b(x)p Fs(\))p Fz(.)6 5641 y(No)n(w,)31 b(for)f(an)n(y)g(W)-7 b(eyl)31 b(system)g Fs(\()p Fl(H)q Fu(;)14 b(W)e Fs(\))31 b Fz(w)n(e)f(de\034ne)h(\(at)f(least\))h(for)f(functions)h Fu(f)37 b Fs(:)28 b(\004)g Fl(!)h Fw(C)51 b Fz(with)31 b(in)n(tegrable)f(symplectic) -118 5740 y(F)-7 b(ourier)26 b(transform)1869 5989 y(5)p eop %%Page: 6 6 bop 1295 355 a Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))24 b(:=)1653 242 y Fq(Z)1699 431 y Ft(\004)1761 355 y Fu(d\030)k Fs(\()p Fl(F)1968 320 y Fm(\000)p Ft(1)1960 379 y(\004)2057 355 y Fu(f)9 b Fs(\)\()p Fu(\030)t Fs(\))24 b Fu(W)12 b Fs(\()p Fu(\030)t Fs(\))p Fu(:)1202 b Fz(\(1.11\))-118 576 y(W)-7 b(e)23 b(do)g(not)f(insist)h (on)g(the)g(precise)f(in)n(terpretation)f(of)i(this)g(form)n(ula;)h(this)f (will)g(b)r(e)g(done)f(later)g(on)h(in)g(the)g(more)f(complicated,)-118 676 y(magnetic)27 b(case.)6 775 y(Once)34 b(again,)i(b)n(y)e(the)h(Stone-v)n (on)e(Neumann)i(Theorem,)g(w)n(e)f(are)g(satis\034ed)g(with)h(the)g(case)e (of)i(the)g(Sc)n(hr\366)r(dinger)d(rep-)-118 875 y(resen)n(tation.)59 b(By)35 b(in)n(tro)r(ducing)f(the)i(explicit)f(form)g(of)h(the)f(Sc)n(hr\366) r(dinger)f(W)-7 b(eyl)36 b(system,)h(one)d(gets)h(immediatly)h(for)f(an)n(y) -118 974 y Fu(u)22 b Fl(2)i Fu(L)88 944 y Ft(2)125 974 y Fs(\()p Fu(X)7 b Fs(\))903 1283 y([)p Fr(Op)o Fs(\()p Fu(f)i Fs(\))p Fu(u)p Fs(]\()p Fu(x)p Fs(\))24 b(=)1443 1170 y Fq(Z)1489 1358 y Fv(X)1566 1283 y Fu(dy)1667 1170 y Fq(Z)1713 1358 y Fv(X)1772 1342 y Fn(?)1825 1283 y Fu(dp)f(e)1972 1248 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)p Fv(p)2230 1283 y Fu(f)2294 1165 y Fq(\022)2365 1226 y Fu(x)c Fs(+)f Fu(y)p 2365 1263 193 4 v 2440 1339 a Fs(2)2567 1283 y Fu(;)c(p)2646 1165 y Fq(\023)2721 1283 y Fu(u)p Fs(\()p Fu(y)s Fs(\))809 b Fz(\(1.12\))-118 1507 y(and)27 b(this)h(is)g(exactly)e(the)i(W)-7 b(eyl)28 b(prescription)f (to)g(quan)n(tize)g(classical)f(sym)n(b)r(ols.)6 1606 y(W)-7 b(e)35 b(note)e(that)i Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))34 b Fz(is)g(an)f(in)n(tegral)g(op)r(erator)f(with)i(k)n(ernel)f Fu(K)2169 1618 y Fv(f)2212 1606 y Fs(\()p Fu(x;)14 b(y)s Fs(\))34 b(:=)f([\(1)23 b Fl(\012)p 2766 1540 68 4 v 22 w(F)2834 1618 y Fv(X)2893 1602 y Fn(?)2932 1606 y Fs(\))p Fu(f)9 b Fs(]\()3079 1569 y Fv(x)p Ft(+)p Fv(y)p 3079 1587 125 4 v 3125 1635 a Ft(2)3213 1606 y Fu(;)14 b(x)23 b Fl(\000)f Fu(y)s Fs(\))p Fz(.)56 b(Then,)36 b(b)n(y)-118 1706 y(an)c(elemen)n(tary)g(application)g(of)h(Sc)n(h)n(w)n (artz's)d(Kernel)i(Theorem,)h(one)g(giv)n(es)e(a)h(sense)h(to)f Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))33 b Fz(for)g(an)n(y)e Fu(f)41 b Fl(2)32 b(S)3573 1676 y Fm(0)3596 1706 y Fs(\(\004\))i Fz(as)e(a)-118 1806 y(con)n(tin)n(uous)h(linear)h(op)r(erator)f(from)h Fl(S)6 b Fs(\()p Fu(X)h Fs(\))35 b Fz(to)f Fl(S)1482 1775 y Fm(0)1506 1806 y Fs(\()p Fu(X)7 b Fs(\))p Fz(.)58 b(In)34 b(fact)h(all)f(these)h(op)r (erators)d(are)i(of)g(the)h(form)f Fr(Op)p Fs(\()p Fu(f)9 b Fs(\))34 b Fz(for)g(some)-118 1905 y(unique)28 b(temp)r(ered)f(distribution)h Fu(f)9 b Fz(.)-118 2137 y Fg(1.4)112 b(The)38 b(Mo)m(y)m(al)f(algebra)-118 2290 y Fz(W)-7 b(e)28 b(turn)g(no)n(w)e(to)i(the)g(sym)n(b)r(olic)f (calculus.)36 b(It)28 b(is)g(easy)e(to)i(see)f(that)h(b)n(y)f(setting)976 2591 y Fs(\()p Fu(f)g Fl(\016)18 b Fu(g)s Fs(\)\()p Fu(\030)t Fs(\))24 b(:=)f(4)1492 2557 y Fv(N)1568 2478 y Fq(Z)1614 2667 y Ft(\004)1676 2591 y Fu(d\021)1777 2478 y Fq(Z)1823 2667 y Ft(\004)1885 2591 y Fu(d\020)30 b(e)2033 2557 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))2496 2591 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))p Fu(g)s Fs(\()p Fu(\020)d Fs(\))883 b Fz(\(1.13\))-118 2834 y(one)27 b(will)h(ha)n(v)n(e)e Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))p Fr(Op)p Fs(\()p Fu(g)s Fs(\))23 b(=)g Fr(Op)o Fs(\()p Fu(f)k Fl(\016)18 b Fu(g)s Fs(\))p Fz(,)28 b(and)f(that)h Fr(Op)o Fs(\()p Fu(f)9 b Fs(\))1896 2803 y Fm(\003)1958 2834 y Fs(=)22 b Fr(Op)o Fs(\()p Fu(f)2237 2803 y Fm(\016)2275 2834 y Fs(\))p Fz(,)28 b(with)g Fu(f)2597 2803 y Fm(\016)2635 2834 y Fs(\()p Fu(x)p Fs(\))c(:=)p 2881 2761 162 4 v 23 w Fu(f)9 b Fs(\()p Fu(x)p Fs(\))q Fz(.)6 2933 y(The)30 b(non-comm)n(utativ)n(e)d(comp)r(osition) h(la)n(w)h Fl(\016)f Fz(is)h(often)h(called)e Fx(the)k(Moyal)g(pr)l(o)l(duct) e Fz(\(or)e Fx(the)j(W)-6 b(eyl)32 b(pr)l(o)l(duct)p Fz(\).)41 b(It)30 b(mak)n(es)-118 3033 y(sense)k(for)h(suitable)g(sym)n(b)r(ols,)h(sa)n (y)d Fu(f)t(;)14 b(g)38 b Fl(2)e(S)6 b Fs(\(\004\))p Fz(.)61 b(F)-7 b(or)34 b(man)n(y)g(purp)r(oses)g(it)i(is)f(useful)g(to)g(extend)g(it) g(to)g(larger)e(classes)h(of)-118 3132 y(functions)k(and)f(distributions.)66 b(The)38 b(standard)e(approac)n(h)g(\(see)h([F)-7 b(o],)40 b([Hor1)o(],)g([Hor2)o(],)g([Sh])e(and)f(man)n(y)g(others\))g(is)g(via)-118 3232 y(oscillatory)25 b(in)n(tegrals.)36 b(Better)27 b(suited)h(to)f(our)g (setting)g(is)g(the)h(approac)n(h)e(b)n(y)h(dualit)n(y)g(of)g([An)n(t)q(],)g ([GBV1])h(and)f([GBV2])g(that)-118 3332 y(w)n(e)g(review)g(no)n(w)g (brie\035y)-7 b(.)6 3431 y(Let)33 b(us)f(denote)f(b)n(y)h Fs(\()p Fl(\001)p Fu(;)14 b Fl(\001)p Fs(\))33 b Fz(the)f(dualit)n(y)g Fl(S)1331 3401 y Fm(0)1355 3431 y Fs(\()p Fu(X)7 b Fs(\))21 b Fl(\002)g(S)6 b Fs(\()p Fu(X)h Fs(\))31 b Fl(!)f Fw(C)15 b Fz(.)56 b(By)32 b(a)g(simple)g(calculation)f(w)n(e)g(see)h(that)g(for)g(an) n(y)f(three)-118 3531 y(functions)d Fu(f)9 b Fz(,)27 b Fu(g)j Fz(and)e Fu(h)f Fz(in)h Fl(S)6 b Fs(\(\004\))29 b Fz(w)n(e)e(ha)n(v)n(e)862 3710 y Fs(\()p Fu(f)t(;)14 b(g)21 b Fl(\016)d Fu(h)p Fs(\))23 b(=)g(\()p Fu(f)k Fl(\016)18 b Fu(g)s(;)c(h)p Fs(\))23 b(=)f(\()p Fu(h;)14 b(f)27 b Fl(\016)18 b Fu(g)s Fs(\))23 b(=)g(\()p Fu(h)18 b Fl(\016)g Fu(f)t(;)c(g)s Fs(\))23 b(=)g(\()p Fu(g)s(;)14 b(h)k Fl(\016)g Fu(f)9 b Fs(\))p Fu(:)-118 3890 y Fz(Th)n(us,)25 b(w)n(e)g(can)f(extend)h Fl(\016)g Fz(to)g(mappings)f Fl(S)6 b Fs(\(\004\))13 b Fl(\002)g(S)1508 3860 y Fm(0)1533 3890 y Fs(\(\004\))24 b Fl(!)f(S)1838 3860 y Fm(0)1862 3890 y Fs(\(\004\))j Fz(and)f Fl(S)2222 3860 y Fm(0)2246 3890 y Fs(\(\004\))13 b Fl(\002)g(S)6 b Fs(\(\004\))25 b Fl(!)e(S)2818 3860 y Fm(0)2842 3890 y Fs(\(\004\))j Fz(b)n(y)f Fs(\()p Fu(f)d Fl(\016)13 b Fu(G;)h(h)p Fs(\))23 b(:=)g(\()p Fu(G;)14 b(h)f Fl(\016)g Fu(f)c Fs(\))-118 3989 y Fz(and)28 b Fs(\()p Fu(F)j Fl(\016)18 b Fu(g)s(;)c(h)p Fs(\))24 b(:=)f(\()p Fu(F)r(;)14 b(g)23 b Fl(\016)18 b Fu(h)p Fs(\))p Fz(,)29 b(for)e Fu(f)t(;)14 b(g)s(;)g(h)23 b Fl(2)i(S)6 b Fs(\(\004\))29 b Fz(and)f Fu(F)r(;)14 b(G)25 b Fl(2)f(S)2017 3959 y Fm(0)2041 3989 y Fs(\(\004\))p Fz(.)40 b(This)28 b(is)g(already)f (useful)h(and)g(allo)n(ws)f(comp)r(osing)g Fu(n)-118 4089 y Fz(sym)n(b)r(ols)g(if)h(all)f(except)h(one)f(are)f(in)i(the)g(Sc)n(h)n(w)n (artz)e(space.)6 4189 y(No)n(w)j(set)g Fl(M)p Fs(\(\004\))d(:=)f Fl(f)p Fu(F)37 b Fl(2)25 b(S)954 4158 y Fm(0)978 4189 y Fs(\(\004\))h Fl(j)g Fu(F)31 b Fl(\016)19 b(S)6 b Fs(\(\004\))27 b Fl(\032)e(S)6 b Fs(\(\004\))54 b Ff(and)g Fl(S)6 b Fs(\(\004\))21 b Fl(\016)e Fu(F)37 b Fl(\032)25 b(S)6 b Fs(\(\004\))p Fl(g)p Fz(.)42 b(Just)29 b(b)n(y)f(some)h(abstract)f(nonsense)-118 4288 y(one)33 b(c)n(hec)n(ks)f (that)h Fl(M)p Fs(\(\004\))h Fz(is)f(a)904 4258 y Fm(\003)942 4288 y Fz(-algebra)e(under)j(the)f(\(extension)g(of)6 b(\))34 b(the)g(Mo)n(y)n(al)d(pro)r(duct)j Fl(\016)e Fz(and)h(the)h(in)n(v)n(olution) 3706 4258 y Fm(\016)3745 4288 y Fz(.)54 b(In)-118 4388 y([GBV1])36 b Fl(M)p Fs(\(\004\))h Fz(is)f(called)g Fx(the)i(Moyal)h(algebr)l(a)f Fz(and)e(some)f(of)i(its)f(prop)r(erties)f(are)g(studied.)64 b(In)36 b(particular)f(it)h(is)h(sho)n(wn)-118 4487 y(that)26 b Fl(M)p Fs(\(\004\))h Fz(is)f(stable)g(under)g(all)g(sort)f(of)h(F)-7 b(ourier)25 b(transforms,)g(it)i(con)n(tains)e(all)g(the)i(distributions)f (with)g(compact)g(supp)r(ort)-118 4587 y(and)33 b(\(th)n(us\))i(large)d (classes)g(of)i(analytic)f(functions.)55 b(It)34 b(also)e(con)n(tains)h(the)h (family)g(of)f Fu(C)2787 4557 y Fm(1)2892 4587 y Fz(functions)g(on)h Fs(\004)g Fz(with)g(all)f(the)-118 4687 y(deriv)-5 b(ativ)n(es)26 b(dominated)i(b)n(y)f(the)h(same)f(\(arbitrary\))f(p)r(olynomial.)6 4786 y(W)-7 b(e)28 b(will)g(reconsider)e(this)i(topic)f(in)h(greater)e (detail)i(in)f(Section)h(4,)f(where)g(the)h(magnetic)f(\034eld)h(will)g(also) e(b)r(e)i(presen)n(t.)-118 5060 y Fy(2)131 b(The)44 b(magnetic)g(W)-11 b(eyl)44 b(system)-118 5242 y Fz(W)-7 b(e)32 b(consider)g(a)f(quan)n(tum)i (particle)e(without)i(in)n(ternal)e(structure)h(mo)n(ving)f(in)h Fu(X)38 b Fs(=)30 b Fw(R)2729 5212 y Fv(N)2798 5242 y Fz(,)j(in)g(the)f (presence)g(of)g(a)g(v)-5 b(ariable)-118 5342 y(magnetic)33 b(\034eld.)54 b(The)33 b Fx(magnetic)j(\034eld)d Fz(is)h(describ)r(ed)f(b)n (y)f(a)h(closed)g(con)n(tin)n(uous)f(\034eld)i(of)f(2-forms)f Fu(B)37 b Fz(de\034ned)d(on)f Fw(R)3676 5312 y Fv(N)3745 5342 y Fz(.)54 b(In)-118 5441 y(the)38 b(standard)f(co)r(ordinate)f(system)i(on)f Fw(R)1275 5411 y Fv(N)1344 5441 y Fz(,)k(it)d(is)f(represen)n(ted)g(b)n(y)g (a)h(con)n(tin)n(uous)e(function)j(taking)e(real)g(an)n(tisymetric)-118 5541 y(matrix)24 b(v)-5 b(alues)24 b(and)h(v)n(erifying)e(the)i(co)r(cycle)f (relation)g Fu(@)1659 5553 y Fv(j)1694 5541 y Fu(B)1757 5553 y Fv(k)q(l)1832 5541 y Fs(+)13 b Fu(@)1954 5553 y Fv(k)1994 5541 y Fu(B)2057 5553 y Fv(lj)2126 5541 y Fs(+)g Fu(@)2248 5553 y Fv(l)2273 5541 y Fu(B)2336 5553 y Fv(j)s(k)2431 5541 y Fs(=)22 b(0)j Fz(in)g(a)f(distributional)g(sense.)35 b(The)25 b(reader)-118 5641 y(will)k(v)n(erify)f(for)h(himself)g(that)g(man)n(y)g (constructions)e(and)i(assertions)e(will)j(still)f(b)r(e)g(v)-5 b(alid)29 b(for)g(lo)r(cally)f(in)n(tegrable)f(\034elds;)j(w)n(e)-118 5740 y(assumed)d(con)n(tin)n(uit)n(y)g(for)g(simplicit)n(y)g(and)h(to)f(ha)n (v)n(e)g(a)g(uniform)g(framew)n(ork.)1869 5989 y(6)p eop %%Page: 7 7 bop 6 154 a Fz(It)35 b(is)e(w)n(ell-kno)n(wn)g(that)h(an)n(y)f(suc)n(h)g (\034eld)i Fu(B)j Fz(ma)n(y)33 b(b)r(e)h(written)g(as)f(the)h (\(distributional\))g(di\033eren)n(tial)g Fu(dA)g Fz(of)g(a)g(\034eld)g(of) -118 254 y(1-forms)25 b Fu(A)p Fz(,)j Fx(the)h(ve)l(ctor)h(p)l(otential)p Fz(,)e(that)f(is)g(highly)f(non-unique)h(\(the)h(gauge)d(am)n(biguit)n(y\);)i (b)n(y)f(using)h(co)r(ordinates,)f(one)g(has)-118 353 y Fu(B)-55 365 y Fv(j)s(k)40 353 y Fs(=)c Fu(@)171 365 y Fv(j)206 353 y Fu(A)268 365 y Fv(k)328 353 y Fl(\000)c Fu(@)455 365 y Fv(k)496 353 y Fu(A)558 365 y Fv(j)621 353 y Fz(for)27 b(eac)n(h)g Fu(j;)14 b(k)26 b Fs(=)c(1)p Fu(;)14 b Fl(\001)g(\001)g(\001)27 b Fu(;)14 b(N)9 b Fz(.)6 453 y(In)40 b(this)f(Section)h(w)n(e)e(shall)h(deduce)h(a)e (form)n(ula)h(for)f(the)i(analog)e(of)h(the)g(W)-7 b(eyl)40 b(system)f(of)g(Subsection)g(1.2,)j(but)d(in)-118 552 y(whic)n(h)c(the)g (magnetic)f(\034eld)i(is)e(also)g(tak)n(en)h(in)n(to)f(accoun)n(t.)58 b(One)35 b(ma)n(y)f(pro)r(ceed)g(as)h(in)g(Subsection)g(1.2,)h(with)f(the)g (single)-118 652 y(mo)r(di\034cation)30 b(whic)n(h)g(consists)f(in)h (replacing)f(the)i(translations)d(b)n(y)i(the)h(magnetic)f(translations)e (\(exp)r(onen)n(tials)i(of)g Fu(q)23 b Fl(\001)d Fs(\005)3820 622 y Fv(A)3875 652 y Fz(,)-118 752 y(where)h Fs(\005)178 722 y Fv(A)256 752 y Fs(:=)h Fu(P)e Fl(\000)7 b Fu(A)p Fs(\()p Fu(Q)p Fs(\))22 b Fz(is)g(the)g(magnetic)g(momen)n(tum\).)35 b(Just)22 b(for)g(a)f(c)n(hange,)h(w)n(e)g(pro)r(ceed)f(in)i(a)e(di\033eren)n (t,)i(but)g(equiv)-5 b(alen)n(t,)-118 851 y(w)n(a)n(y)e(.)33 b(First)22 b(w)n(e)f(get)g(directly)g(the)g(form)n(ula)g(for)g(our)f (magnetic)h(W)-7 b(eyl)22 b(system)f(b)n(y)g(exp)r(onen)n(tiating)f(the)i (self-adjoin)n(t)f(op)r(erators)-118 951 y Fu(\033)s Fs([\()p Fu(q)s(;)14 b(p)p Fs(\))p Fu(;)g Fs(\()p Fu(Q;)g Fs(\005)372 921 y Fv(A)427 951 y Fs(\)])23 b(=)g Fu(Q)16 b Fl(\001)g Fu(p)g Fl(\000)g Fu(q)j Fl(\001)d Fs(\()p Fu(P)28 b Fl(\000)16 b Fu(A)p Fs(\()p Fu(Q)p Fs(\)\))p Fz(,)28 b Fs(\()p Fu(q)s(;)14 b(p)p Fs(\))23 b Fl(2)g Fs(\004)p Fz(.)37 b(Then)27 b(the)g(magnetic)f (translations)f(and)h(the)h(magnetic)f(form)g(of)-118 1051 y(the)i(W)-7 b(eyl)28 b(comm)n(utation)f(relations)f(are)g(deduced)i(as)f (consequences.)6 1150 y(The)c(magnetic)e(translations)g(ha)n(v)n(e)f(app)r (eared)i(since)f(long)h(in)g(the)g(ph)n(ysical)g(literature)f(\(see)h([Lu])g (and)g([Za)o(])g(for)g(example\),)-118 1250 y(esp)r(ecially)28 b(in)h(connection)f(with)h(problems)f(in)h(solid)f(state)g(ph)n(ysics.)39 b(Most)29 b(of)f(the)h(times)g(they)g(w)n(ere)e(used)i(for)f(the)h(case)f(of) -118 1349 y(a)f(constan)n(t)g(\034eld;)h(some)f(references)f(are)h([Be1)o(],) h([Be2)o(],)g([HS1],)f([HS2])h(and)f([Ne)q(].)6 1449 y(W)-7 b(e)36 b(stress)e(that)h(the)g(new)g(ob)5 b(jects)34 b(app)r(earing)g(in)h (the)h(magnetic)e(case)g(are)g(t)n(w)n(o)g(phase)g(factors:)51 b(One)34 b(is)h(de\034ned)g(as)-118 1549 y(the)h(imaginary)d(exp)r(onen)n (tial)i(of)g(the)h(circulation)e(of)h(the)h(v)n(ector)e(p)r(oten)n(tial;)39 b(it)d(en)n(ters)e(the)i(de\034nition)f(of)h(the)f(magnetic)-118 1648 y(translations,)26 b(the)j(magnetic)e(W)-7 b(eyl)28 b(system)g(and)g (\(as)f(a)h(consequence,)f(in)h(Section)g(3\))g(in)g(the)h(expression)d(of)i (the)g(magnetic)-118 1748 y(pseudo)r(di\033eren)n(tial)c(op)r(erators.)34 b(The)25 b(other)f(one,)h(an)f(imaginary)f(exp)r(onen)n(tial)h(of)h(the)g (\035ux)g(of)f(the)h(magnetic)f(\034eld,)i(app)r(ears)-118 1848 y(in)h(connection)e(with)i(m)n(ultiplicativ)n(e)g(prop)r(erties)e(of)h (the)h(magnetic)f(translations)f(and)h(of)g(the)h(magnetic)f(W)-7 b(eyl)27 b(system)f(and)-118 1947 y(\(as)31 b(a)g(consequence,)h(in)g (Section)g(4\))f(in)h(the)g(expression)e(of)i(the)g(comp)r(osition)f(la)n(w)g (de\034ning)h(the)g(sym)n(b)r(olic)f(calculus.)48 b(W)-7 b(e)-118 2047 y(hop)r(e)36 b(that)h(our)e(treatmen)n(t)h(will)h(constitute)g(a)f (source)f(of)h(uni\034cation)g(of)g(the)h(v)-5 b(arious)35 b(\020non-in)n(tegrable)f(phase)i(factors\021)-118 2147 y(scattered)27 b(in)g(the)h(literature)f(on)g(quan)n(tum)h(magnetic)f(\034elds.)-118 2372 y Fg(2.1)112 b(The)38 b(magnetic)e(W)-9 b(eyl)36 b(system)-118 2526 y Fz(Giv)n(en)27 b(a)g Fu(k)s Fz(-form)g Fu(C)34 b Fz(on)27 b Fu(X)34 b Fz(and)28 b(a)f(compact)g Fu(k)s Fz(-surface)f Fu(\015)i Fl(\032)22 b Fu(X)7 b Fz(,)27 b(w)n(e)h(de\034ne)1629 2709 y Fs(\000)1681 2675 y Fv(C)1737 2709 y Fs(\()p Fu(\015)5 b Fs(\))23 b(:=)1983 2596 y Fq(Z)2029 2785 y Fv(\015)2085 2709 y Fu(C)-118 2912 y Fz(\(this)i(in)n(tegral)f(ha)n(ving)f(a)h(w)n (ell-de\034ned)h(in)n(v)-5 b(arian)n(t)23 b(meaning\).)36 b(W)-7 b(e)25 b(shall)f(mainly)h(encoun)n(ter)f(circulations)f(of)i(1-forms)e(along) -118 3012 y(linear)k(segmen)n(ts)f(\()p Fu(\015)i Fs(=)23 b([)p Fu(x;)14 b(y)s Fs(])p Fz(\))28 b(and)f(\035uxes)h(of)f(2-forms)f(through)h (triangles)f(\()p Fu(\015)i Fs(=)p Fu(<)23 b(x;)14 b(y)s(;)g(z)26 b(>)p Fz(\).)6 3111 y(W)-7 b(e)28 b(denote)g(b)n(y)f Fl(H)i Fz(the)f(Hilb)r(ert)g(space)f Fu(L)1339 3081 y Ft(2)1375 3111 y Fs(\()p Fu(X)7 b Fs(\))p Fz(.)37 b(F)-7 b(or)27 b(eac)n(h)g Fu(t)c Fl(2)g Fw(R)34 b Fz(w)n(e)27 b(de\034ne)720 3257 y Fu(W)810 3222 y Fv(A)798 3277 y(t)887 3257 y Fs(:)c(\004)g Fl(!)g(U)8 b Fs(\()p Fl(H)q Fs(\))p Fu(;)181 b(W)1606 3222 y Fv(A)1594 3277 y(t)1660 3257 y Fs(\()p Fu(x;)14 b(p)p Fs(\))24 b(:=)e Fu(e)2023 3222 y Fm(\000)p Fv(it)p Ft(\()p Fv(Q)p Ft(+)p Fv(tx=)p Ft(2\))p Fm(\001)p Fv(p)2467 3257 y Fs(\003)2525 3222 y Fv(A)2578 3257 y Fs(\()p Fu(Q)p Fs(;)14 b Fu(tx)p Fs(\))p Fu(e)2861 3222 y Fv(itx)p Fm(\001)p Fv(P)3023 3257 y Fu(;)640 b Fz(\(2.14\))-118 3402 y(where)27 b(w)n(e)g(in)n(tro)r(duced)g(the)h(exp)r(onen)n(tial)f(of)h (the)g(circulation)e(of)i(the)g(v)n(ector)e(p)r(oten)n(tial)1069 3560 y Fs(\003)1127 3526 y Fv(A)1181 3560 y Fs(\()p Fu(q)s Fs(;)14 b Fu(x)p Fs(\))24 b(:=)f Fu(e)1543 3526 y Fm(\000)p Fv(i)p Ft(\000)1659 3501 y Fn(A)1706 3526 y Ft(\([)p Fv(q)r(;q)r Ft(+)p Fv(x)p Ft(]\))1996 3560 y Fs(=)g Fu(e)2123 3526 y Fm(\000)p Fv(ix)p Fm(\001)2256 3479 y Fh(R)2300 3499 y Fo(1)2288 3548 y(0)2344 3526 y Fv(ds)c(A)p Ft(\()p Fv(q)r Ft(+)p Fv(sx)p Ft(\))2687 3560 y Fu(:)976 b Fz(\(2.15\))6 3705 y(W)-7 b(e)30 b(mak)n(e)d(the)j(con)n(v) n(en)n(tion)d(that)i(the)g(v)n(ector)e(p)r(oten)n(tial)i(will)g(alw)n(a)n(ys) e(b)r(e)i(tak)n(en)f(con)n(tin)n(uous.)40 b(This)28 b(is,)h(indeed,)h(alw)n (a)n(ys)-118 3805 y(p)r(ossible,)d(since)g Fu(B)32 b Fz(is)c(supp)r(osed)f (con)n(tin)n(uous,)g(b)n(y)g Fx(the)j(tr)l(ansversal)g(gauge)1299 4110 y Fu(A)1361 4122 y Fv(i)1389 4110 y Fs(\()p Fu(x)p Fs(\))24 b(=)f Fl(\000)1721 4007 y Fv(N)1690 4031 y Fq(X)1693 4208 y Fv(j)s Ft(=1)1824 3997 y Fq(Z)1907 4018 y Ft(1)1870 4186 y(0)1958 4110 y Fu(ds)g(B)2126 4122 y Fv(ij)2185 4110 y Fs(\()p Fu(sx)p Fs(\))p Fu(sx)2421 4122 y Fv(j)2457 4110 y Fu(:)1206 b Fz(\(2.16\))-118 4339 y(Non-con)n(tin)n(uous)26 b(v)n(ector)g(p)r(oten)n(tials)h(are)g(not)h (really)e(useful)i(in)g(our)f(framew)n(ork,)e(but)k(they)e(could)h(also)e(b)r (e)i(handled)g(either)-118 4439 y(directly)f(or)g(b)n(y)g(exploiting)g(gauge) f(co)n(v)-5 b(ariance.)6 4538 y(The)27 b(next)h(Lemma)e(sa)n(ys)g(that)h Fl(f)p Fu(W)1145 4508 y Fv(A)1133 4559 y(t)1198 4538 y Fs(\()p Fu(x;)14 b(p)p Fs(\))p Fl(g)1430 4550 y Fv(t)p Fm(2)p Fp(R)1574 4538 y Fz(is)27 b(the)g(ev)n(olution)f(group)g(of)h(the)g(self-adjoin)n(t)f (op)r(erator)f Fu(Q)17 b Fl(\001)h Fu(p)f Fl(\000)f Fu(x)i Fl(\001)f Fs(\005)3820 4508 y Fv(A)3875 4538 y Fz(,)-118 4638 y(suitably)27 b(de\034ned.)-118 4746 y FA(Lemma)36 b(2.1.)51 b Fx(W)-6 b(e)35 b(have)h Fu(W)864 4716 y Fv(A)852 4766 y(t)918 4746 y Fs(\()p Fu(x;)14 b(p)p Fs(\))33 b(=)e Fu(e)1276 4716 y Fm(\000)p Fv(it\033)r Ft([\()p Fv(x;p)p Ft(\))p Fv(;)p Ft(\()p Fv(Q;)p Ft(\005)1746 4691 y Fn(A)1793 4716 y Ft(\)])1842 4746 y Fx(,)36 b(wher)l(e)g(the)f(self-adjoint)h(op)l(er)l(ator)g Fu(\033)s Fs([\()p Fu(x;)14 b(p)p Fs(\))p Fu(;)g Fs(\()p Fu(Q;)g Fs(\005)3550 4716 y Fv(A)3605 4746 y Fs(\)])35 b Fx(is)g(the)-118 4846 y(closur)l(e)30 b(of)g(the)g(r)l(estriction)g(at)53 b Fl(S)6 b Fs(\()p Fu(X)h Fs(\))30 b Fx(of)h(the)e(sum)g Fu(S)24 b Fs(+)18 b Fu(T)12 b Fx(,)29 b(with)h Fu(S)e Fs(=)23 b Fu(Q)18 b Fl(\001)g Fu(p)g Fs(+)g Fu(x)h Fl(\001)g Fu(A)p Fs(\()p Fu(Q)p Fs(\))30 b Fx(and)g Fu(T)k Fs(=)23 b Fl(\000)p Fu(x)18 b Fl(\001)h Fu(P)12 b Fx(.)-118 5016 y Fz(P)c(r)g(o)g(o)g(f)g(.)21 b(It)h(is)g(kno)n(wn)f (that)i Fu(S)12 b Fs(+)7 b Fu(T)33 b Fz(is)22 b(indeed)g(essen)n(tially)f (self-adjoin)n(t)h(on)f Fu(C)2298 4986 y Fm(1)2292 5037 y Fv(c)2369 5016 y Fs(\()p Fu(X)7 b Fs(\))p Fz(.)35 b(Th)n(us,)23 b(w)n(e)e(can)h(apply)g (T)-7 b(rotter's)21 b(form)n(ula)-118 5116 y(\(see)27 b([RS)q(],)g(Th.)37 b(VI)r(I.31\).)g(W)-7 b(e)28 b(set)g Fu(S)f Fs(=)c Fu(a)p Fs(\()p Fu(Q)p Fs(\))28 b Fz(and)f(calculate)819 5277 y Fq(\020)868 5369 y Fu(e)907 5335 y Fm(\000)976 5313 y Fn(i)p 969 5322 37 3 v 969 5355 a(n)1015 5335 y Fv(ta)p Ft(\()p Fv(Q)p Ft(\))1185 5369 y Fu(e)1241 5313 y Fn(i)p 1233 5322 V 1233 5355 a(n)1280 5335 y Fv(tx)p Fm(\001)p Fv(P)1418 5277 y Fq(\021)1467 5294 y Fv(n)1563 5369 y Fs(=)51 b Fu(e)1718 5335 y Fm(\000)1786 5313 y Fn(i)p 1779 5322 V 1779 5355 a(n)1826 5335 y Fv(ta)p Ft(\()p Fv(Q)p Ft(\))1995 5369 y Fu(e)2051 5313 y Fn(i)p 2044 5322 V 2044 5355 a(n)2090 5335 y Fv(tx)p Fm(\001)p Fv(P)2228 5369 y Fu(e)2267 5335 y Fm(\000)2336 5313 y Fn(i)p 2328 5322 V 2328 5355 a(n)2375 5335 y Fv(ta)p Ft(\()p Fv(Q)p Ft(\))2544 5369 y Fu(e)2583 5335 y Fm(\000)2652 5313 y Fn(i)p 2645 5322 V 2645 5355 a(n)2691 5335 y Fv(tx)p Fm(\001)p Fv(P)2848 5369 y Fl(\001)507 5551 y(\001)28 b Fu(e)607 5494 y Fo(2)p Fn(i)p 607 5503 51 3 v 614 5537 a(n)667 5517 y Fv(tx)p Fm(\001)p Fv(P)805 5551 y Fu(e)844 5517 y Fm(\000)913 5494 y Fn(i)p 905 5503 37 3 v 905 5537 a(n)952 5517 y Fv(ta)p Ft(\()p Fv(Q)p Ft(\))1121 5551 y Fu(e)1160 5517 y Fm(\000)1222 5494 y Fo(2)p Fn(i)p 1222 5503 51 3 v 1229 5537 a(n)1282 5517 y Fv(tx)p Fm(\001)p Fv(P)1420 5551 y Fu(e)1469 5494 y Fo(3)p Fn(i)p 1469 5503 V 1476 5537 a(n)1529 5517 y Fv(tx)p Fm(\001)p Fv(P)1681 5551 y Fu(:)14 b(:)g(:)f(e)1840 5488 y Fo(\()p Fn(n)p Fi(\000)p Fo(1\))p Fn(i)p 1840 5504 177 3 v 1910 5537 a(n)2027 5517 y Fv(tx)p Fm(\001)p Fv(P)2165 5551 y Fu(e)2204 5517 y Fm(\000)2272 5494 y Fn(i)p 2265 5503 37 3 v 2265 5537 a(n)2312 5517 y Fv(ta)p Ft(\()p Fv(Q)p Ft(\))2481 5551 y Fu(e)2520 5517 y Fm(\000)2582 5488 y Fo(\()p Fn(n)p Fi(\000)p Fo(1\))p Fn(i)p 2582 5504 177 3 v 2652 5537 a(n)2769 5517 y Fv(tx)p Fm(\001)p Fv(P)2906 5551 y Fu(e)2955 5494 y Fn(ni)p 2955 5503 59 3 v 2966 5537 a(n)3024 5517 y Fv(tx)p Fm(\001)p Fv(P)3184 5551 y Fs(=)828 5710 y(=)23 b Fu(e)955 5674 y Fm(\000)1016 5652 y Fn(it)p 1016 5661 46 3 v 1020 5694 a(n)1072 5680 y Fs([)1095 5674 y Fv(a)p Ft(\()p Fv(Q)p Ft(\)+)p Fv(a)1322 5680 y Fs(\()1354 5674 y Fv(Q)p Ft(+)1467 5652 y Fn(tx)p 1467 5661 57 3 v 1477 5694 a(n)1534 5680 y Fs(\))1566 5674 y Ft(+)p Fm(\001\001\001)o Ft(+)p Fv(a)1763 5680 y Fs(\()1795 5674 y Fv(Q)p Ft(+)1908 5645 y Fo(\()p Fn(n)p Fi(\000)p Fo(1\))p Fn(tx)p 1908 5661 212 3 v 1996 5694 a(n)2129 5680 y Fs(\))q(])2189 5710 y Fu(e)2228 5676 y Fv(itx)p Fm(\001)p Fv(P)2389 5710 y Fu(:)1869 5989 y Fz(7)p eop %%Page: 8 8 bop -118 154 a Fz(One)28 b(notes)f(the)i(app)r(earance)d(of)i(a)g(Riemman)g (sum)g(at)g(the)g(exp)r(onen)n(t,)g(hence)g(the)h(last)e(expression)g(con)n (v)n(erges)e(strongly)i(to)-118 254 y Fu(e)-79 223 y Fm(\000)p Fv(i)7 176 y Fh(R)52 196 y Fn(t)39 245 y Fo(0)91 223 y Fv(ds)19 b(a)p Ft(\()p Fv(Q)p Ft(+)p Fv(sx)p Ft(\))440 254 y Fu(e)479 223 y Fv(itx)p Fm(\001)p Fv(P)640 254 y Fz(.)37 b(The)27 b(pro)r(of)g(is)h (ended)g(b)n(y)f(remarking)f(that)648 376 y Fq(Z)731 397 y Fv(t)694 565 y Ft(0)774 489 y Fu(ds)e Fl(f)p Fs(\()p Fu(y)c Fs(+)e Fu(sx)p Fs(\))i Fl(\001)e Fu(p)g Fs(+)g Fu(x)h Fl(\001)g Fu(A)p Fs(\()p Fu(y)i Fs(+)d Fu(sx)p Fs(\))p Fl(g)24 b Fs(=)e Fu(ty)f Fl(\001)e Fu(p)f Fs(+)2325 433 y Fu(t)2355 403 y Ft(2)p 2325 470 68 4 v 2338 546 a Fs(2)2402 489 y Fu(x)h Fl(\001)g Fu(p)f Fs(+)g(\000)2705 455 y Fv(A)2759 489 y Fs([)p Fu(y)s(;)c(y)21 b Fs(+)d Fu(tx)p Fs(])p Fu(:)3833 712 y Fe(\004)6 871 y Fz(Another,)44 b(more)39 b(anno)n(ying,)j(pro)r(of)d(w)n(ould)h(consist)g(in)g(sho)n(wing)f (that)i(for)e(all)h Fu(\030)48 b Fl(2)d Fs(\004)p Fz(,)57 b Fu(t)45 b Fl(7!)f Fu(W)3252 883 y Fv(t)3281 871 y Fs(\()p Fu(\030)t Fs(\))d Fz(is)f(a)g(strongly)-118 971 y(con)n(tin)n(uous)26 b(unitary)h(group)g(in)h Fl(H)g Fz(and)g(then)g(doing)f(the)h(necessary)d (deriv)-5 b(ations.)6 1070 y(W)e(e)28 b(note)g(the)g(ob)n(vious)e(form)n(ula) h Fu(W)1172 1040 y Fv(A)1160 1091 y(t)1226 1070 y Fs(\()p Fu(\030)t Fs(\))c(=)g Fu(W)1531 1040 y Fv(A)1519 1091 y Ft(1)1585 1070 y Fs(\()p Fu(t\030)t Fs(\))p Fz(.)38 b(The)27 b(op)r(erator)f Fu(W)2375 1040 y Fv(A)2363 1091 y Ft(1)2429 1070 y Fs(\()p Fu(\030)t Fs(\))i Fz(will)g(b)r(e)g(denoted)g(simply)f(b)n(y)h Fu(W)3616 1040 y Fv(A)3670 1070 y Fs(\()p Fu(\030)t Fs(\))p Fz(.)-118 1170 y FA(De\034nition)42 b(2.2.)64 b Fz(The)37 b(family)h Fl(f)p Fu(W)1129 1140 y Fv(A)1182 1170 y Fs(\()p Fu(\030)t Fs(\))p Fl(g)1328 1182 y Fv(\030)r Fm(2)p Ft(\004)1491 1170 y Fz(will)f(b)r(e)h(called)f Fx(the)h(magnetic)i(W)-6 b(eyl)38 b(system)h(asso)l(ciate)l(d)h(to)f(the)f(ve)l(ctor)-118 1270 y(p)l(otential)30 b Fu(A)p Fz(.)37 b(W)-7 b(e)28 b(write)g(do)n(wn)f(here,)g (for)g(further)g(use,)h(the)g(action)f(of)g Fu(W)2270 1239 y Fv(A)2324 1270 y Fs(\()p Fu(\030)t Fs(\))i Fz(on)e(v)n(ectors)f Fu(u)d Fl(2)g(H)h Fs(=)f Fu(L)3242 1239 y Ft(2)3278 1270 y Fs(\()p Fu(X)7 b Fs(\))p Fz(:)938 1485 y Fq(\002)973 1552 y Fu(W)1063 1518 y Fv(A)1117 1552 y Fs(\()p Fu(x;)14 b(p)p Fs(\))p Fu(u)1355 1485 y Fq(\003)1403 1552 y Fs(\()p Fu(y)s Fs(\))24 b(=)e Fu(e)1661 1518 y Fm(\000)p Fv(i)p Ft(\()p Fv(y)r Ft(+)p Fv(x=)p Ft(2\))p Fm(\001)p Fv(p)2038 1552 y Fu(e)2077 1518 y Fm(\000)p Fv(i)p Ft(\000)2193 1493 y Fn(A)2240 1518 y Ft(\([)p Fv(y)r(;y)r Ft(+)p Fv(x)p Ft(]\))2514 1552 y Fu(u)p Fs(\()p Fu(y)f Fs(+)d Fu(x)p Fs(\))p Fu(:)845 b Fz(\(2.17\))6 1735 y(The)30 b(usual)f(W)-7 b(eyl)29 b(system)g(w)n(as)f(a)h(pro)5 b(jectiv)n(e)28 b(represen)n(tation)f(of)j Fs(\004)p Fz(.)42 b(No)n(w)29 b(the)g(situation)g(is)g(of)h(the)f(same)g(nature,)g(but)-118 1834 y(more)e(in)n(v)n(olv)n(ed.)35 b(F)-7 b(or)27 b Fu(x;)14 b(y)s(;)g(q)26 b Fl(2)d Fu(X)7 b Fz(,)27 b(let)h(us)g(de\034ne)1236 2116 y Fs(\012)1296 2082 y Fv(B)1353 2116 y Fs(\()p Fu(q)s Fs(;)14 b Fu(x;)g(y)s Fs(\))23 b(:=)g Fu(e)1795 2082 y Fm(\000)p Fv(i)p Ft(\000)1911 2057 y Fn(B)1960 2082 y Ft(\()p Fv()p Ft(\))2521 2116 y Fu(:)1142 b Fz(\(2.18\))-118 2299 y(W)-7 b(e)26 b(note)g(that)h(this)f (is)g(a)g(con)n(tin)n(uous)f(function)h(of)g Fu(q)j Fz(for)d(\034xed)g Fu(x)h Fz(and)e Fu(y)s Fz(,)i(th)n(us)f(it)g(de\034nes)g(a)g(m)n (ultiplication)g(op)r(erator)e(in)j Fl(H)q Fz(.)-118 2399 y FA(Prop)s(osition)j(2.3.)37 b Fx(F)-6 b(or)30 b(any)59 b Fu(\030)28 b Fs(=)22 b(\()p Fu(x;)14 b(k)s Fs(\))p Fu(;)g(\021)27 b Fs(=)c(\()p Fu(y)s(;)14 b(l)r Fs(\))22 b Fl(2)i Fs(\004)30 b Fx(one)g(has)1033 2681 y Fu(W)1123 2647 y Fv(A)1177 2681 y Fs(\()p Fu(\030)t Fs(\))p Fu(W)1371 2647 y Fv(A)1426 2681 y Fs(\()p Fu(\021)s Fs(\))24 b(=)e Fu(e)1697 2624 y Fn(i)p 1694 2633 29 3 v 1694 2667 a Fo(2)1732 2647 y Fv(\033)r Ft(\()p Fv(\030)r(;\021)r Ft(\))1917 2681 y Fs(\012)1977 2647 y Fv(B)2034 2681 y Fs(\()p Fu(Q)p Fs(;)14 b Fu(x;)g(y)s Fs(\))p Fu(W)2419 2647 y Fv(A)2473 2681 y Fs(\()p Fu(\030)23 b Fs(+)18 b Fu(\021)s Fs(\))p Fu(:)940 b Fz(\(2.19\))-118 2864 y(P)8 b(r)g(o)g(o)g(f)g(.)38 b(By)i(Stok)n(es)f (Theorem)f(coupled)i(with)g(the)g(relation)f Fu(B)47 b Fs(=)c Fu(dA)p Fz(,)h(one)39 b(gets)g(for)g(an)n(y)g Fu(x;)14 b(y)s(;)g(q)47 b Fl(2)c Fu(X)j Fz(the)40 b(equalit)n(y)-118 2976 y Fs(\012)-58 2945 y Fv(B)-1 2976 y Fs(\()p Fu(q)s Fs(;)14 b Fu(x;)g(y)s Fs(\))24 b(=)e(\003)437 2945 y Fv(A)491 2976 y Fs(\()p Fu(q)s Fs(;)14 b Fu(x)p Fs(\)\003)737 2945 y Fv(A)792 2976 y Fs(\()p Fu(q)21 b Fs(+)d Fu(x)p Fs(;)c Fu(y)s Fs(\))1139 2908 y Fq(\002)1174 2976 y Fs(\003)1232 2945 y Fv(A)1286 2976 y Fs(\()p Fu(q)s Fs(;)g Fu(x)19 b Fs(+)f Fu(y)s Fs(\))1620 2908 y Fq(\003)1655 2926 y Fm(\000)p Ft(1)1744 2976 y Fz(.)37 b(Then)27 b(\(2.19\))g(follo)n(ws)g (b)n(y)g(a)g(routine)g(calculation.)414 b Fe(\004)6 3134 y Fz(Let)34 b(us)f(denote)g(b)n(y)g Fu(C)6 b Fs(\()p Fu(X)h Fs(;)14 b Fu(U)9 b Fs(\(1\)\))33 b Fz(the)g(group)f(\(with)i(p)r(oin)n(t)n(wise)f(m)n (ultiplication\))g(of)g(all)g(con)n(tin)n(uous)f(functions)h(on)g 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b(are)g(all)h(in)n(tegral)f(op)r(erators)g(and)h(b)n(y)g(the)h(Kernel)e (Theorem)h(they)g(can)g(also)g(b)r(e)g(de\034ned)h(for)f(sym)n(b)r(ols)f (whic)n(h)i(are)-118 2889 y(temp)r(ered)e(distributions.)39 b(The)28 b(problem)g(of)g(iden)n(tifying)g(\034nite-rank,)g(Hilb)r(ert-Sc)n (hmidt)h(and)f(compact)g(op)r(erators)e(is)i(also)-118 2989 y(addressed.)47 b(F)-7 b(or)31 b(this,)h(an)g(extension)e(of)i(the)g (classical)d(F)-7 b(ourier-Wigner)30 b(transform)g(\(cf.)49 b([F)-7 b(o]\))32 b(is)f(of)g(great)f(help.)49 b(It)32 b(also)-118 3088 y(sho)n(ws)20 b(a)g(p)r(osteriori)f(the)i(irreducibilit)n(y)f(of)h(our)f (magnetic)g(W)-7 b(eyl)21 b(system.)35 b(In)21 b(the)g(\034nal)g(part)f(of)h (the)g(Section)f(w)n(e)h(compare)e(the)-118 3188 y(magnetic)25 b(W)-7 b(eyl)26 b(calculus)f(and)h(the)g(comp)r(osition)f(of)h(the)g(usual)g (W)-7 b(eyl)26 b(caculus)f(with)h(the)h(minimal)f(coupling)f(prescription.) -118 3288 y(They)37 b(are)e(di\033eren)n(t)i(but,)j(striking)c(enough,)i (they)f(giv)n(e)f(the)h(same)f(result)h(in)g(man)n(y)f(imp)r(ortan)n(t)g (cases.)64 b(This)36 b(explains)-118 3387 y(p)r(erhaps)23 b(the)h(fact)g (that)h(the)f(quan)n(tization)f(of)h(observ)-5 b(ables)22 b(in)i(a)g (magnetic)f(\034eld)h(has)f(not)h(b)r(een)h(treated)e(prop)r(erly)g(b)r (efore)g(in)-118 3487 y(a)29 b(suitable)g(generalit)n(y)-7 b(.)39 b(W)-7 b(e)30 b(note,)f(ho)n(w)n(ev)n(er,)f(that)h(in)g([Lu])h(one)e (\034nds)i(\(in)f(a)g(non-systematic)f(setting\))h(the)h(righ)n(t)e(attitude) -118 3587 y(for)f(the)h(case)f(of)g(p)r(erio)r(dic)h(sym)n(b)r(ols)f(dep)r (ending)g(only)h(on)f Fu(p)p Fz(.)37 b(W)-7 b(e)28 b(thank)f(Professor)f (Nenciu)i(for)f(dra)n(wing)g(our)f(atten)n(tion)i(to)-118 3686 y(this)g(reference.)6 3786 y(Let)35 b(us)g(\034nally)f(remark)f(that)i(for)f (a)h(constan)n(t)f(magnetic)g(\034eld,)i(using)f(a)f(linear)g(v)n(ector)f(p)r (oten)n(tial,)j(one)f(is)f(lead)h(to)f(a)-118 3885 y(c)n(hange)d(of)i(the)g (canonic)f(symplectic)g(form)h(of)f(the)h(space)f 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5989 y(9)p eop %%Page: 10 10 bop -118 155 a Fz(where)137 134 y Fs(~)129 155 y(\003)187 124 y Fv(A)241 155 y Fs(\()p Fu(x;)14 b(y)s Fs(\))36 b(:=)f Fu(e)631 124 y Fm(\000)p Fv(i)p Ft(\000)747 99 y Fn(A)794 124 y Ft(\([)p Fv(x;y)r Ft(]\))1016 155 y Fs(=)g(\003)1174 124 y Fv(A)1228 155 y Fs(\()p Fu(x)p Fs(;)14 b Fu(y)26 b Fl(\000)d Fu(x)p Fs(\))p Fz(.)60 b(F)-7 b(or)34 b Fu(A)i Fs(=)f(0)f Fz(this)h(is)g(the)h(usual)e(W)-7 b(eyl)35 b(prescription)f(to)h(quan)n(tize)f(a)-118 254 y(classical)27 b(sym)n(b)r(ol,)i(encoun)n(tered)e(in)i(the)g(theory)f(of)h(pseudo)r (di\033eren)n(tial)f(op)r(erators.)38 b(F)-7 b(or)28 b(general)g(\(con)n(tin) n(uous\))g Fu(A)h Fz(this)g(is,)-118 354 y(in)f(our)f(opinion,)g(the)h(righ)n (t)e(form)n(ula)h(that)h(should)f(stand)h(for)f Fu(f)9 b Fs(\()p Fu(Q;)14 b Fs(\005)2137 324 y Fv(A)2190 354 y Fs(\))p Fz(.)6 453 y(In)28 b(fact,)f(the)h(precise)f(sense)f(of)i(\(3.27\))e(and)h(\(3.28\)) g(and)g(of)g(their)g(equiv)-5 b(alence)27 b(dep)r(end)h(on)f(our)f (assumptions)h(on)g Fu(f)36 b Fz(and)-118 553 y Fu(u)p Fz(.)g(In)28 b(the)f(next)h(paragraphs,)c(under)j(certain)g(h)n(yp)r(othesis)f(on)h(the)h (magnetic)f(\034eld,)g(w)n(e)g(shall)g(co)n(v)n(er)e(the)j(v)n(ery)e(general) g(case)-118 653 y(in)g(whic)n(h)g Fu(f)34 b Fz(is)26 b(a)f(temp)r(erate)g (distribution;)i(then)f(b)r(oth)g(form)n(ulae)f(will)h(mak)n(e)f(sense)g (with)h(a)f(suitable)h(rein)n(terpretation)e(and)-118 752 y(actually)i (de\034ne)h(the)h(same)e(ob)5 b(ject.)37 b(If)27 b Fu(f)36 b Fz(is)27 b(sub)5 b(ject)27 b(to)g(suitable)g(strong)f(deca)n(y)g (assumptions,)g(then)i(no)e(sp)r(ecial)h(condition)-118 852 y(is)35 b(needed)h(\(except)g(our)e(standing)h(con)n(v)n(en)n(tion)f(that)i Fu(A)g Fz(is)f(con)n(tin)n(uous\).)60 b(All)36 b(is)f(smo)r(oth,)i(for)e (example,)i(if)f Fu(f)44 b Fz(is)36 b(in)f(the)-118 952 y(Sc)n(h)n(w)n(artz) 26 b(class)h Fl(S)6 b Fs(\(\004\))p Fz(.)40 b(On)27 b(the)i(other)e(hand,)h (once)g(again)e(without)j(an)n(y)e(assumption)g(on)h(the)g(magnetic)g (\034eld,)g(\(3.27\))f(can)-118 1051 y(b)r(e)i(extended)g(straigh)n(tforw)n (ardly)c(to)k Fu(f)9 b Fz('s)28 b(that)h(are)e(F)-7 b(ourier)28 b(transforms)f(of)i(b)r(ounded)g(complex)f(measures)f(on)i Fs(\004)p Fz(.)40 b(No)n(w,)29 b(of)-118 1151 y(course,)d(\(3.28\))h(needs)g (a)h(rein)n(terpretation.)6 1250 y(T)-7 b(o)28 b(adv)n(o)r(cate)e(our)h(c)n (hoice)f(of)i(the)g(mapping)f Fr(Op)1560 1214 y Fv(A)1614 1250 y Fz(,)h(an)f(imp)r(ortan)n(t)g(p)r(oin)n(t)h(is)g(to)f(note)h Fx(gauge)i(c)l(ovarianc)l(e)p Fz(:)-118 1350 y FA(Prop)s(osition)37 b(3.4.)55 b Fx(L)l(et)35 b Fu(A)h Fx(and)g Fu(A)1086 1320 y Fm(0)1145 1350 y Fx(b)l(e)g(two)g(c)l(ontinuous)e(ve)l(ctor)i(p)l(otentials)h (de\034ning)e(the)h(same)g(c)l(ontinuous)f(magnetic)-118 1450 y(\034eld:)57 b Fu(dA)39 b Fs(=)g Fu(B)k Fs(=)38 b Fu(dA)674 1420 y Fm(0)698 1450 y Fx(.)65 b(Then)39 b(ther)l(e)g(exists)f(a)h(r)l(e)l (al)g Fu(C)1778 1420 y Ft(1)1815 1450 y Fx(-function)f Fu(\032)h Fx(on)f Fu(X)45 b Fx(such)39 b(that)f Fu(A)2941 1420 y Fm(0)3004 1450 y Fs(=)g Fu(A)26 b Fs(+)e Fl(r)p Fu(\032)39 b Fx(and)g(we)f(have)-118 1549 y Fu(e)-79 1519 y Fv(i\032)p Ft(\()p Fv(Q)p Ft(\))87 1549 y Fu(W)177 1519 y Fv(A)231 1549 y Fs(\()p Fu(\030)t Fs(\))p Fu(e)374 1519 y Fm(\000)p Fv(i\032)p Ft(\()p Fv(Q)p Ft(\))615 1549 y Fs(=)23 b Fu(W)793 1519 y Fv(A)p Ft(+)p Fm(r)p Fv(\032)988 1549 y Fs(\()p Fu(\030)t Fs(\))30 b Fx(for)h(al)t(l)g Fu(\030)c Fl(2)c Fs(\004)30 b Fx(and)h Fu(e)1801 1519 y Fv(i\032)p Ft(\()p Fv(Q)p Ft(\))1966 1549 y Fr(Op)2077 1512 y Fv(A)2131 1549 y Fs(\()p Fu(f)9 b Fs(\))p Fu(e)2284 1519 y Fm(\000)p Fv(i\032)p Ft(\()p Fv(Q)p Ft(\))2525 1549 y Fs(=)22 b Fr(Op)2723 1512 y Fv(A)p Ft(+)p Fm(r)p Fv(\032)2918 1549 y Fs(\()p Fu(f)9 b Fs(\))29 b Fx(for)i(al)t(l)g Fu(f)g Fl(2)24 b Fr(F)3515 1561 y Ft(\004)3563 1549 y Fu(L)3620 1519 y Ft(1)3656 1549 y Fs(\(\004\))p Fx(.)-118 1720 y Fz(P)8 b(r)g(o)g(o)g(f)g(.)21 b(It)i(is)g(w)n(ell-kno)n(wn)e (\(cf.)36 b([Le])23 b(for)f(example\))g(that)h(if)g Fu(dA)h Fs(=)e Fu(B)28 b Fs(=)22 b Fu(dA)2284 1690 y Fm(0)2331 1720 y Fz(and)g Fu(A;)14 b(A)2648 1690 y Fm(0)2695 1720 y Fz(ha)n(v)n(e)21 b Fu(L)2938 1690 y Ft(1)2938 1743 y Fd(lo)r(c)3025 1720 y Fz(-comp)r(onen)n (ts,)i(then)g(there)-118 1819 y(exists)e Fu(\032)g Fz(\(in)h(some)e(suitable) i(lo)r(cal)e(Sob)r(olev)h(space)f(that)i(do)r(es)f(not)g(matter)g(here\))g (suc)n(h)g(that)g Fu(A)2888 1789 y Fm(0)2918 1819 y Fl(\000)6 b Fu(A)22 b Fs(=)h Fl(r)p Fu(\032)45 b Fz(in)21 b(distributional)-118 1919 y(sense.)46 b(No)n(w,)31 b(since)g(in)g(our)f(case)g Fu(A)h Fz(and)g Fu(A)1310 1889 y Fm(0)1364 1919 y Fz(are)f(con)n(tin)n(uous,)h Fu(\032)g Fz(will)g(b)r(e)g(of)g(class)f Fu(C)2660 1889 y Ft(1)2728 1919 y Fz(b)n(y)h(a)f(simple)h(argumen)n(t.)46 b(The)31 b(t)n(w)n(o)-118 2019 y(iden)n(tities)d(are)e(v)n(eri\034ed)h(b)n(y)g(trivial)g(calculations)g (based)g(on)g(the)h(relation)656 2205 y Fu(e)695 2170 y Fv(i\032)p Ft(\()p Fv(Q)p Ft(\))861 2205 y Fu(e)900 2170 y Fv(ix)p Fm(\001)p Fv(P)1036 2205 y Fu(e)1075 2170 y Fm(\000)p Fv(i\032)p Ft(\()p Fv(Q)p Ft(\))1316 2205 y Fs(=)22 b Fu(e)1442 2170 y Fm(\000)p Fv(i)p Ft([)p Fv(\032)p Ft(\()p Fv(Q)p Ft(+)p Fv(x)p Ft(\))p Fm(\000)p Fv(\032)p Ft(\()p Fv(Q)p Ft(\)])1977 2205 y Fu(e)2016 2170 y Fv(ix)p Fm(\001)p Fv(P)2175 2205 y Fs(=)h Fu(e)2302 2170 y Fm(\000)p Fv(ix)p Fm(\001)2435 2123 y Fh(R)2479 2144 y Fo(1)2466 2192 y(0)2522 2170 y Fv(ds)c Fm(r)p Fv(\032)p Ft(\()p Fv(Q)p Ft(+)p Fv(sx)p Ft(\))2925 2205 y Fu(e)2964 2170 y Fv(ix)p Fm(\001)p Fv(P)3100 2205 y Fu(:)3833 2378 y Fe(\004)6 2537 y FA(Remark.)35 b Fz(One)26 b(implemen)n(ts)g(Planc)n(k's)f(constan)n(t)h(at) f(the)i(lev)n(el)e(of)h(the)g(ph)n(ysical)f(momen)n(tum,)i(b)n(y)f(setting)g Fu(P)34 b Fs(=)23 b Fw(~)p Fu(D)i Fs(:=)-118 2637 y Fl(\000)p Fu(i)p Fw(~)p Fl(r)p Fz(.)36 b(This)28 b(giv)n(es)e(for)h(the)h(magnetic)f(W) -7 b(eyl)28 b(system)1056 2836 y Fu(W)1146 2802 y Fv(A)1134 2856 y Fp(~)1200 2836 y Fs(\()p Fu(x;)14 b(p)p Fs(\))24 b(=)f Fu(e)1541 2800 y Fm(\000)p Fv(i)1616 2806 y Fs(\()1648 2800 y Fv(Q)p Ft(+)1761 2778 y Fc(~)p 1761 2787 33 3 v 1763 2820 a Fo(2)1804 2800 y Fv(x)1842 2806 y Fs(\))1874 2800 y Fm(\001)p Fv(p)1932 2836 y Fu(e)1971 2802 y Fm(\000)2038 2779 y Fn(i)p 2032 2788 V 2032 2822 a Fc(~)2075 2802 y Ft(\000)2116 2777 y Fn(A)2164 2802 y Ft(\([)p Fv(Q;Q)p Ft(+)p Fp(~)p Fv(x)p Ft(]\))2507 2836 y Fu(e)2546 2802 y Fv(i)p Fp(~)p Fv(x)p Fm(\001)p Fv(D)-118 2993 y Fz(and)33 b(the)g Fw(~)p Fz(-dep)r(enden)n(t)h(magnetic)e Fs(2)p Fz(-co)r(cycle)g(will)h(b)r(e)h Fs(\012)1739 2963 y Fv(B)1739 3016 y Fp(~)1796 2993 y Fs(\()p Fu(q)s Fs(;)14 b Fu(x;)g(y)s Fs(\))33 b(=)f Fu(e)2234 2963 y Fm(\000)2301 2941 y Fn(i)p 2295 2950 V 2295 2983 a Fc(~)2338 2963 y Ft(\000)2379 2938 y Fn(B)2428 2963 y Ft(\()p Fv()p Ft(\))3099 2993 y Fz(.)54 b(W)-7 b(e)33 b(collect)g(here,)h(for)-118 3093 y(the)28 b(con)n(v)n(enience)e(of)i(the)g(reader,)e(form)n(ulae)g(for)h(the)h (magnetic)f(W)-7 b(eyl)28 b(calculus)526 3229 y Fq(\020)576 3321 y Fr(Op)686 3284 y Fv(A)686 3341 y Fp(~)740 3321 y 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3706 y Fs(2)p 905 3743 45 4 v 905 3819 a Fw(~)960 3645 y Fq(\023)1021 3662 y Ft(2)p Fv(N)1131 3649 y Fq(Z)1177 3838 y Ft(\004)1239 3762 y Fu(d\021)1340 3649 y Fq(Z)1386 3838 y Ft(\004)1448 3762 y Fu(d\020)30 b(e)1596 3728 y Fm(\000)p Ft(2)1696 3706 y Fn(i)p 1691 3715 33 3 v 1691 3748 a Fc(~)1733 3728 y Fv(\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))2088 3762 y Fu(e)2127 3728 y Fm(\000)2194 3706 y Fn(i)p 2189 3715 V 2189 3748 a Fc(~)2232 3728 y Ft(\000)2273 3703 y Fn(B)2322 3728 y Ft(\()p Fv()p Ft(\))3147 3762 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))p Fu(g)s Fs(\()p Fu(\020)d Fs(\))p Fu(:)6 3951 y Fz(In)28 b(the)g(sequel)f Fw(~)h Fz(will)g(alw)n(a)n(ys)d(b)r(e)j Fs(1)p Fz(.)6 4051 y FA(Remark.)36 b Fz(One)27 b(often)h(uses)f(instead)g(of)h(\(0.1\))f Fx(the)j Fu(\034)9 b Fx(-quantizations)28 b Fz(\()p Fu(\034)33 b Fl(2)24 b Fs([0)p Fu(;)14 b Fs(1])p Fz(\),)27 b(giv)n(en)g(b)n(y)g(\(cf.)37 b([Sh)q(]\))764 4267 y Fs([)p Fr(Op)897 4287 y Ft(\()p Fv(\034)7 b Ft(\))991 4267 y Fs(\()p Fu(f)i Fs(\))p Fu(u)p Fs(]\()p Fu(x)p Fs(\))24 b(:=)1421 4154 y Fq(Z)1467 4343 y Fp(R)1514 4326 y Fo(2)p Fn(N)1609 4267 y Fu(dy)i(dp)d(e)1866 4233 y Fv(i)p Ft(\()p Fv(x)p Fm(\000)p Fv(y)r Ft(\))p Fm(\001)p Fv(p)2124 4267 y Fu(f)g Fs(\(\(1)18 b Fl(\000)g Fu(\034)9 b Fs(\))p Fu(x)20 b Fs(+)e Fu(\034)9 b(y)s(;)14 b(p)p Fs(\))g Fu(u)p Fs(\()p Fu(y)s Fs(\))p Fu(:)-118 4487 y Fz(They)35 b(are)e(someho)n(w)h(connected)g (with)h(the)h(ordering)d(of)h Fu(Q)h Fz(and)f Fu(P)47 b Fz(in)35 b(the)g(expression)f(of)g Fu(f)9 b Fs(\()p Fu(Q;)14 b(P)e Fs(\))p Fz(.)58 b(The)35 b(cases)f Fu(\034)45 b Fs(=)34 b(0)-118 4587 y Fz(and)c Fu(\034)38 b Fs(=)28 b(1)i Fz(are)g(called)g(resp)r(ectiv)n(ely)g Fx(the)i(right)g Fz(and)e Fx(the)j(left)g(quantization)p Fz(.)46 b(Rather)30 b(often,)i(in)f(texb)r(o)r(oks,)f(only)h(the)g(case)-118 4687 y Fu(\034)40 b Fs(=)30 b(0)h Fz(is)h(treated.)49 b(But)32 b(the)h(W)-7 b(eyl)32 b(prescription)e Fr(Op)g Fl(\021)g Fr(Op)1883 4706 y Ft(\(1)p Fv(=)p 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h(stage)e(should)h(this)-118 1553 y(replacemen)n(t)h(b)r(e)h(p)r(erformed)f (when)h(quan)n(tization)f(of)g(a)h(classical)e(observ)-5 b(able)30 b Fu(f)41 b Fz(is)31 b(in)n(tended.)50 b(The)32 b(wrong)e(answ)n(er)g(is)i (to)-118 1653 y(comp)r(ose)d Fu(f)35 b Fs(:)27 b(\004)g Fl(!)f Fw(C)51 b Fz(with)30 b(the)g(c)n(hange)f(of)g(v)-5 b(ariables)29 b Fs(\()p Fu(x;)14 b(p)p Fs(\))27 b Fl(7!)g Fs(\()p Fu(x;)14 b(p)20 b Fl(\000)f Fu(A)p Fs(\()p Fu(x)p Fs(\)\))32 b Fz(and)d(then)i(apply)e (the)h(W)-7 b(eyl)30 b(calculus.)43 b(As)-118 1752 y(seen)30 b(in)h(the)g(In)n(tro)r(duction,)g(this)g(w)n(ould)f(giv)n(e)g(a)g(gauge)g (non-co)n(v)-5 b(arian)n(t)28 b(form)n(ula.)45 b(The)30 b(righ)n(t)g(approac) n(h)f(is)i(to)f(apply)h(to)f Fu(f)-118 1852 y Fz(itself)24 b(a)f(mo)r(di\034ed)h(\(magnetic\))g(W)-7 b(eyl)24 b(calculus.)34 b(And)25 b(this)e(mo)r(di\034cation)h(is)f(go)n(v)n(erned)f(actually)g(b)n(y) i(the)g(sound,)g(elemen)n(tary)-118 1952 y(form)d(of)g(the)h(minimal)f (coupling)g(principle:)33 b(the)22 b(quan)n(tum)f(observ)-5 b(able)20 b Fu(P)33 b Fz(is)21 b(replaced)g(b)n(y)g Fs(\005)2844 1921 y Fv(A)2921 1952 y Fs(=)i Fu(P)17 b Fl(\000)6 b Fu(A)p Fs(\()p Fu(Q)p Fs(\))21 b Fz(and)g(this)h(ob)5 b(ject)-118 2051 y(determines)22 b(the)h(expression)f(of)g(the)h(W)-7 b(eyl)23 b(system)g Fu(W)1624 2021 y Fv(A)1678 2051 y Fz(,)h(used)e(in)h(the)g (de\034nition)g(of)g Fu(f)9 b Fs(\()p Fu(Q;)14 b Fs(\005)2840 2021 y Fv(A)2894 2051 y Fs(\))p Fz(.)35 b(One)23 b(could)f(sa)n(y)g(that)h (this)g(is)-118 2151 y(correct,)k(since)h Fu(W)477 2121 y Fv(A)560 2151 y Fz(summarizes)f(the)i(comm)n(utation)f(relations)f(of)h(the)h(family)f (of)h(op)r(erators)d Fs(\()p Fu(Q)3052 2163 y Ft(1)3089 2151 y Fu(;)14 b(:)g(:)g(:)g(;)g(Q)3340 2163 y Fv(N)3402 2151 y Fs(;)g(\005)3501 2121 y Fv(A)3501 2171 y Ft(1)3556 2151 y Fu(;)g(:)g(:)g(:)f (;)h Fs(\005)3802 2121 y Fv(A)3802 2174 y(N)3865 2151 y Fs(\))-118 2250 y Fz(for)27 b(whic)n(h)g(a)h(functional)f(calculus)g(is)h(requested.)6 2350 y(Ho)n(w)n(ev)n(er,)21 b(one)h(could)g(ask)f(for)g(a)g(more)g (sophisticated)h(\(and)g(not)f(so)h(clear)e(ideologically\))h(form)g(of)h (the)g(minimal)g(coupling)-118 2450 y(principle:)36 b(\034nd)28 b(a)f(transformation)e Fu(T)1116 2420 y Fv(A)1196 2450 y Fz(acting)i(on)f (phase-space)g(functions)h(suc)n(h)g(that,)g(for)g(an)n(y)f Fu(f)9 b Fz(,)27 b Fu(f)9 b Fs(\()p Fu(Q;)14 b Fs(\005)3388 2420 y Fv(A)3442 2450 y Fs(\))27 b Fz(is)g(obtained)-118 2549 y(\(also\))g(b)n(y)g(W)-7 b(eyl)28 b(quan)n(tizing)f(the)h(sym)n(b)r(ol)f Fu(T)1332 2519 y Fv(A)1385 2549 y Fu(f)32 b Fl(\021)22 b Fu(f)1595 2519 y Fv(A)1649 2549 y Fz(.)37 b(A)28 b(brief)f(examination)g(of)g(this)h (topic)g(follo)n(ws.)6 2649 y(Let)34 b(us)f(assume,)g(for)g(con)n(v)n (enience,)g(that)h Fu(B)j Fz(and)c Fu(A)g Fz(are)f(of)h(class)g Fu(C)2264 2619 y Fm(1)2258 2672 y Fd(p)r(ol)2352 2649 y Fz(.)54 b(Both)33 b Fr(Op)2751 2612 y Fv(A)2838 2649 y Fz(and)g Fr(Op)g Fz(are)f(one-to-one)f(\(ev)n(en)-118 2749 y(isomorphic\))26 b(from)i Fl(S)587 2718 y Fm(0)610 2749 y Fs(\(\004\))h Fz(to)f Fl(L)p Fs([)p Fl(S)6 b Fs(\()p Fu(X)h Fs(\))p Fu(;)14 b Fl(S)1229 2718 y Fm(0)1253 2749 y Fs(\()p Fu(X)7 b Fs(\)])p Fz(.)37 b(Using)27 b(notations)g(from)g(Subsection)h(3.2,)e(one)h(has)720 2948 y Fr(Op)830 2911 y Fv(A)884 2948 y Fs(\()p Fu(f)9 b Fs(\))24 b(=)e Fr(Op)o Fs(\()p Fu(f)1301 2914 y Fv(A)1355 2948 y Fs(\))i Fl(,)f Fr(Int)o Fs(\()p Fu(K)1743 2914 y Fv(A)1797 2948 y Fu(f)9 b Fs(\))23 b(=)g Fr(Int)o Fs(\()p Fu(K)6 b(f)2266 2914 y Fv(A)2319 2948 y Fs(\))24 b Fl(,)f Fu(f)2531 2914 y Fv(A)2608 2948 y Fs(=)f Fu(K)2772 2914 y Fm(\000)p Ft(1)2861 2948 y Fu(K)2938 2914 y Fv(A)2991 2948 y Fu(f)t(:)-118 3097 y Fz(By)27 b(using)g(explicit)h (form)n(ulae)f(for)g Fu(K)1068 3067 y Fv(A)1149 3097 y Fz(and)g Fu(K)33 b Fz(and)28 b(the)g(iden)n(tit)n(y)f Fu(S)2089 3076 y Fs(~)2081 3097 y(\003)2139 3067 y Fv(A)2193 3097 y Fu(S)2249 3067 y Fm(\000)p Ft(1)2361 3097 y Fs(=)2456 3076 y(~)2448 3097 y(\003)2506 3067 y Fv(A)2578 3097 y 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Fs(\()p Fu(x;)14 b(y)s Fs(\))24 b(=)e Fu(e)527 4746 y Fv(iy)r Fm(\001)p Fv(A)p Ft(\()p Fv(x)p Ft(\))750 4777 y Fz(,)27 b Fu(x;)14 b(y)26 b Fl(2)e Fu(X)7 b Fz(.)6 4876 y(A)20 b(comparison)e(of)h(the)h(explicit)f(form)n(ulae)f(for)h Fu(T)1537 4846 y Fv(A)1610 4876 y Fz(and)g Fu(M)1853 4846 y Fv(A)1926 4876 y Fz(sho)n(ws)f(once)h(again)f(the)i(di\033erence)f(b)r(et)n(w)n(een)g (the)h(correct)e(and)-118 4976 y(the)29 b(mistak)n(en)f(quan)n(tizations)g (in)h(a)f(magnetic)g(\034eld:)40 b Fx(The)32 b(c)l(orr)l(e)l(ct)e(one)h (involves)h(cir)l(culations)g(of)g(the)f(magnetic)g(p)l(otential)-118 5075 y Fu(A)f Fx(thr)l(ough)g(se)l(gments)g Fs([)p Fu(x)693 5087 y Ft(1)730 5075 y Fu(;)14 b(x)814 5087 y Ft(2)852 5075 y Fs(])p Fx(,)31 b(while)g(for)g(the)f(wr)l(ong)g(one)g(the)h(same)f(cir)l (culations)h(ar)l(e)f(c)l(alculate)l(d)h(by)g(using)e(the)h(c)l(onstant)-118 5175 y(value)e Fu(A)156 5187 y Fv(x)194 5195 y Fo(1)226 5187 y Fv(;x)284 5195 y Fo(2)343 5175 y Fs(:=)23 b Fu(A)530 5108 y Fq(\000)578 5141 y Fv(x)616 5149 y Fo(1)648 5141 y Ft(+)p Fv(x)737 5149 y Fo(2)p 578 5156 192 4 v 657 5204 a Ft(2)779 5108 y Fq(\001)817 5175 y Fz(,)28 b Fx(taken)i(at)f(the)h(midd)t(le)i(of)e (the)g(r)l(esp)l(e)l(ctive)h(se)l(gment)p Fz(.)6 5275 y(One)d(has)f(a)g (complete)g(c)n(haracterization)e(of)j(the)g(v)n(ector)e(p)r(oten)n(tials)h (for)g(whic)n(h)h(the)f(wrong)g(quan)n(tization)f(is)i(go)r(o)r(d:)-118 5374 y FA(Lemma)j(3.11.)40 b Fx(One)31 b(has)h Fu(T)851 5344 y Fv(A)929 5374 y Fs(=)25 b Fu(M)1109 5344 y Fv(A)1194 5374 y Fx(\(which)33 b(is)e(e)l(quivalent)h(to)f Fr(Op)2152 5337 y Fv(A)2206 5374 y Fs(\()p Fu(f)9 b Fs(\))26 b(=)f Fr(Op)o Fs(\()p Fu(M)2668 5344 y Fv(A)2722 5374 y Fu(f)9 b Fs(\))p Fx(,)32 b Fl(8)p Fu(f)h Fl(2)26 b(S)3119 5344 y Fm(0)3142 5374 y Fs(\(\004\))p Fx(\))32 b(if)g(and)g(only)f(if)h Fu(A)-118 5474 y Fx(is)e(line)l(ar.)1848 5989 y Fz(13)p eop %%Page: 14 14 bop -118 154 a Fz(P)8 b(r)g(o)g(o)g(f)g(.)27 b Fb(1)18 b Fl(\012)g(F)379 166 y Fv(X)470 154 y Fz(and)28 b Fb(1)18 b Fl(\012)p 782 87 68 4 v 19 w(F)849 166 y Fv(X)5 b(?)975 154 y Fz(b)r(eing)28 b(one-to-one,)e(w)n(e)i(ha)n(v)n(e)e Fu(T)2003 124 y Fv(A)2080 154 y Fs(=)d Fu(M)2258 124 y Fv(A)2340 154 y Fz(if)29 b(and)e(only)h(if)2845 133 y Fs(~)2837 154 y(\003)2895 124 y Fv(A)2968 154 y Fl(\016)18 b Fu(S)28 b Fs(=)c(\006)3256 124 y Fv(A)3310 154 y Fz(,)k(i.e.)38 b(if)28 b(and)g(only)-118 268 y(if)66 b Fu(y)25 b Fl(\001)107 201 y Fq(R)163 222 y Ft(1)p Fv(=)p Ft(2)146 298 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)317 268 y Fu(dt)46 b Fs([)p Fu(A)p Fs(\()p Fu(x)20 b Fs(+)e Fu(ty)s Fs(\))g Fl(\000)g Fu(A)p Fs(\()p Fu(x)p Fs(\)])q Fz(=0,)34 b Fl(8)p Fu(x;)14 b(y)35 b Fl(2)e Fu(X)7 b Fz(.)53 b(A)33 b(simple)h(application)e(of)h(T)-7 b(a)n(ylor's)32 b(form)n(ula)g(sho)n(ws)g(that)h(this)h(is)-118 378 y(equiv)-5 b(alen)n(t)27 b(to)h(the)g(ann)n(ulation)e(of)i(all)f(the)h(second)f(deriv)-5 b(ativ)n(es)26 b(of)i(all)f(the)h(comp)r(onen)n(ts)f(of)h Fu(A)p Fz(.)876 b Fe(\004)6 537 y Fz(One)31 b(of)f(the)h(most)f(imp)r(ortan)n(t)g (examples)g(is)g(the)h(constan)n(t)f(magnetic)g(\034eld.)46 b(In)31 b(this)f(case,)h(ev)n(eryb)r(o)r(dy)e(w)n(ould)h(c)n(ho)r(ose)-118 636 y(a)j(linear)g(p)r(oten)n(tial)h(v)n(ector)e Fu(A)i Fz(and)g(no)f(care)g (is)h(needed)f(in)h(the)h(c)n(hoice)d(of)i(the)g(quan)n(tization)f(pro)r (cedure.)54 b(Most)34 b(articles)-118 736 y(in)n(v)n(olving)24 b(a)i(functional)g(calculus)f(in)h(a)g(magnetic)f(\034eld)h(are)f(written)h (for)g(constan)n(t)f Fu(B)30 b Fx(and)c Fz(linear)f Fu(A)p Fz(.)37 b(Note,)26 b(ho)n(w)n(ev)n(er,)e(that)-118 835 y(the)k(iden)n(tit)n (y)f Fu(T)392 805 y Fv(A)469 835 y Fs(=)22 b Fu(M)646 805 y Fv(A)728 835 y Fx(is)30 b(not)f(gauge)h(invariant)p Fz(.)6 935 y(Ho)n(w)n(ev)n(er,)h(one)f(can)h(ha)n(v)n(e)f Fu(T)936 905 y Fv(A)989 935 y Fu(f)38 b Fs(=)29 b Fu(M)1252 905 y Fv(A)1306 935 y Fu(f)39 b Fz(for)31 b(an)n(y)g Fu(A)g Fz(for)g(certain)f(sp)r(ecial)h (functions)h Fu(f)9 b Fz(.)47 b(This)31 b(is)h(ob)n(viously)d(true)i(if)h Fu(f)-118 1035 y Fz(dep)r(ends)24 b(only)g(on)g(the)g(v)-5 b(ariable)23 b Fu(x)h Fl(2)f Fu(X)7 b Fz(.)35 b(A)n(ctually)-7 b(,)25 b(in)f(this)g(case)f Fr(Op)2102 998 y Fv(A)2156 1035 y Fs(\()p Fu(f)9 b Fs(\))23 b(=)g Fr(Op)o Fs(\()p Fu(M)2613 1005 y Fv(A)2667 1035 y Fu(f)9 b Fs(\))23 b(=)f Fu(f)9 b Fs(\()p Fu(Q)p Fs(\))p Fz(.)36 b(Let)24 b(us)g(giv)n(e)f(some)h(more)-118 1134 y(in)n(teresting)j(examples.)-118 1234 y FA(Prop)s(osition)h(3.12.)36 b Fx(L)l(et)28 b Fu(f)37 b Fx(b)l(e)28 b(a)g(p)l(olynomial)j(of)e(or)l(der)g Fu(m)f Fx(in)g Fu(p)p Fx(,)h(not)f(dep)l(ending)h(on)g(the)f(variable)i(in)f Fu(X)7 b Fx(.)37 b(If)29 b Fu(m)23 b Fl(\024)f Fs(2)p Fx(,)29 b(then)-118 1334 y Fu(T)-57 1303 y Fv(A)-4 1334 y Fu(f)j Fs(=)22 b Fu(M)246 1303 y Fv(A)300 1334 y Fu(f)9 b Fx(,)30 b(henc)l(e)g Fr(Op)744 1297 y Fv(A)798 1334 y Fs(\()p Fu(f)9 b Fs(\))23 b(=)g Fr(Op)o Fs(\()p Fu(M)1255 1303 y Fv(A)1309 1334 y Fu(f)9 b Fs(\))p Fx(.)39 b(This)30 b(is)h(no)e(longer)i(true)e(for)h Fu(m)23 b Fs(=)g(3)p Fx(.)-118 1504 y Fz(P)8 b(r)g(o)g(o)g(f)g(.)19 b(Let)i(us)g(consider)e(the)i(monomial)f Fu(f)1273 1516 y Fv(\013)1320 1504 y Fs(\()p Fu(x;)14 b(p)p Fs(\))24 b(:=)e Fu(p)1686 1474 y Fv(\013)1733 1504 y Fz(.)35 b(Then)21 b(w)n(e)f(ha)n(v)n(e)g Fs(\()p Fb(1)5 b Fl(\012)p 2456 1437 V 5 w(F)2522 1516 y Fv(X)2581 1500 y Fn(?)2620 1504 y Fs(\))p Fu(f)2693 1516 y Fv(\013)2763 1504 y Fs(=)23 b(\()p Fl(\000)p Fu(i@)5 b Fs(\))3058 1474 y Fv(\013)3105 1504 y Fu(\016)s Fz(,)22 b(th)n(us)3366 1437 y Fq(\000)3404 1504 y Fu(T)3465 1474 y Fv(A)3518 1504 y Fu(f)3568 1437 y Fq(\001)3619 1504 y Fs(\()p Fu(x;)14 b(p)p Fs(\))24 b(=)-118 1545 y Fq(h)-79 1637 y Fs(\()p Fu(i@)26 1649 y Fv(y)66 1637 y Fs(\))98 1607 y Fv(\013)146 1637 y Fu(e)185 1607 y Fm(\000)p Fv(i\034)298 1582 y Fn(A)344 1607 y Ft(\()p Fv(x;p)p Ft(;)p Fv(y)r Ft(\))546 1545 y Fq(i)599 1637 y Fl(j)622 1649 y Fv(y)r Ft(=0)820 1637 y Fz(and)990 1570 y Fq(\000)1028 1637 y Fu(M)1118 1607 y Fv(A)1171 1637 y Fu(f)1221 1570 y Fq(\001)1273 1637 y Fs(\()p Fu(x;)14 b(p)p Fs(\))38 b(=)1602 1545 y Fq(h)1642 1637 y Fs(\()p Fu(i@)1747 1649 y Fv(y)1786 1637 y Fs(\))1818 1607 y Fv(\013)1866 1637 y Fu(e)1905 1607 y Fm(\000)p Fv(i\026)2020 1582 y Fn(A)2068 1607 y Ft(\()p Fv(x;p)p Ft(;)p Fv(y)r Ft(\))2270 1545 y Fq(i)2323 1637 y Fl(j)2346 1649 y Fv(y)r Ft(=0)2470 1637 y Fz(,)g(where)e(the)g(t)n(w)n(o)f(phases)h(are)f(de\034ned)h(b)n(y)-118 1795 y Fu(\034)-73 1765 y Fv(A)-18 1795 y Fs(\()p Fu(x;)14 b(p)p Fs(;)g Fu(y)s Fs(\))43 b(:=)g Fu(y)29 b Fl(\001)547 1703 y Fq(h)586 1795 y Fu(p)18 b Fl(\000)729 1728 y Fq(R)785 1748 y Ft(1)p Fv(=)p Ft(2)769 1824 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)939 1795 y Fu(dt)23 b(A)p Fs(\()p Fu(x)c Fs(+)f Fu(ty)s Fs(\))1384 1703 y Fq(i)1503 1795 y Fz(and)40 b Fu(\026)1727 1765 y Fv(A)1781 1795 y Fs(\()p Fu(x;)14 b(p)p Fs(;)g Fu(y)s Fs(\))43 b(:=)g Fu(y)29 b Fl(\001)e Fs([)p Fu(p)f Fl(\000)g Fu(A)p Fs(\()p Fu(x)p Fs(\)])p Fz(.)75 b(W)-7 b(e)40 b(concen)n(trate)e(on)i(the)g(cases) -118 1914 y Fu(m)23 b Fs(=)f(1)p Fu(;)14 b Fs(2)p Fu(;)g Fs(3)p Fz(.)36 b(The)27 b(follo)n(wing)g(list)h(of)f(relations)f(is)i(needed:)602 2113 y Fu(i@)675 2125 y Fv(y)709 2133 y Fn(j)743 2113 y Fu(e)782 2079 y Fm(\000)p Fv(i')928 2113 y Fs(=)1016 2046 y Fq(\000)1054 2113 y Fu(@)1098 2125 y Fv(y)1132 2133 y Fn(j)1167 2113 y Fu(')1221 2046 y Fq(\001)1273 2113 y Fu(e)1312 2079 y Fm(\000)p Fv(i')1435 2113 y Fu(;)97 b(i)1584 2079 y Ft(2)1621 2113 y Fu(@)1665 2125 y Fv(y)1699 2134 y Fn(k)1739 2113 y Fu(@)1783 2125 y Fv(y)1817 2133 y Fn(j)1852 2113 y Fu(e)1891 2079 y Fm(\000)p Fv(i')2037 2113 y Fs(=)2124 2046 y Fq(\000)2162 2113 y Fu(i@)2235 2125 y Fv(y)2269 2134 y Fn(k)2309 2113 y Fu(@)2353 2125 y Fv(y)2387 2133 y Fn(j)2422 2113 y Fu(')19 b Fs(+)f Fu(@)2622 2125 y Fv(y)2656 2134 y Fn(k)2696 2113 y Fu(')24 b(@)2818 2125 y Fv(y)2852 2133 y Fn(j)2886 2113 y Fu(')2940 2046 y Fq(\001)2993 2113 y Fu(e)3032 2079 y Fm(\000)p Fv(i')3155 2113 y Fu(;)-49 2445 y(i)-20 2411 y Ft(3)17 2445 y Fu(@)61 2457 y Fv(y)95 2466 y Fn(l)123 2445 y Fu(@)167 2457 y Fv(y)201 2466 y Fn(k)241 2445 y Fu(@)285 2457 y Fv(y)319 2465 y Fn(j)354 2445 y Fu(e)393 2411 y Fm(\000)p Fv(i')539 2445 y Fs(=)626 2378 y Fq(\000)664 2445 y Fl(\000)p Fu(@)773 2457 y Fv(y)807 2466 y Fn(l)835 2445 y Fu(@)879 2457 y Fv(y)913 2466 y Fn(k)953 2445 y Fu(@)997 2457 y Fv(y)1031 2465 y Fn(j)1066 2445 y Fu(')19 b Fs(+)f Fu(i@)1295 2457 y Fv(y)1329 2465 y Fn(j)1363 2445 y Fu(')24 b(@)1485 2457 y Fv(y)1519 2466 y Fn(l)1547 2445 y Fu(@)1591 2457 y Fv(y)1625 2466 y Fn(k)1665 2445 y Fu(')19 b Fs(+)f Fu(i@)1894 2457 y Fv(y)1928 2466 y Fn(k)1967 2445 y Fu(')24 b(@)2089 2457 y Fv(y)2123 2466 y Fn(l)2151 2445 y Fu(@)2195 2457 y Fv(y)2229 2465 y Fn(j)2264 2445 y Fu(')18 b Fs(+)g Fu(i@)2492 2457 y Fv(y)2526 2466 y Fn(l)2554 2445 y Fu(')23 b(@)2675 2457 y Fv(y)2709 2466 y Fn(k)2750 2445 y Fu(@)2794 2457 y Fv(y)2828 2465 y Fn(j)2862 2445 y Fu(')c Fs(+)f Fu(@)3062 2457 y Fv(y)3096 2466 y Fn(l)3152 2445 y Fu(')23 b(@)3273 2457 y Fv(y)3307 2466 y Fn(k)3347 2445 y Fu(')h(@)3469 2457 y Fv(y)3503 2465 y Fn(j)3538 2445 y Fu(')3592 2378 y Fq(\001)3644 2445 y Fu(e)3683 2411 y Fm(\000)p Fv(i')3806 2445 y Fu(:)6 2628 y Fz(Note)k(that)g Fu(@)431 2640 y Fv(y)465 2648 y Fn(j)500 2628 y Fu(\026)550 2598 y Fv(A)604 2628 y Fs(\()p Fu(x;)14 b(p)p Fs(;)g Fu(y)s Fs(\))23 b(=)g Fu(p)1028 2640 y Fv(j)1081 2628 y Fl(\000)18 b Fu(A)1226 2640 y Fv(j)1261 2628 y Fs(\()p Fu(x)p Fs(\))p Fz(,)29 b(while)f(the)g(higher-order)d(deriv)-5 b(ativ)n(es)26 b(v)-5 b(anish.)6 2728 y(A)28 b(simple)g(calculation)f(giv)n (es)533 2987 y Fu(@)577 2999 y Fv(y)611 3007 y Fn(j)646 2987 y Fu(\034)691 2953 y Fv(A)746 2987 y Fs(\()p Fu(x;)14 b(p)p Fs(;)g Fu(y)s Fs(\))23 b(=)g Fu(p)1170 2999 y Fv(j)1223 2987 y Fl(\000)1306 2874 y Fq(Z)1389 2895 y Ft(1)p Fv(=)p Ft(2)1352 3063 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)1522 2987 y Fu(dt)g(A)1680 2999 y Fv(j)1716 2987 y Fs(\()p Fu(x)c Fs(+)f Fu(ty)s Fs(\))g Fl(\000)2137 2884 y Fv(N)2107 2909 y Fq(X)2104 3084 y Fv(n)p Ft(=1)2243 2987 y Fu(y)2284 2999 y Fv(n)2343 2874 y Fq(Z)2426 2895 y Ft(1)p Fv(=)p Ft(2)2389 3063 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)2559 2987 y Fu(tdt)23 b Fs(\()p Fu(@)2761 2999 y Fv(j)2797 2987 y Fu(A)2859 2999 y Fv(n)2904 2987 y Fs(\)\()p Fu(x)d Fs(+)e Fu(ty)s Fs(\))p Fu(;)-118 3221 y Fz(and)36 b(b)n(y)g(taking)f(the)h(v)-5 b(alue)36 b(in)h Fu(y)j Fs(=)c(0)g Fz(one)g(gets)1520 3154 y Fq(\002)1555 3221 y Fu(@)1599 3233 y Fv(y)1633 3241 y Fn(j)1668 3221 y Fu(\034)1713 3191 y Fv(A)1768 3221 y Fs(\()p Fu(x;)14 b(p)p Fs(;)g Fu(y)s Fs(\))2039 3154 y Fq(\003)2087 3221 y Fl(j)2110 3233 y Fv(y)r Ft(=0)2272 3221 y Fs(=)2373 3154 y Fq(\002)2408 3221 y Fu(@)2452 3233 y Fv(y)2486 3241 y Fn(j)2521 3221 y Fu(\026)2571 3191 y Fv(A)2625 3221 y Fs(\()p Fu(x;)g(p)p Fs(;)g Fu(y)s Fs(\))2896 3154 y Fq(\003)2945 3221 y Fl(j)2968 3233 y Fv(y)r Ft(=0)3129 3221 y Fs(=)37 b Fu(p)3273 3233 y Fv(j)3332 3221 y Fl(\000)23 b Fu(A)3482 3233 y Fv(j)3518 3221 y Fs(\()p Fu(x)p Fs(\))p Fz(.)63 b(Th)n(us)-118 3335 y Fu(T)-57 3305 y Fv(A)-4 3335 y Fu(f)44 b Fs(=)34 b Fu(M)270 3305 y Fv(A)324 3335 y Fu(f)44 b Fz(for)34 b(an)n(y)g(\034rst-order)f(p)r(olynomial.)57 b(This)35 b(is)g(not)g(amaizing:)50 b Fr(Op)2535 3298 y Fv(A)2589 3335 y Fs(\()p Fu(p)2663 3347 y Fv(j)2698 3335 y Fs(\))36 b(=)e(\005)2927 3305 y Fv(A)3017 3335 y Fz(w)n(as)f(accepted)i(as)f(a)g(basic)-118 3435 y(principle.)6 3535 y(One)28 b(also)e(has)231 3794 y Fu(@)275 3806 y Fv(y)309 3815 y Fn(k)349 3794 y Fu(@)393 3806 y Fv(y)427 3814 y Fn(j)462 3794 y Fu(\034)507 3760 y Fv(A)562 3794 y Fs(\()p Fu(x;)14 b(p)p Fs(;)g Fu(y)s Fs(\))23 b(=)g Fl(\000)1023 3681 y Fq(Z)1105 3702 y Ft(1)p Fv(=)p Ft(2)1068 3870 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)1239 3794 y Fu(tdt)g Fs(\()p Fu(@)1441 3806 y Fv(j)1476 3794 y Fu(A)1538 3806 y Fv(k)1598 3794 y Fs(+)18 b Fu(@)1725 3806 y Fv(k)1766 3794 y Fu(A)1828 3806 y Fv(j)1863 3794 y Fs(\)\()p Fu(x)h Fs(+)f Fu(ty)s Fs(\))h Fl(\000)2317 3691 y Fv(N)2286 3715 y Fq(X)2284 3891 y Fv(n)p Ft(=1)2423 3794 y Fu(y)2464 3806 y Fv(n)2522 3681 y Fq(Z)2606 3702 y Ft(1)p Fv(=)p Ft(2)2569 3870 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)2739 3794 y Fu(t)2769 3760 y Ft(2)2806 3794 y Fu(dt)k Fs(\()p Fu(@)2978 3806 y Fv(k)3019 3794 y Fu(@)3063 3806 y Fv(j)3098 3794 y Fu(A)3160 3806 y Fv(n)3206 3794 y Fs(\)\()p Fu(x)c Fs(+)f Fu(ty)s Fs(\))p Fu(:)-118 4028 y Fz(By)k(\020miracle\021)27 b(this)c(term)f(v)-5 b(anishes)21 b(in)i Fu(y)j Fs(=)c(0)p Fz(;)i(then)e(straigh)n(tforw)n(ardly)d Fu(T)2249 3997 y Fv(A)2303 4028 y Fu(f)31 b Fs(=)23 b Fu(M)2553 3997 y Fv(A)2606 4028 y Fu(f)31 b Fz(also)21 b(for)h(second-order)e(p)r (olynomials.)-118 4127 y(This)34 b(is)f(signi\034can)n(t,)i(since)e(most)h (of)f(the)h(time)g(p)r(eople)g(considered)f(the)h(case)f Fu(f)9 b Fs(\()p Fu(x;)14 b(p)p Fs(\))33 b(=)g Fl(j)p Fu(p)p Fl(j)2926 4097 y Ft(2)2963 4127 y Fz(,)j(leading)d(to)g(the)h(magnetic)-118 4227 y(Laplacian)26 b Fs(\001)330 4197 y Fv(A)407 4227 y Fs(=)d(\(\005)589 4197 y Fv(A)644 4227 y Fs(\))676 4197 y Ft(2)713 4227 y Fz(;)28 b(no)f(care)g(is)g(needed)h(in)g(this)g(case.)6 4326 y(The)g(situation)f(c)n (hanges)f(drastically)h(for)g(third)g(order)g(p)r(olynomials.)35 b(One)28 b(has)-118 4669 y Fu(@)-74 4681 y Fv(y)-40 4690 y Fn(l)-12 4669 y Fu(@)32 4681 y Fv(y)66 4690 y Fn(k)106 4669 y Fu(@)150 4681 y Fv(y)184 4689 y Fn(j)219 4669 y Fu(\034)264 4635 y Fv(A)319 4669 y Fs(\()p Fu(x;)14 b(p)p Fs(;)g Fu(y)s Fs(\))23 b(=)g Fl(\000)780 4556 y Fq(Z)862 4577 y Ft(1)p Fv(=)p Ft(2)825 4745 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)995 4669 y Fu(t)1025 4635 y Ft(2)1063 4669 y Fu(dt)37 b Fs(\()p Fu(@)1249 4681 y Fv(k)1290 4669 y Fu(@)1334 4681 y Fv(j)1369 4669 y Fu(A)1431 4681 y Fv(l)1475 4669 y Fs(+)18 b Fu(@)1602 4681 y Fv(l)1628 4669 y Fu(@)1672 4681 y Fv(j)1707 4669 y Fu(A)1769 4681 y Fv(k)1828 4669 y Fs(+)g Fu(@)1955 4681 y Fv(l)1981 4669 y Fu(@)2025 4681 y Fv(k)2066 4669 y Fu(A)2128 4681 y Fv(j)2163 4669 y Fs(\))c(\()p Fu(x)f Fs(+)g Fu(ty)s Fs(\))g Fl(\000)2608 4565 y Fv(N)2578 4590 y Fq(X)2576 4766 y Fv(n)p Ft(=1)2714 4669 y Fu(y)2755 4681 y Fv(n)2814 4556 y Fq(Z)2897 4577 y Ft(1)p Fv(=)p Ft(2)2860 4745 y Fm(\000)p Ft(1)p Fv(=)p Ft(2)3030 4669 y Fu(t)3060 4635 y Ft(3)3097 4669 y Fu(dt)23 b Fs(\()p Fu(@)3269 4681 y Fv(l)3295 4669 y Fu(@)3339 4681 y Fv(k)3380 4669 y Fu(@)3424 4681 y Fv(j)3459 4669 y Fu(A)3521 4681 y Fv(n)3567 4669 y Fs(\)\()p Fu(x)13 b Fs(+)g Fu(ty)s Fs(\))p Fu(;)-118 4936 y Fz(whic)n(h)27 b(in)h Fu(y)e Fs(=)d(0)k Fz(tak)n(es)f(the)i(v)-5 b(alue)28 b Fl(\000)1101 4903 y Ft(1)p 1084 4917 67 4 v 1084 4964 a(12)1174 4936 y Fs(\()q Fu(@)1251 4948 y Fv(k)1291 4936 y Fu(@)1335 4948 y Fv(j)1370 4936 y Fu(A)1432 4948 y Fv(l)1477 4936 y Fs(+)18 b Fu(@)1604 4948 y Fv(l)1629 4936 y Fu(@)1673 4948 y Fv(j)1708 4936 y Fu(A)1770 4948 y Fv(k)1830 4936 y Fs(+)g Fu(@)1957 4948 y Fv(l)1982 4936 y Fu(@)2026 4948 y Fv(k)2067 4936 y Fu(A)2129 4948 y Fv(j)2165 4936 y Fs(\))c(\()p Fu(x)p Fs(\))p Fz(.)38 b(In)27 b(this)h(case)f Fu(T)2885 4906 y Fv(A)2938 4936 y Fu(f)32 b Fl(6)p Fs(=)22 b Fu(M)3188 4906 y Fv(A)3242 4936 y Fu(f)9 b Fz(.)518 b Fe(\004)-118 5210 y Fy(4)131 b(The)44 b(magnetic)g(Mo)l(y)l(al)h(algebra)-118 5392 y Fz(W)-7 b(e)22 b(come)g(no)n(w)f(to)h(an)g(imp)r(ortan)n(t)f(p)r(oin)n (t)h(in)g(the)h(dev)n(elopmen)n(t)e(of)h(our)f(functional)h(calculus.)35 b(The)22 b(pro)r(duct)g(of)f(the)i(op)r(erators)-118 5492 y Fr(Op)-8 5455 y Fv(A)46 5492 y Fs(\()p Fu(f)9 b Fs(\))32 b Fz(and)g Fr(Op)468 5455 y Fv(A)522 5492 y Fs(\()p Fu(g)s Fs(\))g Fz(is)g(again)f(an)g(in)n(tegral)g(op)r(erator)f(with)i(a)g(k)n(ernel)f Fu(K)2333 5462 y Fv(A)2386 5492 y Fs(\()p Fu(f)f Fl(\016)2531 5462 y Fv(B)2609 5492 y Fu(g)s Fs(\))p Fz(,)j(that)f(can)g(formally)f(b)r(e)h (asso)r(ciated)-118 5591 y(to)f(the)h(function)g(on)f Fs(\004)g Fz(obtained)g(b)n(y)g(the)h(follo)n(wing)e(non-comm)n(utativ)n(e)g(comp)r (osition)g(la)n(w,)i(called)f Fx(the)i(magnetic)h(Moyal)-118 5691 y(pr)l(o)l(duct)c(of)g(the)g(functions)e Fu(f)36 b Fx(and)28 b Fu(g)s Fz(:)1848 5989 y(14)p eop %%Page: 15 15 bop 389 358 a Fs(\()p Fu(f)27 b Fl(\016)531 324 y Fv(B)606 358 y Fu(g)s Fs(\)\()p Fu(\030)t Fs(\))d(=)e(4)938 324 y Fv(N)1015 245 y Fq(Z)1061 434 y Ft(\004)1123 358 y Fu(d\021)1224 245 y Fq(Z)1270 434 y Ft(\004)1332 358 y Fu(d\020)30 b(e)1480 324 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))1942 358 y Fu(e)1981 324 y Fm(\000)p Fv(i)p Ft(\000)2097 299 y Fn(B)2147 324 y Ft(\()p Fv()p Ft(\))2972 358 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))p Fu(g)s Fs(\()p Fu(\020)d Fs(\))24 b(=)318 b Fz(\(4.36\))585 638 y Fs(=)22 b(4)714 604 y Fv(N)790 525 y Fq(Z)837 714 y Ft(\004)898 638 y Fu(d\021)1000 525 y Fq(Z)1046 714 y Ft(\004)1108 638 y Fu(d\020)29 b(e)1255 604 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\021)r(;\020)t Ft(\))1550 638 y Fu(e)1589 604 y Fm(\000)p Fv(i)p Ft(\000)1705 579 y Fn(B)1754 604 y Ft(\()p Fv()p Ft(\))2581 638 y Fu(f)9 b Fs(\()p Fu(\030)22 b Fl(\000)c Fu(\021)s Fs(\))p Fu(g)s Fs(\()p Fu(\030)23 b Fl(\000)18 b Fu(\020)6 b Fs(\);)-118 849 y Fz(here)32 b Fu(\030)j Fs(=)c(\()p Fu(q)s(;)14 b(p)p Fs(\))p Fu(;)46 b(\021)34 b Fs(=)d(\()p Fu(x;)14 b(k)s Fs(\))p Fu(;)46 b(\020)37 b Fs(=)31 b(\()p Fu(y)s(;)14 b(l)r Fs(\))p Fz(.)52 b(Th)n(us)32 b(w)n(e)g(ha)n(v)n(e)f Fr(Op)1984 812 y Fv(A)2038 849 y Fs(\()p Fu(f)g Fl(\016)2184 819 y Fv(B)2262 849 y Fu(g)s Fs(\))h(=)e Fr(Op)2575 812 y Fv(A)2629 849 y Fs(\()p Fu(f)9 b Fs(\))p Fr(Op)2853 812 y Fv(A)2907 849 y Fs(\()p Fu(g)s Fs(\))p Fz(.)52 b(W)-7 b(e)33 b(can)f(also)f(de\034ne)i(an)-118 948 y(in)n(v)n(olution)27 b(\(the)h(same)f(as)g(in)g(the)h(non-magnetic)f (case\))g(b)n(y)g Fu(f)1886 918 y Fm(\016)1924 948 y Fs(\()p Fu(\030)t Fs(\))d(:=)p 2162 876 155 4 v 22 w Fu(f)9 b Fs(\()p Fu(\030)t Fs(\))28 b Fz(suc)n(h)g(that)f Fr(Op)2822 911 y Fv(A)2876 948 y Fs(\()p Fu(f)2958 918 y Fm(\016)2996 948 y Fs(\))c(=)g Fr(Op)3249 911 y Fv(A)3303 948 y Fs(\()p Fu(f)9 b Fs(\))3417 918 y Fm(\003)3455 948 y Fz(.)6 1048 y(The)34 b(in)n(tegral)e(de\034ning)i Fu(f)d Fl(\016)925 1018 y Fv(B)1004 1048 y Fu(g)36 b Fz(is)e(absolutely)e (con)n(v)n(ergen)n(t)f(only)j(for)f(restricted)f(classes)g(of)i(sym)n(b)r (ols.)54 b(One)33 b(seeks)g(to)-118 1148 y(extend)e(the)h(comp)r(osition)e (la)n(w)g Fl(\016)964 1117 y Fv(B)1052 1148 y Fz(to)h(large)f(classes)g(of)h (distributions)g(in)g(suc)n(h)g(a)f(w)n(a)n(y)g(as)h(to)g(obtain)f (\(together)h(with)g(the)-118 1247 y(in)n(v)n(olution)269 1217 y Fm(\016)307 1247 y Fz(\))365 1217 y Fm(\003)403 1247 y Fz(-algebras.)i(F)-7 b(or)25 b(an)n(y)f(c)n(hoice)g(of)h(a)g(magnetic)g(p)r(oten)n(tial,)g(the)g (functional)h(calculus)e(with)i(magnetic)e(\034eld)i(will)-118 1347 y(b)r(e)e(a)g(represen)n(tation)e(of)i(these)896 1317 y Fm(\003)935 1347 y Fz(-algebras.)33 b(But)24 b(the)h(algebras)d(themselv)n (es)h(are)g(completely)h(in)n(trinsic,)h(b)r(eing)f(de\034ned)g(only)-118 1446 y(in)e(terms)g(of)f(the)i(magnetic)e(\034eld.)35 b(W)-7 b(e)22 b(shall)g(do)f(this)h(extension)g(b)n(y)f(dualit)n(y)-7 b(,)23 b(follo)n(wing)e(the)h(approac)n(h)e(of)i([An)n(t])g(and)g([GBV1])-118 1546 y(\(see)28 b(also)g([GBV2])h(and)f([EGBV)q(]\))h(v)-5 b(alid)29 b(in)g(the)g(absence)f(of)g(the)h(magnetic)f(\034eld.)41 b(W)-7 b(e)29 b(obtain)f(a)h(magnetic)f(analog)f(of)h(the)-118 1646 y(Mo)n(y)n(al)c(algebra)h(outlined)h(in)h(Subsection)f(1.4;)f(the)i (terminology)e(is)h(that)g(of)g(the)h(references)e(ab)r(o)n(v)n(e)f(and)i(it) h(is)f(suggested)f(b)n(y)-118 1745 y(some)i(early)f(fundamen)n(tal)i(w)n(ork) e(of)i(Mo)n(y)n(al)e(\(cf.)37 b([Mo]\).)6 1845 y(The)j(standard)f(tec)n (hnique)g(of)h(extending)g(the)g(comp)r(osition)f(la)n(w,)j(based)d(on)g (oscillatory)f(in)n(tegrals)g(and)h(classes)g(of)-118 1945 y(sym)n(b)r(ols,)33 b(is)g(less)g(appropriate)e(for)h(our)h(presen)n(t)f (purp)r(oses.)52 b(But)33 b(w)n(e)g(in)n(tend)g(to)g(deal)g(with)g(this)g (topic)g(in)h(a)e(subsequen)n(t)-118 2044 y(publication.)-118 2277 y Fg(4.1)112 b(The)38 b(magnetic)e(Mo)m(y)m(al)h(pro)s(duct)-118 2430 y Fz(Before)28 b(discussing)f(rigorously)f(the)j(sense)f(of)h(form)n (ula)f(\(4.36\))f(for)h(v)-5 b(arious)28 b(assumptions)g(on)g Fu(f)9 b Fz(,)28 b Fu(g)k Fz(and)c Fu(B)t Fz(,)h(w)n(e)g(mak)n(e)e(some)-118 2529 y(formal)33 b(remarks.)54 b(Note)34 b(that)h(if)f Fu(B)k Fs(=)c(0)p Fz(,)h(\(4.36\))e(reduces)g(to)h(the)h(usual)e(comp)r(osition)g (of)h(sym)n(b)r(ols)g(\(1.13\))f(in)h(the)g(W)-7 b(eyl)-118 2629 y(quan)n(tization.)34 b(The)21 b(magnetic)g(correction)f(consists)g(of)h (a)g(phase)g(factor)g(de\034ned)g(in)h(terms)f(of)g(the)h(\035ux)g(of)f(the)h (magnetic)e(\034eld)-118 2729 y(through)26 b(suitable)h(triangles.)35 b(The)27 b(asso)r(ciativit)n(y)e(of)i(the)h(ab)r(o)n(v)n(e)d(comp)r(osition)h (la)n(w)g(comes)h(from)f(the)i(2-co)r(cycle)d(condition,)-118 2837 y(the)f(second)e(iden)n(tit)n(y)h(in)h(\(2.20\).)35 b(F)-7 b(or)22 b(this)i(just)g(notice)f(that)g Fu(e)1866 2806 y Fm(\000)p Fv(i)p Ft(\000)1982 2781 y Fn(B)2032 2806 y Ft(\()p Fv()p Ft(\))2880 2837 y Fs(=)g(\012)3028 2806 y Fv(B)3085 2837 y Fs(\()p Fu(q)13 b Fl(\000)d Fu(y)j Fs(+)d Fu(x)p Fs(;)k(2)p Fu(y)f Fl(\000)d Fs(2)p Fu(q)s(;)k Fs(2)p Fu(q)f Fl(\000)-118 2936 y Fs(2)p Fu(x)p Fs(\))35 b Fz(and)h(do)e(the)i(righ)n(t)f(calculation.)58 b(Finally)-7 b(,)38 b(it)d(is)g(easy)g(to)g(c)n(hec)n(k)f(that)i Fr(Op)2483 2899 y Fv(A)2537 2936 y Fs(\()p Fu(f)c Fl(\016)2684 2906 y Fv(B)2764 2936 y Fu(g)s Fs(\))k(=)g Fr(Op)3086 2899 y Fv(A)3140 2936 y Fs(\()p Fu(f)9 b Fs(\))p Fr(Op)3364 2899 y Fv(A)3418 2936 y Fs(\()p Fu(g)s Fs(\))36 b Fz(whenev)n(er)-118 3036 y(ev)n(erything)26 b(is)i(w)n(ell-de\034ned.)6 3136 y(In)d(general,)e(if)i(no)e(sp)r(ecial)h (assumption)f(on)h Fu(B)k Fz(is)c(imp)r(osed,)h(it)g(is)f(not)g(so)f(easy)g (to)h(de\034ne)g(and)g(use)g(sharply)f(the)h(magnetic)-118 3235 y(Mo)n(y)n(al)36 b(pro)r(duct.)69 b(F)-7 b(or)37 b Fu(f)t(;)14 b(g)43 b Fl(2)e(F)997 3247 y Ft(\004)1045 3235 y Fu(L)1102 3205 y Ft(1)1139 3235 y Fs(\(\004\))e Fz(b)r(oth)f Fr(Op)1614 3198 y Fv(A)1668 3235 y Fs(\()p Fu(f)9 b Fs(\))38 b Fz(and)g Fr(Op)2102 3198 y Fv(A)2156 3235 y Fs(\()p Fu(g)s Fs(\))h Fz(are)e(de\034ned) h(as)g(b)r(ounded)g(linear)f(op)r(erators)f(in)-118 3335 y Fl(H)g Fs(=)f Fu(L)145 3305 y Ft(2)182 3335 y Fs(\()p Fu(X)7 b Fs(\))p Fz(,)36 b(but)g(it)f(is)g(not)g(clear)e(if)j(their)f(pro)r(duct)f (is)h(of)g(the)g(form)g Fr(Op)2347 3298 y Fv(A)2401 3335 y Fs(\()p Fu(h)p Fs(\))g Fz(for)g(some)f Fu(h)h Fz(\(ev)n(en)n(tually)f(in)h Fl(F)3581 3347 y Ft(\004)3629 3335 y Fu(L)3686 3305 y Ft(1)3722 3335 y Fs(\(\004\))p Fz(\).)-118 3434 y(On)29 b(the)h(other)f(hand,)h(if)g Fu(f)t(;)14 b(g)29 b Fl(2)e Fu(L)987 3404 y Ft(1)1024 3434 y Fs(\(\004\))p Fz(,)k(then)f(the)g(in)n(tegral)e(in)i(\(4.36\))f(is)g (absolutely)g(con)n(v)n(ergen)n(t)e(and)j(de\034nes)f(a)g(b)r(ounded)-118 3534 y(con)n(tin)n(uous)22 b(function)i(on)f Fs(\004)p Fz(.)36 b(Ho)n(w)n(ev)n(er,)22 b(w)n(e)i(do)f(not)g(see)g(wh)n(y)g(should)g(this)h (function)g(b)r(e)g(in)n(tegrable)e(and,)i(an)n(yw)n(a)n(y)-7 b(,)22 b(applying)-118 3634 y Fr(Op)-8 3597 y Fv(A)74 3634 y Fz(to)27 b(all)g(these)h(is)f(problematic.)36 b(One)27 b(can)g(also)g(tak)n (e)g(adv)-5 b(an)n(tage)25 b(of)j(Prop)r(osition)f(3.10)f(\(b\))i(to)f(endo)n (w)g Fu(L)3409 3604 y Ft(2)3446 3634 y Fs(\(\004\))h Fz(with)g(the)-118 3733 y(structure)f(of)g(a)402 3703 y Fm(\003)440 3733 y Fz(-algebra,)e(the)j (comp)r(osition)f(la)n(w)g(coinciding)g(with)h(\(4.36\))f(on)g(suitable)g (subsets.)6 3833 y(In)37 b(fact)f(man)n(y)g(other)f(solutions)h(exist)g(but)h Fx(they)h(do)g(not)g(se)l(em)f(to)h(b)l(e)g(natur)l(al)f(enough)h(in)g(the)g (pr)l(esent)f(fr)l(amework)p Fz(.)-118 3933 y(In)29 b(a)g(future)h (publication)f(w)n(e)g(will)g(giv)n(e)f(another)g(\(equiv)-5 b(alen)n(t\))30 b(form)f(of)g(the)g(magnetic)g(Mo)n(y)n(al)f(pro)r(duct)h (\(a)g(\020v)n(ery)f(t)n(wisted)-118 4032 y(con)n(v)n(olution\021\))18 b(for)i(whic)n(h)g(it)h(will)f(b)r(e)h(relativ)n(ely)e(easy)g(to)h(de\034ne)h (nice)f(normed)2372 4002 y Fm(\003)2411 4032 y Fz(-algebras.)32 b(Their)19 b(pull-bac)n(ks)g(in)i(the)g(pseudo-)-118 4132 y(di\033eren)n (tial)i(represen)n(tation)e(in)n(v)n(olv)n(e)g(a)i(partial)f(F)-7 b(ourier)22 b(transformation)f(whic)n(h)i(has)f(not)h(an)g(explicitly)g (expressible)f(range.)-118 4231 y(So)27 b(w)n(e)h(p)r(ostp)r(one)g(the)g (study)g(of)g(magnetic)f(comp)r(osition)g(la)n(ws)g(for)h(general)e(\(con)n (tin)n(uous\))h(magnetic)h(\034elds)g(and)f(turn)h(to)g(a)-118 4331 y(sp)r(ecial)f(case)g(whic)n(h)g(is,)h(ho)n(w)n(ev)n(er,)d(v)n(ery)i (comprehensiv)n(e.)-118 4431 y FA(Prop)s(osition)j(4.13.)36 b Fx(Assume)29 b(that)h(the)g(c)l(omp)l(onents)f(of)i(the)f(magnetic)g (\034eld)g Fu(B)k Fx(ar)l(e)c(of)h(class)f Fu(C)3065 4401 y Fm(1)3059 4454 y Fa(p)m(ol)3148 4431 y Fx(.)6 4543 y(\(a\))40 b(F)-6 b(or)40 b(any)f Fu(f)t(;)14 b(g)43 b Fl(2)e(S)6 b Fs(\(\004\))41 b Fx(one)e(has)i Fu(f)34 b Fl(\016)1408 4513 y Fv(B)1490 4543 y Fu(g)43 b Fl(2)e(S)6 b Fs(\(\004\))p Fx(.)69 b(The)40 b(map)g Fl(\016)2351 4513 y Fv(B)2448 4543 y Fs(:)h Fl(S)6 b Fs(\(\004\))27 b Fl(\002)e(S)6 b Fs(\(\004\))41 b Fl(!)g(S)6 b Fs(\(\004\))41 b Fx(is)e(biline)l(ar)i(and)-118 4643 y(c)l(ontinuous.)6 4743 y(\(b\))30 b(F)-6 b(or)30 b(any)g(c)l(ontinuous)f(ve)l(ctor)h(p)l(otential)g Fu(A)g Fx(such)g(that)g Fu(dA)23 b Fs(=)g Fu(B)t Fx(,)30 b(one)g(has)54 b Fr(Op)2686 4706 y Fv(A)2740 4743 y Fs(\()p Fu(f)27 b Fl(\016)2882 4712 y Fv(B)2958 4743 y Fu(g)s Fs(\))22 b(=)h Fr(Op)3253 4706 y Fv(A)3307 4743 y Fs(\()p Fu(f)9 b Fs(\))p Fr(Op)3532 4706 y Fv(A)3586 4743 y Fs(\()p Fu(g)s Fs(\))p Fx(.)-118 4913 y Fz(P)f(r)g(o)g(o)g(f)g(.)22 b(One)h(can)g(pro)n(v)n(e)f(that)i Fl(S)6 b Fs(\(\004\))k Fl(\016)1107 4883 y Fv(B)1175 4913 y Fl(S)c Fs(\(\004\))25 b Fl(\032)e(S)6 b Fs(\(\004\))25 b Fz(directly)-7 b(,)24 b(estimating)f Fu(\030)2420 4883 y Fv(\013)2467 4913 y Fu(@)2516 4883 y Fv(\014)2561 4913 y Fs(\()p Fu(f)c Fl(\016)2695 4883 y Fv(B)2762 4913 y Fu(g)s Fs(\))k Fz(b)n(y)h(in)n(v)n(olv)n(ed)e (manipulations)g(in)-118 5013 y(\(4.36\).)36 b(This)27 b(also)g(giv)n(es)f (the)i(required)f(con)n(tin)n(uit)n(y)-7 b(.)36 b(W)-7 b(e)28 b(prefer)f(to)g(outline)h(a)f(simpler)g(pro)r(of,)g(based)h(on)f(in)n(tegral) f(k)n(ernels.)6 5112 y(Let)f(us)g(c)n(ho)r(ose)f Fu(A)f Fl(2)g Fu(C)745 5082 y Fm(1)739 5135 y Fd(p)r(ol)834 5112 y Fs(\()p Fu(X)r(;)14 b(X)1050 5082 y Fv(?)1087 5112 y Fs(\))26 b Fz(suc)n(h)e(that)h Fu(dA)f Fs(=)e Fu(B)t Fz(;)k(this)f(is)g(p)r(ossible)f(b)n(y)h(\(2.16\).)35 b(W)-7 b(e)25 b(kno)n(w)f(that)h(the)g(maps)f Fu(K)3733 5082 y Fv(A)3810 5112 y Fs(:=)-110 5211 y(~)-118 5232 y(\003)-60 5202 y Fv(A)-6 5232 y Fu(S)50 5202 y Fm(\000)p Ft(1)138 5232 y Fs(\()p Fb(1)12 b Fl(\012)p 307 5165 68 4 v 12 w(F)375 5244 y Fv(X)434 5227 y Fn(?)472 5232 y Fs(\))24 b(:)f Fl(S)6 b Fs(\(\004\))24 b Fl(!)f(S)6 b Fs(\()p Fu(X)19 b Fl(\002)12 b Fu(X)1208 5202 y Fv(?)1245 5232 y Fs(\))p Fz(,)50 b Fr(Int)22 b Fs(:)h Fl(S)6 b Fs(\()p Fu(X)19 b Fl(\002)12 b Fu(X)1865 5202 y Fv(?)1902 5232 y Fs(\))23 b Fl(!)h(L)p Fs([)p Fl(S)2200 5202 y Fm(0)2224 5232 y Fs(\()p Fu(X)7 b Fs(\))p Fu(;)14 b Fl(S)6 b Fs(\()p Fu(X)h Fs(\)])24 b Fz(and)g Fr(Op)2912 5195 y Fv(A)2990 5232 y Fs(:)f Fl(S)6 b Fs(\(\004\))24 b Fl(!)f(L)p Fs([)p Fl(S)3477 5202 y Fm(0)3501 5232 y Fs(\()p Fu(X)7 b Fs(\))p Fu(;)14 b Fl(S)6 b Fs(\()p Fu(X)h Fs(\)])-118 5331 y Fz(are)26 b(top)r(ological)f (isomorphisms)g(and)h(that)h Fr(Op)1408 5294 y Fv(A)1485 5331 y Fs(=)22 b Fr(Int)17 b Fl(\016)f Fu(K)1842 5301 y Fv(A)1895 5331 y Fz(.)37 b(One)26 b(c)n(hec)n(ks)g(easily)g(that)h Fu(K)2869 5301 y Fv(A)2922 5331 y Fs(\()p Fu(f)f Fl(\016)3063 5301 y Fv(B)3136 5331 y Fu(g)s Fs(\))d(=)g Fu(K)3399 5301 y Fv(A)3452 5331 y Fs(\()p Fu(f)9 b Fs(\))17 b Fl(z)g Fu(K)3714 5301 y Fv(A)3767 5331 y Fs(\()p Fu(g)s Fs(\))p Fz(,)-118 5431 y(where)27 b Fs(\()p Fu(L)19 b Fl(z)g Fu(M)9 b Fs(\)\()p Fu(x;)14 b(y)s Fs(\))24 b(:=)736 5364 y Fq(R)775 5460 y Fv(X)852 5431 y Fu(dz)k(L)p Fs(\()p Fu(x;)14 b(z)t Fs(\))p Fu(M)9 b Fs(\()p Fu(z)t(;)14 b(y)s Fs(\))27 b Fz(is)h(the)g(comp)r(osition)g(rule)g(of)g(in)n(tegral)f(k)n (ernels)f(\(leading)i(to)g(the)h(pro)r(duct)-118 5531 y(of)24 b(the)h(in)n(tegral)f(op)r(erators)e(in)n(v)n(olv)n(ed\).)35 b(These)24 b(facts)g(and)h(the)g(con)n(tin)n(uit)n(y)f(of)g Fl(z)f Fs(:)g Fl(S)6 b Fs(\()p Fu(X)20 b Fl(\002)12 b Fu(X)2811 5500 y Fv(?)2848 5531 y Fs(\))g Fl(\002)g(S)6 b Fs(\()p Fu(X)20 b Fl(\002)12 b Fu(X)3299 5500 y Fv(?)3337 5531 y Fs(\))23 b Fl(!)g(S)6 b Fs(\()p Fu(X)20 b Fl(\002)12 b Fu(X)3828 5500 y Fv(?)3865 5531 y Fs(\))-118 5630 y Fz(imply)28 b(b)r(oth)g(the)g(p)r(oin)n (ts)f(\(a\))h(and)f(\(b\))i(for)e Fu(A)c Fl(2)g Fu(C)1492 5600 y Fm(1)1486 5653 y Fd(p)r(ol)1581 5630 y Fs(\()p Fu(X)r(;)14 b(X)1797 5600 y Fv(?)1835 5630 y Fs(\))p Fz(.)1848 5989 y(15)p eop %%Page: 16 16 bop 6 154 a Fz(The)33 b(general)d(case)i(of)g(a)g(con)n(tin)n(uous)f Fu(A)h Fz(in)h(\(b\))g(is)f(solv)n(ed)f(b)n(y)h(gauge)f(co)n(v)-5 b(ariance,)31 b(cf.)51 b(Prop)r(osition)31 b(3.4)h(\(w)n(e)g(consider)-118 254 y(here)27 b Fl(L)p Fs([)p Fl(S)198 223 y Fm(0)222 254 y Fs(\()p Fu(X)7 b Fs(\))p Fu(;)14 b Fl(S)6 b Fs(\()p Fu(X)h Fs(\)])28 b Fz(em)n(b)r(edded)g(in)g Fl(B)1206 186 y Fq(\002)1240 254 y Fu(L)1297 223 y Ft(2)1334 254 y Fs(\()p Fu(X)7 b Fs(\))1474 186 y Fq(\003)1508 254 y Fz(\).)2270 b Fe(\004)6 412 y Fz(Since)22 b Fl(S)6 b Fs(\(\004\))24 b Fz(is)e(ob)n(viously)e(stable)i(under)f(in)n(v)n (olution,)i Fs(\()p Fl(S)6 b Fs(\(\004\))p Fu(;)14 b Fl(\016)2014 382 y Fv(B)2072 412 y Fu(;)2095 382 y Fm(\016)2147 412 y Fs(\))22 b Fz(is)g(a)2342 382 y Fm(\003)2381 412 y Fz(-algebra)d(\(all)j(the)h(axioms) d(are)h(easily)g(v)n(eri\034ed\))-118 512 y(and)32 b Fr(Op)158 475 y Fv(A)243 512 y Fs(:)f Fl(S)6 b Fs(\(\004\))33 b Fl(!)d(L)p Fs([)p Fl(S)754 482 y Fm(0)779 512 y Fs(\()p Fu(X)7 b Fs(\))p Fu(;)14 b Fl(S)6 b Fs(\()p Fu(X)h Fs(\)])32 b Fz(is)g(a)1369 482 y Fm(\003)1407 512 y Fz(-isomorphism.)50 b(Unfortunately)-7 b(,)33 b Fl(S)6 b Fs(\(\004\))34 b Fz(is)e(to)r(o)g(small)g(for)g(man)n(y)g (purp)r(oses.)-118 612 y(F)-7 b(or)27 b(instance,)g(functions)h(dep)r(ending) g(only)f(on)g Fu(x)h Fz(or)f(on)g Fu(p)h Fz(are)e(not)i(included.)-118 844 y Fg(4.2)112 b(Extension)37 b(b)m(y)g(dualit)m(y)-118 997 y Fz(W)-7 b(e)33 b(start)e(extending)i(b)n(y)f(dualit)n(y)g(the)g(magnetic)g (Mo)n(y)n(al)f(pro)r(duct)h(with)h(an)f(asymmetric)g(v)n(ersion:)45 b(w)n(e)32 b(shall)g(comp)r(ose)f(a)-118 1097 y(Sc)n(h)n(w)n(artz)22 b(test)h(function)h(with)g(a)f(temp)r(ered)h(distribution.)35 b(The)24 b(result)f(is)g(a)g(priori)f(a)h(temp)r(ered)h(distribution,)g(but)g (w)n(e)f(shall)-118 1196 y(b)r(e)28 b(able)f(to)h(get)f(more)g(precise)g (informations)g(in)h(certain)f(cases.)36 b(The)28 b(comp)r(onen)n(ts)f(of)g (the)i(magnetic)e(\034eld)h(will)g(b)r(e)g(alw)n(a)n(ys)-118 1296 y(considered)e(to)i(b)r(e)g(in)g Fu(C)665 1266 y Fm(1)659 1319 y Fd(p)r(ol)753 1296 y Fz(,)g(th)n(us)g(the)g(conclusions)e(of)i (Theorem)e(4.13)h(hold.)6 1396 y(The)h(dualit)n(y)f(approac)n(h)f(is)h (facilitated)h(b)n(y)f(the)h(next)g(Lemma:)-118 1495 y FA(Lemma)h(4.14.)37 b Fx(F)-6 b(or)30 b(any)g(functions)f Fu(f)39 b Fx(and)30 b Fu(g)i Fx(in)e Fl(S)6 b Fs(\(\004\))31 b Fx(we)f(have)564 1607 y Fq(Z)610 1796 y Ft(\004)672 1720 y Fu(d\030)d Fs(\()p Fu(f)h Fl(\016)921 1686 y Fv(B)996 1720 y Fu(g)s Fs(\)\()p Fu(\030)t Fs(\))23 b(=)1286 1607 y Fq(Z)1332 1796 y Ft(\004)1394 1720 y Fu(d\030)k Fs(\()p Fu(g)22 b Fl(\016)1636 1686 y Fv(B)1711 1720 y Fu(f)9 b Fs(\)\()p Fu(\030)t Fs(\))23 b(=)2008 1607 y Fq(Z)2054 1796 y Ft(\004)2116 1720 y Fu(d\030)k(f)9 b Fs(\()p Fu(\030)t Fs(\))p Fu(g)s Fs(\()p Fu(\030)t Fs(\))24 b(=)p Fu(<)p 2699 1653 50 4 v 22 w(f)9 b(;)14 b(g)25 b(>)p Fl(\021)e Fs(\()p Fu(f)t(;)14 b(g)s Fs(\))p Fu(:)-118 1957 y Fz(P)8 b(r)g(o)g(o)g(f)g(.)22 b(Of)i(course,)g(one)f(needs)h(only)f(to)h(sho)n(w)f(that)1579 1890 y Fq(R)1618 1987 y Ft(\004)1680 1957 y Fu(d\030)28 b Fs(\()p Fu(f)20 b Fl(\016)1922 1927 y Fv(B)1989 1957 y Fu(g)s Fs(\)\()p Fu(\030)t Fs(\))k(=)2280 1890 y Fq(R)2319 1987 y Ft(\004)2381 1957 y Fu(d\030)j(f)9 b Fs(\()p Fu(\030)t Fs(\))p Fu(g)s Fs(\()p Fu(\030)t Fs(\))p Fz(;)26 b(the)e(other)f(iden)n(tities)i(are)d(trivial)-118 2057 y(consequences.)6 2156 y(The)28 b(calculation)f(will)g(b)r(e)h(straigh)n (tforw)n(ard)d(b)n(y)i(regularization.)35 b(W)-7 b(e)28 b(c)n(ho)r(ose)e (sequences)h Fs(\()p Fu(a)2984 2168 y Fv(n)3029 2156 y Fs(\))3061 2168 y Fv(n)p Fm(2)p Fp(N)3216 2156 y Fl(2)d(S)6 b Fs(\()p Fu(X)h Fs(\))p Fz(,)28 b Fs(\()p Fu(b)3610 2168 y Fv(n)3655 2156 y Fs(\))3687 2168 y Fv(n)p Fm(2)p Fp(N)3842 2156 y Fl(2)-118 2256 y(S)6 b Fs(\()p Fu(X)46 2226 y Fv(?)84 2256 y Fs(\))28 b Fz(suc)n(h)f(that)h Fu(a)555 2268 y Fv(n)623 2256 y Fl(!)23 b Fs(1)28 b Fz(in)f Fl(S)951 2226 y Fm(0)975 2256 y Fs(\()p Fu(X)7 b Fs(\))28 b Fz(and)f Fu(b)1340 2268 y Fv(n)1408 2256 y Fl(!)c Fs(1)k Fz(in)h Fl(S)1736 2226 y Fm(0)1760 2256 y Fs(\()p Fu(X)1868 2226 y Fv(?)1906 2256 y Fs(\))p Fz(.)37 b(Then)28 b(it)g(is)f(sho)n(wn)g(easily)g(that)663 2364 y Fq(Z)710 2553 y Fv(X)787 2364 y Fq(Z)833 2553 y Fv(X)892 2536 y Fn(?)944 2477 y Fu(dq)g(dp)c Fs(\()p Fu(f)k Fl(\016)1301 2443 y 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Fz(and)f(Prop)r(osition)-118 3599 y(3.5,)34 b Fu(T)44 b Fz(is)33 b(in)h Fr(Op)441 3562 y Fv(A)495 3599 y Fs([)p Fl(M)p Fs(\(\004\)])g Fz(if)g(and)f(only)g(if)h Fu(T)12 b Fl(L)p Fs(\()p Fl(S)1519 3569 y Fm(0)1542 3599 y Fu(;)i Fl(S)6 b Fs(\))34 b Fl(\032)e(L)p Fs(\()p Fl(S)1943 3569 y Fm(0)1967 3599 y Fu(;)14 b Fl(S)6 b Fs(\))34 b Fz(and)f Fl(L)p Fs(\()p Fl(S)2438 3569 y Fm(0)2463 3599 y Fu(;)14 b Fl(S)6 b Fs(\))p Fu(T)44 b Fl(\032)32 b(L)p Fs(\()p Fl(S)2923 3569 y Fm(0)2948 3599 y Fu(;)14 b Fl(S)6 b Fs(\))p Fz(.)55 b(The)33 b(last)g(inclusion)g(is) -118 3699 y(equiv)-5 b(alen)n(t)29 b(to)h Fu(T)443 3669 y Fm(\003)480 3699 y Fl(L)p Fs(\()p Fl(S)625 3669 y Fm(0)649 3699 y Fu(;)14 b Fl(S)6 b Fs(\))27 b Fl(\032)f(L)p Fs(\()p Fl(S)1037 3669 y Fm(0)1062 3699 y Fu(;)14 b Fl(S)6 b Fs(\))p Fz(.)43 b(It)30 b(is)g(easy)e(to)i(see)f(that)h Fu(T)12 b Fl(L)p Fs(\()p Fl(S)2243 3669 y Fm(0)2266 3699 y Fu(;)i Fl(S)6 b Fs(\))27 b Fl(\032)f(L)p Fs(\()p Fl(S)2654 3669 y Fm(0)2678 3699 y Fu(;)14 b Fl(S)6 b Fs(\))31 b Fz(if)f(and)f(only)g(if)i Fu(T)37 b Fl(2)27 b(L)p Fs(\()p Fl(S)6 b Fs(\))p Fz(.)44 b(One)-118 3799 y(implication)30 b(is)f(trivial)h(and)f(the)i(other)e(one)h(follo)n(ws)e(once)i(again)f(from)g (the)h(Closed)g(Graph)f(Theorem)g(and)h(from)f(the)i(fact)-118 3898 y(that)25 b(for)g(an)n(y)f Fu(v)j Fl(2)c(S)32 b Fz(there)25 b(exist)g Fu(u)e Fl(2)g Fu(S)1174 3868 y Fm(0)1222 3898 y Fz(and)i Fu(S)j Fl(2)c(L)p Fs(\()p Fl(S)1684 3868 y Fm(0)1708 3898 y Fu(;)14 b Fl(S)6 b Fs(\))26 b Fz(suc)n(h)f(that)g Fu(S)5 b(u)23 b Fs(=)f Fu(v)s Fz(.)37 b(Then,)25 b(b)n(y)g(taking)g(also)f Fu(T)3367 3868 y Fm(\003)3430 3898 y Fz(in)n(to)h(accoun)n(t,)-118 3998 y(w)n(e)i(see)g(that)h Fr(Op)428 3961 y Fv(A)482 3998 y Fs([)p Fl(M)p Fs(\(\004\)])c(=)f Fl(L)p Fs(\()p Fl(S)6 b Fs(\))20 b Fl(\\)e(L)p Fs(\()p Fl(S)1274 3968 y Fm(0)1299 3998 y Fs(\))p Fz(.)6 4097 y(Let)28 b Fu(F)r(;)14 b(G)24 b Fl(2)f(M)p Fs(\(\004\))p Fz(.)38 b(W)-7 b(e)28 b(calculate)f(for)g Fu(u)22 b Fl(2)i(S)1515 4067 y Fm(0)1566 4097 y Fz(and)k Fu(h)23 b Fl(2)g(S)6 b Fs(\(\004\))-64 4305 y Fr(Op)46 4268 y Fv(A)100 4305 y Fs(\()p Fu(F)31 b Fl(\016)258 4271 y Fv(B)333 4305 y Fu(G)p Fs(\))444 4213 y Fq(h)484 4305 y Fr(Op)594 4268 y Fv(A)648 4305 y Fs(\()p Fu(h)p Fs(\))p Fu(u)808 4213 y Fq(i)870 4305 y Fs(=)23 b Fr(Op)1068 4268 y Fv(A)1122 4305 y Fs(\()p Fu(F)31 b Fl(\016)1280 4271 y Fv(B)1355 4305 y Fu(G)19 b Fl(\016)1481 4271 y Fv(B)1556 4305 y Fu(h)p Fs(\))p Fu(u)k Fs(=)f Fr(Op)1904 4268 y Fv(A)1958 4305 y Fs(\()p Fu(F)12 b Fs(\))2101 4213 y Fq(h)2141 4305 y Fr(Op)2251 4268 y Fv(A)2305 4305 y Fs(\()p Fu(G)19 b Fl(\016)2463 4271 y Fv(B)2538 4305 y Fu(h)p Fs(\))p Fu(u)2666 4213 y Fq(i)2728 4305 y Fs(=)2816 4213 y Fq(h)2855 4305 y Fr(Op)2965 4268 y Fv(A)3019 4305 y Fs(\()p Fu(F)12 b Fs(\))p Fr(Op)3259 4268 y Fv(A)3313 4305 y Fs(\()p Fu(G)p Fs(\))3442 4213 y Fq(i)3496 4305 y Fr(Op)3606 4268 y Fv(A)3660 4305 y Fs(\()p Fu(h)p Fs(\))p Fu(u;)-118 4527 y Fz(where)30 b(w)n(e)h(used)g(Prop)r (osition)f(4.18.)45 b(Since)31 b(an)n(y)f Fu(v)i Fl(2)d(S)37 b Fz(can)31 b(b)r(e)g(written)g(as)f Fr(Op)2516 4490 y Fv(A)2570 4527 y Fs(\()p Fu(h)p Fs(\))p Fu(u)h Fz(for)f(some)g Fu(u)e Fl(2)h(S)3318 4497 y Fm(0)3373 4527 y Fz(and)i Fu(h)d Fl(2)h(S)6 b Fs(\(\004\))p Fz(,)-118 4627 y(the)28 b(m)n(ultiplicativ)n(e)f(prop)r(ert)n (y)g(of)g Fr(Op)1092 4590 y Fv(A)1174 4627 y Fz(on)g Fl(M)p Fs(\(\004\))i Fz(is)e(sho)n(wn.)6 4726 y(The)h(in)n(v)n(olutivit)n(y)f(of)g Fr(Op)824 4689 y Fv(A)906 4726 y Fz(is)h(v)-5 b(alid)27 b(on)h Fl(S)1362 4696 y Fm(0)1385 4726 y Fs(\(\004\))p Fz(,)h(as)e(remark)n(ed)f(b)r (efore.)1565 b Fe(\004)6 4885 y FA(Remark.)33 b Fz(Prop)r(ositions)19 b(4.20)f(and)h(4.21)f(are)h(the)h(most)f(imp)r(ortan)n(t)g(results.)34 b(W)-7 b(e)20 b(note)f(here)g(rapidly)g(some)g(extra)f(results)-118 4984 y(concerning)28 b(the)i(magnetic)g(Mo)n(y)n(al)e(algebras,)g(all)i(of)f (an)h(elemen)n(tary)f(nature.)43 b(One)29 b(also)g(de\034nes)g(b)n(y)h (dualit)n(y)f(pro)r(ducts)h(of)-118 5084 y(the)g(form)g Fu(F)279 5096 y Ft(1)336 5084 y Fl(\016)378 5054 y Fv(B)455 5084 y Fu(G)20 b Fl(\016)582 5054 y Fv(B)659 5084 y Fu(F)712 5096 y Ft(2)777 5084 y Fl(2)27 b(S)915 5054 y Fm(0)939 5084 y Fs(\(\004\))k Fz(for)e Fu(F)1271 5096 y Ft(1)1336 5084 y Fl(2)e(M)1518 5096 y Fv(R)1573 5084 y Fs(\(\004\))p Fz(,)k Fu(F)1799 5096 y Ft(2)1864 5084 y Fl(2)c(M)2046 5096 y Fv(L)2096 5084 y Fs(\(\004\))k Fz(and)e Fu(G)f Fl(2)f(S)2640 5054 y Fm(0)2664 5084 y Fs(\(\004\))p Fz(;)32 b Fl(S)2894 5054 y Fm(0)2918 5084 y Fs(\(\004\))f Fz(is)f(a)f Fs(\()p Fl(M)3357 5096 y Fv(R)3412 5084 y Fs(\(\004\))p Fu(;)14 b Fl(M)3668 5096 y Fv(L)3718 5084 y Fs(\(\004\)\))p Fz(-)-118 5184 y(bimo)r(dule.)38 b(In)27 b(fact)h Fl(M)644 5196 y Fv(L)712 5184 y Fl(\016)754 5154 y Fv(B)829 5184 y Fl(M)929 5196 y Fv(L)1002 5184 y Fl(\032)23 b(M)1190 5196 y Fv(L)1267 5184 y Fz(and)28 b Fl(M)1529 5196 y Fv(R)1601 5184 y Fl(\016)1643 5154 y Fv(B)1718 5184 y Fl(M)1818 5196 y Fv(R)1896 5184 y Fl(\032)23 b(M)2084 5196 y Fv(R)2138 5184 y Fz(,)28 b(hence)f Fl(M)2519 5196 y Fv(L)2597 5184 y Fz(and)g Fl(M)2858 5196 y Fv(R)2940 5184 y Fz(are)g(algebras.)35 b(But)28 b(they)g(are)-118 5283 y(di\033eren)n(t)d(and) g(corresp)r(ond)f(to)h(eac)n(h)f(other)h(b)n(y)g(complex)f(conjugation,)h(so) g Fl(M)g Fz(is)g(optimally)f(de\034ned)i(as)e(a)3315 5253 y Fm(\003)3353 5283 y Fz(-algebra)f(b)n(y)i(the)-118 5383 y(presen)n(t)i(metho) r(ds.)37 b(The)27 b(pro)r(of)g(of)h(Prop)r(osition)f(4.21)f(also)g(leads)h (to)h Fr(Op)2228 5346 y Fv(A)2282 5383 y Fl(M)2382 5395 y Fv(L)2432 5383 y Fs(\(\004\))c(=)e Fl(L)p Fs(\()p Fl(S)6 b Fs(\))29 b Fz(and)f Fr(Op)3140 5346 y Fv(A)3194 5383 y Fl(M)3294 5395 y Fv(R)3348 5383 y Fs(\(\004\))c(=)f Fl(L)p Fs(\()p Fl(S)3724 5353 y Fm(0)3748 5383 y Fs(\))p Fz(.)6 5483 y(The)28 b(next)g(striking)f (result)g(sho)n(ws)f(once)h(more)g(the)h(imp)r(ortance)f(of)g(the)h(magnetic) f(Mo)n(y)n(al)f(algebras.)-118 5582 y FA(Prop)s(osition)k(4.22.)36 b Fx(One)30 b(has)g Fl(S)1015 5552 y Fm(0)1039 5582 y Fs(\(\004\))19 b Fl(\016)1219 5552 y Fv(B)1294 5582 y Fl(S)6 b Fs(\(\004\))25 b Fl(\032)d(M)1681 5594 y Fv(R)1736 5582 y Fs(\(\004\))30 b Fx(and)g Fl(S)6 b Fs(\(\004\))20 b Fl(\016)2283 5552 y Fv(B)2358 5582 y Fl(S)2414 5552 y Fm(0)2438 5582 y Fs(\(\004\))k Fl(\032)f(M)2769 5594 y Fv(L)2818 5582 y Fs(\(\004\))p Fx(.)1848 5989 y Fz(17)p eop %%Page: 18 18 bop -118 154 a Fz(P)8 b(r)g(o)g(o)g(f)g(.)282 309 y Fr(Op)392 272 y Fv(A)447 309 y Fs([)p Fl(S)526 275 y Fm(0)549 309 y Fs(\(\004\))20 b Fl(\016)730 275 y Fv(B)805 309 y Fl(S)6 b Fs(\(\004\)])24 b(=)f Fr(Op)1225 272 y Fv(A)1279 309 y Fs([)p Fl(S)1358 275 y Fm(0)1382 309 y Fs(\(\004\)])p Fr(Op)1635 272 y Fv(A)1689 309 y Fs([)p Fl(S)6 b Fs(\(\004\)])25 b(=)e Fl(L)p Fs(\()p Fl(S)6 b Fu(;)14 b Fl(S)2261 275 y Fm(0)2285 309 y Fs(\))p Fl(L)p Fs(\()p Fl(S)2462 275 y Fm(0)2487 309 y Fu(;)g Fl(S)6 b Fs(\))24 b(=)e Fl(L)p Fs(\()p Fl(S)2868 275 y Fm(0)2893 309 y Fs(\))h Fl(\032)g Fr(Op)3146 272 y Fv(A)3200 309 y Fl(M)3300 321 y Fv(R)3354 309 y Fs(\(\004\))p Fu(;)-118 464 y Fz(th)n(us)k Fl(S)120 434 y Fm(0)144 464 y Fs(\(\004\))20 b Fl(\016)325 434 y Fv(B)400 464 y Fl(S)6 b Fs(\(\004\))24 b Fl(\032)f(M)787 476 y Fv(R)841 464 y Fs(\(\004\))p Fz(.)38 b(The)28 b(other)e(inclusion)i(is) f(pro)n(v)n(ed)f(analogously)-7 b(.)1284 b Fe(\004)6 623 y Fz(W)-7 b(e)39 b(note)f(that)h(b)r(oth)g(the)g(inclusions)e(are)h(strict.)69 b(F)-7 b(or)38 b(zero)f(magnetic)h(\034eld)h Fu(f)34 b Fl(\016)25 b Fu(G)39 b Fz(is)f(smo)r(oth)g(if)h Fu(f)50 b Fl(2)41 b(S)6 b Fs(\(\004\))40 b Fz(and)-118 723 y Fu(G)23 b Fl(2)h(S)105 692 y Fm(0)128 723 y Fs(\(\004\))p Fz(,)29 b(cf.)37 b([GBV1].)-118 950 y Fg(4.4)112 b(Some)37 b(imp)s(ortan)m(t)f(sub)s(classes)-118 1103 y Fz(W)-7 b(e)32 b(k)n(eep)g(the)g(setting)g(of)g(the)g(preceding)f (paragraphs,)f(i.e.)50 b(the)32 b(comp)r(onen)n(ts)g(of)f Fu(B)37 b Fz(\(and)32 b(those)f(of)h Fu(A)g Fz(when)g(necessary\))-118 1203 y(are)e(of)h(class)e Fu(C)384 1173 y Fm(1)378 1226 y Fd(p)r(ol)473 1203 y Fz(.)47 b(Simple)31 b(examples)f(sho)n(w)g(readily)g(that)h Fl(M)p Fs(\(\004\))h Fz(is)f(m)n(uc)n(h)f(larger)f(than)i Fl(S)6 b Fs(\(\004\))p Fz(.)48 b(One)31 b(sho)n(ws)e(easily)h(that)-118 1322 y(if)g Fu(f)9 b Fs(\()p Fu(x;)14 b(p)p Fs(\))27 b(=)g Fu(f)360 1334 y Ft(1)397 1322 y Fs(\()p Fu(x)p Fs(\))k Fz(dep)r(ends)f(only)g (on)f(the)h(v)-5 b(ariable)29 b(in)h Fu(X)7 b Fz(,)30 b(then)h Fr(Op)2153 1286 y Fv(A)2207 1322 y Fs(\()p Fu(f)9 b Fs(\))27 b(=)f Fu(f)2480 1334 y Ft(1)2517 1322 y Fs(\()p Fu(Q)p Fs(\))p Fz(.)44 b(If)31 b Fu(f)2841 1334 y Ft(1)2908 1322 y Fz(has)e(temp)r(erate)h (gro)n(wth)e(then)-118 1422 y Fu(f)-77 1434 y Ft(1)-40 1422 y Fs(\()p Fu(Q)p Fs(\))j Fl(2)h(L)p Fs(\()p Fl(S)6 b Fs(\))23 b Fl(\\)f(L)p Fs(\()p Fl(S)630 1392 y Fm(0)655 1422 y Fs(\))p Fz(,)34 b(th)n(us)f Fu(f)40 b Fl(2)31 b(M)p Fs(\(\004\))j Fz(b)n(y)e(Prop)r (osition)g(4.21.)50 b(It)33 b(is)g(also)e(quite)i(ob)n(vious)e(that)i Fl(F)3266 1434 y Ft(\004)3314 1422 y Fu(L)3371 1392 y Ft(1)3408 1422 y Fs(\(\004\))f Fl(\032)f(M)p Fs(\(\004\))p Fz(,)-118 1522 y(since)h Fu(W)180 1492 y Fv(A)234 1522 y Fs(\()p Fu(\030)t Fs(\))h Fz(is)g(a)f(con)n(tin)n(uous)f(op)r(erator)f(in)j Fl(S)39 b Fz(for)32 b(all)g Fu(\030)j Fl(2)d Fs(\004)p Fz(.)52 b(A)n(ctually)-7 b(,)33 b(the)g(same)f(argumen)n(t)f(w)n(ould)h(also)f(sho)n(w)g(that)-118 1621 y(F)-7 b(ourier)28 b(transforms)h(of)g(b)r(ounded,)i(complex)e(measures) f(on)i Fs(\004)g Fz(are)e(also)h(in)h(the)g(magnetic)f(Mo)n(y)n(al)f (algebra.)41 b(In)30 b(the)g(sequel)-118 1721 y(w)n(e)d(shall)g(outline)h(a)f (less)g(eviden)n(t)h(example.)6 1821 y(Let)f Fu(C)219 1790 y Fm(1)213 1844 y Fd(p)r(ol,u)363 1821 y Fs(\(\004\))d Fl(\032)f(S)650 1790 y Fm(0)674 1821 y Fs(\(\004\))k Fz(b)r(e)h(the)e(space)g(of)h (inde\034nitely)g(deriv)-5 b(able)26 b(complex)g(functions)h(on)f Fs(\004)i Fz(ha)n(ving)d(uniform)i(p)r(olyno-)-118 1920 y(mial)j(gro)n(wth)e (at)i(in\034nit)n(y;)h(i.e.)44 b Fu(f)35 b Fl(2)28 b Fu(C)1140 1890 y Fm(1)1134 1943 y Fd(p)r(ol,u)1284 1920 y Fs(\(\004\))j Fz(when)f(it)g(is)g(inde\034nitely)g(deriv)-5 b(able)30 b(and)f(there)h (exists)f Fu(m)e Fl(2)g Fw(N)40 b Fz(\(dep)r(ending)-118 2033 y(on)27 b Fu(f)9 b Fz(\))28 b(suc)n(h)f(that)h(for)f(an)n(y)g(m)n(ulti-index) g Fu(a)c Fl(2)h Fw(N)1400 2003 y Ft(2)p Fv(N)1529 2033 y Fz(one)j(has)g Fl(j)p Fs(\()p Fu(@)1933 2003 y Fv(a)1974 2033 y Fu(f)9 b Fs(\)\()p Fu(\030)t Fs(\))p Fl(j)23 b(\024)g Fu(C)2353 2045 y Fv(a)2417 2033 y Fu(<)f(\030)27 b(>)2632 2003 y Fv(m)2723 2033 y Fz(for)g(all)g Fu(\030)g Fl(2)d Fs(\004)p Fz(.)-118 2132 y FA(Prop)s(osition)30 b(4.23.)36 b Fu(C)697 2102 y Fm(1)691 2156 y Fd(p)r(ol,u)842 2132 y Fs(\(\004\))24 b Fl(\032)e(M)p Fs(\(\004\))p Fx(.)-118 2303 y Fz(P)8 b(r)g(o)g(o)g(f)g(.)25 b(First)g(w)n(e)h(reduce)f(our)h(pro)r (of)f(to)h(a)f(precise)g(estimate.)37 b(Classes)24 b(of)i(functions)g Fl(T)c Fs(\(\004\))27 b Fz(will)f(b)r(e)g(denoted)g(brie\035y)g(b)n(y)f Fl(T)d Fz(.)6 2403 y(F)-7 b(or)30 b(an)n(y)g Fu(m)e Fl(2)g Fw(R)37 b Fz(w)n(e)30 b(set)g Fu(S)906 2372 y Fv(m)901 2423 y Ft(0)997 2403 y Fs(:=)e Fl(f)p Fu(f)36 b Fl(2)28 b Fu(C)1380 2372 y Fm(1)1478 2403 y Fl(j)g Fu(<)g Fl(\001)g Fu(>)1738 2372 y Fm(\000)p Fv(m)1880 2403 y Fu(@)1929 2372 y Fv(a)1969 2403 y Fu(f)37 b Fl(2)28 b Fu(L)2187 2372 y Fm(1)2257 2403 y Fu(;)42 b Fl(8)p Fu(a)27 b Fl(2)h Fw(N)2577 2372 y Ft(2)p Fv(N)2679 2403 y Fl(g)i Fz(and)g Fu(R)2979 2372 y Fv(m)2978 2423 y Ft(0)3070 2403 y Fs(:=)e Fl(f)p Fu(f)36 b Fl(2)28 b Fu(C)3453 2372 y Fm(1)3552 2403 y Fl(j)p Fu(<)f Fl(\001)h Fu(>)3783 2372 y Fm(\000)p Fv(m)-118 2502 y Fu(@)-69 2472 y Fv(a)-29 2502 y Fu(f)j Fl(2)24 b Fu(L)179 2472 y Ft(1)216 2502 y Fu(;)37 b Fl(8)p Fu(a)22 b Fl(2)h Fw(N)521 2472 y Ft(2)p Fv(N)623 2502 y Fl(g)p Fz(.)35 b(The)24 b(reason)f(for)g(in)n(tro)r(ducing)h(these)g(function)g(spaces)f(is) h(the)h(fact)f(that)g Fu(C)3111 2472 y Fm(1)3105 2525 y Fd(p)r(ol,u)3256 2502 y Fs(\(\004\))f(=)g Fl([)3541 2514 y Fv(m)p Fm(2)p Fp(R)3691 2502 y Fu(S)3747 2472 y Fv(m)3742 2523 y Ft(0)3833 2502 y Fs(=)-118 2620 y Fl([)-63 2632 y Fv(m)p Fm(2)p Fp(R)87 2620 y Fu(R)151 2590 y Fv(m)150 2641 y Ft(0)214 2620 y Fz(.)49 b(This)32 b(follo)n(ws)f(from) g Fu(R)1021 2590 y Fv(m)1020 2641 y Ft(0)1114 2620 y Fl(\032)f Fu(S)1265 2590 y Fv(m)1260 2641 y Ft(0)1357 2620 y Fl(\032)g Fu(R)1516 2585 y Fv(m)p Ft(+2)p Fv(N)6 b Ft(+)p Fv(")1515 2642 y Ft(0)1804 2620 y Fz(,)33 b(v)-5 b(alid)32 b(for)f(an)n(y)g Fu(m)f Fl(2)g Fw(R)38 b Fz(and)32 b(an)n(y)f Fu(")f(>)f Fs(0)p Fz(,)k(whic)n(h)e(is)h(sho)n(wn)f(b)n(y)-118 2720 y(trivial)c(estimates.)36 b(So)28 b(the)g(Prop)r(osition)e(will)i(b)r(e)g(pro)n(v)n(ed)e(if)i(w)n(e)f (sho)n(w)g(that)h Fu(R)2419 2690 y Fv(m)2418 2740 y Ft(0)2505 2720 y Fl(\032)22 b(M)p Fz(,)28 b Fl(8)p Fu(m)22 b Fl(2)h Fw(R)p Fz(.)6 2820 y(In)33 b(fact)g(the)h(spaces)e Fu(S)749 2789 y Fv(m)744 2840 y Ft(0)844 2820 y Fz(w)n(ere)g(in)n(tro)r(duced)h(only)f(for)g (comparison.)51 b(They)33 b(are)f(the)h(\034rst)g(constituen)n(ts)f(of)h Fu(C)3591 2789 y Fm(1)3585 2843 y Fd(p)r(ol,u)3769 2820 y Fz(y)n(ou)-118 2930 y(w)n(ould)23 b(think)h(of,)g(but)h(tec)n(hnically)e(the)h(classes)e Fu(R)1472 2900 y Fv(m)1471 2950 y Ft(0)1558 2930 y Fz(are)h(b)r(etter)h (suited,)g(since)g Fu(L)2457 2900 y Ft(1)2493 2930 y Fz(-inequalities)f(in)h (initial)g(spaces)f(are)f(within)-118 3029 y(reac)n(h)k(\(for)i(example)f(w)n (e)g(ha)n(v)n(e)f Fl(k)d Fu(f)k Fl(\016)1074 2999 y Fv(B)1149 3029 y Fu(g)f Fl(k)1257 3041 y Fm(1)1350 3029 y Fl(\024k)c Fu(f)31 b Fl(k)1593 3041 y Ft(1)1630 3029 y Fl(k)23 b Fu(g)i Fl(k)1802 3041 y Ft(1)1839 3029 y Fz(\).)38 b Fu(R)1996 2999 y Fv(m)1995 3050 y Ft(0)2086 3029 y Fz(is)27 b(a)h(lo)r(cally)e(con)n(v)n(ex) h(space)f(with)i(the)g(family)g(of)g(norms)-118 3129 y Fl(f)p Fu(r)-37 3099 y Fv(m)-39 3150 y(n)26 3129 y Fl(g)68 3141 y Fv(n)p Fm(2)p Fv(N)216 3129 y Fz(,)h(where)f Fu(r)548 3099 y Fv(m)546 3150 y(n)612 3129 y Fs(\()p Fu(f)9 b Fs(\))25 b(:=)863 3067 y Fq(P)950 3154 y Fm(j)p Fv(a)p Fm(j\024)p Fv(n)1148 3129 y Fl(k)f Fu(@)1263 3099 y Fv(a)1302 3129 y Fs([)p Fu(<)h Fl(\001)f Fu(>)1527 3099 y Fm(\000)p Fv(m)1666 3129 y Fu(f)9 b Fs(])24 b Fl(k)1805 3141 y Ft(1)1871 3129 y Fz(\(other,)k(equiv)-5 b(alen)n(t,)28 b(family)h(is)f(obtained)g(b)n(y)h(writing)f Fu(@)3696 3099 y Fv(a)3764 3129 y Fz(and)-118 3243 y Fu(<)23 b Fl(\001)g Fu(>)81 3213 y Fm(\000)p Fv(m)223 3243 y Fz(in)28 b(rev)n(ersed)d(order.\))6 3343 y(Since)h Fl(S)31 b Fz(is)25 b(dense)g(in)g Fu(R)765 3312 y Fv(m)764 3363 y Ft(0)828 3343 y Fz(,)h(to)e(sho)n(w)h(that)g Fu(R)1420 3312 y Fv(m)1419 3363 y Ft(0)1506 3343 y Fl(\032)d(M)j Fz(it)h(will)f(b)r(e)g(su\036cien)n(t)g(to)g (pro)n(v)n(e)e(that)j(for)e(an)n(y)g Fu(g)i Fl(2)d(S)32 b Fz(the)25 b(mappings)-118 3442 y Fl(S)32 b(3)27 b Fu(f)34 b Fl(7!)26 b Fu(f)i Fl(\016)341 3412 y Fv(B)418 3442 y Fu(g)g Fl(2)f(S)35 b Fz(and)30 b Fl(S)i(3)26 b Fu(f)35 b Fl(7!)26 b Fu(g)c Fl(\016)1269 3412 y Fv(B)1345 3442 y Fu(f)35 b Fl(2)26 b(S)36 b Fz(are)29 b(con)n(tin)n(uous)f(if)i(on)f(the)g(initial)h(space)e Fl(S)36 b Fz(w)n(e)29 b(cosider)f(the)i(top)r(ology)-118 3542 y(induced)25 b(from)f Fu(R)445 3512 y Fv(m)444 3562 y Ft(0)508 3542 y Fz(.)36 b(W)-7 b(e)25 b(shall)f(treat)g(the)h(\034rst)f(mapping;)h(in)g(fact)g(this)f (is)h(enough,)f(since)h Fu(R)2833 3512 y Fv(m)2832 3562 y Ft(0)2920 3542 y Fz(is)f(left)i(in)n(v)-5 b(arian)n(t)23 b(b)n(y)h(complex)-118 3641 y(conjugation.)6 3741 y(In)41 b(the)h(sequel)e(w)n(e)g(shall)h(alw)n(a)n (ys)e(write)h Fu(\030)49 b Fs(=)c(\()p Fu(q)s(;)14 b(p)p Fs(\))p Fz(,)45 b Fu(\021)j Fs(=)c(\()p Fu(x;)14 b(k)s Fs(\))42 b Fz(and)f Fu(\020)51 b Fs(=)45 b(\()p Fu(y)s(;)14 b(l)r Fs(\))p Fz(.)76 b(W)-7 b(e)41 b(also)f(set)h Fu(\036)3438 3711 y Fv(B)3495 3741 y Fs(\()p Fu(q)s(;)14 b(x;)g(y)s Fs(\))46 b(:=)-118 3854 y Fu(e)-79 3824 y Fm(\000)p Fv(i)p Ft(\000)37 3799 y Fn(B)86 3824 y Ft(\()p Fv()p Ft(\))912 3854 y Fz(;)23 b(it)e(is)f(a)g(function)h(in)g Fu(C)1645 3824 y Fm(1)1639 3877 y Fd(p)r(ol)1754 3854 y Fz(with)g Fl(j)p Fu(\036)2008 3824 y Fv(B)2066 3854 y Fs(\()p Fu(q)s(;)14 b(x;)g(y)s Fs(\))p Fl(j)23 b Fs(=)g(1)p Fz(.)34 b(By)20 b(taking)g(in)n(to)g(accoun)n(t) g(the)h(discussion)-118 3966 y(ab)r(o)n(v)n(e)j(and)h(the)g(form)g(of)h(the)f (seminorms)f(on)h Fl(S)6 b Fz(,)27 b(w)n(e)e(see)f(that)i(it)g(is)f(enough)f (to)h(pro)n(v)n(e)f(that)i(for)e(an)n(y)h Fu(g)g Fl(2)f(S)6 b Fz(,)26 b Fu(\013;)14 b(\014)t(;)g(\015)5 b(;)14 b(\016)26 b Fl(2)e Fw(N)3829 3936 y Fv(N)-118 4066 y Fz(and)k Fu(m)23 b Fl(2)h Fw(R)34 b Fz(\(I)28 b(shall)f(tak)n(e)h Fu(m)f Fz(to)h(b)r(e)g(an)g (ev)n(en)f(p)r(ositiv)n(e)h(in)n(teger\))f(there)g(exist)h Fu(n)c Fl(2)f Fw(N)38 b Fz(and)28 b Fu(C)i(<)23 b Fl(1)28 b Fz(\(they)g(b)r(oth)h(dep)r(end)f(on)-118 4166 y(ev)n(erything\))e(suc)n(h)i (that)1261 4365 y Fl(k)22 b Fu(q)1365 4331 y Fv(\013)1413 4365 y Fu(p)1455 4331 y Fv(\014)1499 4365 y Fu(@)1548 4331 y Fv(\015)1543 4385 y(q)1590 4365 y Fu(@)1639 4331 y Fv(\016)1634 4385 y(p)1676 4365 y Fs(\()p Fu(f)27 b Fl(\016)1818 4331 y Fv(B)1893 4365 y Fu(g)s Fs(\))c Fl(k)2033 4377 y Fm(1)2126 4365 y Fl(\024)f Fu(C)6 b(r)2317 4331 y Fv(m)2315 4385 y(n)2382 4365 y Fs(\()p Fu(f)j Fs(\))p Fu(:)-118 4498 y Fz(No)n(w)27 b(the)h(pro)r(of)f(will)h(pro)r (ceed)f(in)g(sev)n(eral)f(steps:)6 4597 y Fx(Step)k(1.)38 b Fz(A)28 b(simple)f(calculation)g(giv)n(es)1391 4685 y Fq(\002)1426 4753 y Fu(q)1466 4718 y Fv(\013)1513 4753 y Fu(p)1555 4718 y Fv(\014)1600 4753 y Fu(@)1649 4718 y Fv(\015)1644 4773 y(q)1691 4753 y Fu(@)1740 4718 y Fv(\016)1735 4773 y(p)1776 4753 y Fs(\()p Fu(f)h Fl(\016)1919 4718 y Fv(B)1994 4753 y Fu(g)s Fs(\))2069 4685 y Fq(\003)2117 4753 y Fs(\()p Fu(q)s(;)14 b(p)p Fs(\))23 b(=)303 4961 y(=)406 4882 y Fq(X)390 5061 y Fv(\015)429 5044 y Fi(0)451 5061 y Fm(\024)p Fv(\015)555 4961 y Fu(C)620 4921 y Fv(\013\014)s(\015)t(\016)614 4986 y(\015)653 4969 y Fi(0)793 4848 y Fq(Z)839 5037 y Ft(\004)901 4848 y Fq(Z)947 5037 y Ft(\004)1009 4961 y Fu(d\021)k(d\020)i(q)1268 4927 y Fv(\013)1316 4961 y Fu(p)1358 4927 y Fv(\014)1402 4961 y Fs(\()p Fu(k)22 b Fl(\000)c Fu(l)r Fs(\))1641 4927 y Fv(\015)1680 4902 y Fi(0)1706 4961 y Fs(\()p Fu(y)j Fl(\000)d Fu(x)p Fs(\))1962 4927 y Fv(\016)1999 4961 y Fu(e)2038 4927 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))2501 4961 y Fs(\()p Fu(@)2582 4927 y Fv(\015)t Fm(\000)p Fv(\015)2712 4902 y Fi(0)2577 4982 y Fv(q)2737 4961 y Fu(\036)2786 4927 y Fv(B)2844 4961 y Fs(\)\()p Fu(q)s(;)c(x;)g(y)s Fs(\))p Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))p Fu(g)s Fs(\()p Fu(\020)d Fs(\))p Fu(:)-118 5196 y Fz(The)31 b(Leibnitz)g(rule)g(w)n(as)f(used,)i(as)f(w)n(ell) f(as)h(the)g(t)n(w)n(o)f(iden)n(tities)61 b Fu(@)2062 5165 y Fv(\016)2057 5216 y(p)2098 5196 y Fu(e)2137 5165 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))2629 5196 y Fs(=)28 b(\(2)p Fu(i)p Fs(\))2857 5165 y Fm(j)p Fv(\016)r Fm(j)2933 5196 y Fs(\()p Fu(y)23 b Fl(\000)e Fu(x)p Fs(\))3194 5165 y Fv(\016)3231 5196 y Fu(e)3270 5165 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r 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b(dev)n(elopping,)-118 5513 y(the)i(factor)f Fu(<)22 b(\021)k(>)482 5483 y Fv(m)573 5513 y Fz(con)n(tributes)h(with)h (terms)f(of)g(the)h(form)g Fu(x)1906 5483 y Fv(\026)1951 5513 y Fu(k)1997 5483 y Fv(\027)2038 5513 y Fz(.)37 b(Th)n(us,)27 b(w)n(e)g(need)h(to)f(estimate)h(ob)5 b(jects)27 b(as)291 5593 y Fq(Z)338 5782 y Ft(\004)399 5593 y Fq(Z)445 5782 y Ft(\004)507 5706 y Fu(d\021)g(d\020)i(q)766 5672 y Fv(\013)809 5680 y Fo(1)846 5706 y Fu(x)893 5672 y Fv(\013)936 5680 y Fo(2)973 5706 y Fs(\()p Fu(y)22 b Fl(\000)c Fu(x)p Fs(\))1230 5672 y Fv(\013)1273 5680 y Fo(3)1310 5706 y Fu(p)1352 5672 y Fv(\013)1395 5680 y Fo(4)1432 5706 y Fu(k)1478 5672 y Fv(\013)1521 5680 y Fo(5)1557 5706 y Fs(\()p Fu(k)k Fl(\000)c Fu(l)r Fs(\))1796 5672 y Fv(\013)1839 5680 y Fo(6)1875 5706 y Fu(e)1914 5672 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))2377 5706 y Fu(')p Fs(\()p Fu(q)s(;)c(x;)g(y)s Fs(\))2714 5639 y Fq(\002)2749 5706 y Fu(<)23 b(\021)j(>)2969 5672 y Fm(\000)p Fv(m)3107 5706 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))3265 5639 y Fq(\003)3314 5706 y Fu(g)s Fs(\()p Fu(\021)s Fs(\))p Fu(:)1848 5989 y Fz(18)p eop %%Page: 19 19 bop -118 154 a Fz(Here)35 b Fu(')h Fz(is)g Fu(C)333 124 y Fm(1)327 177 y Fd(p)r(ol)457 154 y Fz(in)g(all)f(the)h(v)-5 b(ariables;)39 b(the)d(zero)e(order)h(deriv)-5 b(ativ)n(e)34 b(is)i(no)f(longer)f(b)r (ounded)i(and)g(this)g(will)f(cause)g(some)-118 254 y(complications.)6 353 y Fx(Step)24 b(3.)35 b Fz(The)21 b(heart)f(of)h(the)g(pro)r(of)f(lies)h (in)g(exploiting)f(the)i(nice)e(prop)r(erties)g(of)h(the)g(factor)f Fu(e)2878 323 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))3362 353 y Fz(b)n(y)g(in)n(tegrations)-118 453 y(b)n(y)27 b(parts)f(\(an)h (oscillatory)e(in)n(tegral)g(tec)n(hnique\).)37 b(This)27 b(w)n(orks)e (e\036cien)n(tly)i(only)g(with)g(resp)r(ect)g(to)g(certain)f(of)h(the)g(v)-5 b(ariables.)-118 552 y(W)e(e)36 b(pro)r(duce)f(these)h(v)-5 b(ariables)34 b(b)n(y)i(making)f(linear)f(com)n(binations)h(of)h(other,)h (non-con)n(v)n(enien)n(t)d(ones.)60 b(Let)36 b(us)g(write,)h(for)-118 652 y(instance,)27 b Fu(p)c Fs(=)g(\()p Fu(p)c Fl(\000)f Fu(l)r Fs(\))g(+)g Fu(l)r Fz(,)28 b Fu(k)21 b Fl(\000)d Fu(l)25 b Fs(=)e(\()p Fu(k)f Fl(\000)c Fu(p)p Fs(\))g(+)g(\()p Fu(p)h Fl(\000)f Fu(l)r Fs(\))28 b Fz(and)f Fu(k)f Fs(=)d(\()p Fu(k)f Fl(\000)c Fu(l)r Fs(\))g(+)g Fu(l)25 b Fs(=)e(\()p Fu(k)f Fl(\000)c Fu(p)p Fs(\))g(+)h(\()p Fu(p)f Fl(\000)g Fu(l)r Fs(\))h(+)f Fu(l)r Fz(.)36 b(Then,)28 b(plainly)-7 b(,)28 b(w)n(e)-118 752 y(are)e(reduced)i(to)f(estimating)g(terms)h(of)f(the)h(form)105 935 y Fq(Z)151 1123 y Ft(\004)212 935 y Fq(Z)259 1123 y Ft(\004)320 1048 y Fu(d\021)f(d\020)j(q)580 1014 y Fv(\014)618 1022 y Fo(1)654 1048 y Fu(x)701 1014 y Fv(\014)739 1022 y Fo(2)775 1048 y Fs(\()p Fu(y)22 b Fl(\000)c Fu(x)p Fs(\))1032 1014 y Fv(\014)1070 1022 y Fo(3)1107 1048 y Fu(l)1134 1014 y Fv(\014)1172 1022 y Fo(4)1207 1048 y Fs(\()p Fu(p)h Fl(\000)f Fu(k)s Fs(\))1461 1014 y Fv(\014)1499 1022 y Fo(5)1535 1048 y Fs(\()p Fu(p)h Fl(\000)f Fu(l)r Fs(\))1770 1014 y Fv(\014)1808 1022 y Fo(6)1844 1048 y Fu(e)1883 1014 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))2373 1048 y Fu(v)s(ar)r(phi)p Fs(\()p Fu(q)s(;)c(x;)g(y)s Fs(\))2901 981 y Fq(\002)2936 1048 y Fu(<)23 b(\021)j(>)3156 1014 y Fm(\000)p Fv(m)3294 1048 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))3452 981 y Fq(\003)3501 1048 y Fu(g)s Fs(\()p Fu(\021)s Fs(\))p Fu(:)6 1264 y Fx(Step)30 b(4.)38 b Fz(W)-7 b(e)28 b(use)494 1459 y Fs(\()p Fu(p)19 b Fl(\000)f Fu(k)s Fs(\))748 1425 y Fv(\014)786 1433 y Fo(5)822 1459 y Fs(\()p Fu(p)h Fl(\000)f Fu(l)r Fs(\))1057 1425 y Fv(\014)1095 1433 y Fo(6)1131 1459 y Fu(e)1170 1425 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))1656 1459 y Fs(=)k(\(2)p Fu(i)p Fs(\))1878 1425 y Fm(\000j)p Fv(\014)1988 1433 y Fo(5)2020 1425 y Fm(j)2043 1459 y Fs(\()p Fl(\000)p Fs(2)p Fu(i)p Fs(\))2243 1425 y Fm(\000j)p Fv(\014)2353 1433 y Fo(6)2384 1425 y Fm(j)2408 1459 y Fu(@)2457 1425 y Fv(\014)2495 1433 y Fo(5)2452 1480 y Fv(y)2531 1459 y Fu(@)2580 1425 y Fv(\014)2618 1433 y Fo(6)2575 1480 y Fv(x)2668 1367 y Fq(h)2707 1459 y Fu(e)2746 1425 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))3209 1367 y Fq(i)3262 1459 y Fu(:)-118 1654 y Fz(After)29 b(an)f(in)n(tegration)f(b)n(y)i(parts)f (and)g(an)g(application)g(of)h(Leibnitz's)g(rule)f(w)n(e)g(see)h(that)f(w)n (e)h(are)e(reduced)i(to)f(b)r(ound)h(terms)-118 1753 y(as)539 1762 y Fq(Z)585 1950 y Ft(\004)647 1762 y Fq(Z)693 1950 y Ft(\004)755 1875 y Fu(d\021)e(d\020)i( )s Fs(\()p Fu(q)s(;)14 b(x;)g(y)s Fs(\))p Fu(e)1339 1840 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))1803 1875 y Fu(@)1852 1840 y Fv(\015)1887 1848 y Fo(1)1847 1895 y Fv(x)1937 1807 y Fq(\002)1971 1875 y Fu(<)23 b(\021)j(>)2191 1840 y Fm(\000)p Fv(m)2329 1875 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))2487 1807 y Fq(\003)2536 1875 y Fu(l)2563 1840 y Fv(\015)2598 1848 y Fo(2)2647 1807 y Fq(\000)2713 1875 y Fu(par)r(tial)2968 1840 y Fv(\015)3003 1848 y Fo(3)2966 1895 y Fv(y)3039 1875 y Fu(g)3082 1807 y Fq(\001)3133 1875 y Fs(\()p Fu(\020)d Fs(\))-118 2061 y Fz(for)27 b(some)g(un)n(b)r(ounded)h Fu(C)711 2031 y Fm(1)705 2085 y Fd(p)r(ol)800 2061 y Fz(-function)f Fu( )s Fz(.)37 b(A)n(ctually)-7 b(,)28 b(only)f(the)h(p)r(olynomial)f(estimate)g(on) g Fu( )k Fz(itself)d(will)g(coun)n(t)f(no)n(w.)6 2161 y Fx(Step)37 b(5.)60 b Fz(P)n(olynomial)34 b(b)r(ounds)h(are)f(v)n(ery)g(demo)r(cratic)g (with)h(resp)r(ect)g(to)g(the)g(c)n(hoice)g(of)f(v)-5 b(ariables;)38 b(use)d(for)f(example)-118 2261 y(inequalities)27 b(of)h(the)g(form)f Fu(<)22 b(x)d Fs(+)f Fu(y)26 b(>)p Fl(\024)c Fs(2)1249 2231 y Ft(1)p Fv(=)p Ft(2)1376 2261 y Fu(<)h(x)g(><)g(y)i(>)p Fz(.)37 b(Th)n(us)27 b(w)n(e)g(can)g(write)516 2481 y Fu( )s Fs(\()p Fu(q)s(;)14 b(x;)g(y)s Fs(\))23 b(=)953 2364 y Fq(\024)1368 2424 y Fu( )s Fs(\()p Fu(q)s(;)14 b(x;)g(y)s Fs(\))p 1007 2462 1049 4 v 1007 2538 a Fu(<)22 b(y)k(>)1226 2514 y Fv(j)1261 2538 y Fu(<)c(q)g Fl(\000)c Fu(x)23 b(>)1625 2514 y Fv(j)1660 2538 y Fu(<)g(q)e Fl(\000)d Fu(y)26 b(>)2021 2514 y Fv(j)2065 2364 y Fq(\025)2146 2413 y(\002)2181 2481 y Fu(<)c(y)k(>)2400 2446 y Fv(j)2435 2481 y Fu(<)c(q)g Fl(\000)c Fu(x)23 b(>)2799 2446 y Fv(j)2834 2481 y Fu(<)f(q)g Fl(\000)c Fu(y)26 b(>)3195 2446 y Fv(j)3229 2413 y Fq(\003)-118 2700 y Fz(and)h(the)h(\034rst)g(factor)e (will)i(b)r(e)g(b)r(ounded)g(for)f Fu(j)33 b Fz(large)26 b(enough.)36 b(By)27 b(dev)n(elopping,)g(w)n(e)g(need)h(to)f(estimate)285 2800 y Fq(Z)331 2988 y Ft(\004)393 2800 y Fq(Z)439 2988 y Ft(\004)501 2913 y Fu(d\021)f(d\020)k(\032)p Fs(\()p Fu(q)s(;)14 b(x;)g(y)s Fs(\)\()p Fu(q)22 b Fl(\000)c Fu(x)p Fs(\))1285 2878 y Fv(\016)1315 2886 y Fo(1)1353 2913 y Fs(\()p Fu(q)j Fl(\000)e Fu(y)s Fs(\))1603 2878 y Fv(\016)1633 2886 y Fo(2)1669 2913 y Fu(e)1708 2878 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))2171 2913 y Fu(@)2220 2878 y Fv(\016)2250 2886 y Fo(3)2215 2933 y Fv(x)2300 2846 y Fq(\002)2335 2913 y Fu(<)k(\021)j(>)2555 2878 y Fm(\000)p Fv(m)2692 2913 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))2850 2846 y Fq(\003)2899 2913 y Fu(y)2943 2878 y Fv(\016)2973 2886 y Fo(4)3010 2913 y Fu(l)3037 2878 y Fv(\016)3067 2886 y Fo(5)3117 2846 y Fq(\000)3155 2913 y Fu(@)3204 2878 y Fv(\016)3234 2886 y Fo(6)3199 2933 y Fv(y)3270 2913 y Fu(g)3313 2846 y Fq(\001)3365 2913 y Fs(\()p Fu(\020)d Fs(\))p Fu(;)-118 3129 y Fz(where)27 b(no)n(w)g Fu(\032)g Fz(is)h(b)r (ounded.)6 3229 y Fx(Step)i(6.)38 b Fz(But)28 b(one)f(has)562 3403 y Fs(\()p Fu(q)22 b Fl(\000)c Fu(x)p Fs(\))815 3369 y Fv(\016)845 3377 y Fo(1)882 3403 y Fs(\()p Fu(q)k Fl(\000)c Fu(y)s Fs(\))1132 3369 y Fv(\016)1162 3377 y Fo(2)1199 3403 y Fu(e)1238 3369 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))1724 3403 y Fs(=)23 b(\()p Fl(\000)p Fs(2)p Fu(i)p Fs(\))2012 3369 y Fm(\000j)p Fv(\016)2114 3377 y Fo(1)2145 3369 y Fm(j)2169 3403 y Fs(\(2)p Fu(i)p Fs(\))2304 3369 y Fm(\000j)p Fv(\016)2406 3377 y Fo(2)2438 3369 y Fm(j)2462 3403 y Fu(@)2511 3367 y Fv(\016)2541 3375 y Fo(1)2506 3428 y Fv(l)2577 3403 y Fu(@)2626 3367 y Fv(\016)2656 3375 y Fo(2)2621 3428 y Fv(k)2692 3403 y Fu(e)2731 3369 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))3194 3403 y Fu(:)-118 3578 y Fz(W)-7 b(e)28 b(p)r(erform)f(our)g(last)g(in)n(tegration)f(b)n(y)h(parts,)g (reducing)g(ourselv)n(es)f(to)h(estimate)442 3678 y Fq(Z)488 3866 y Ft(\004)550 3678 y Fq(Z)596 3866 y Ft(\004)658 3791 y Fu(d\021)f(d\020)k(\032)p Fs(\()p Fu(q)s(;)14 b(x;)g(y)s Fs(\))p Fu(e)1228 3757 y Fm(\000)p Ft(2)p Fv(i\033)r Ft(\()p Fv(\030)r Fm(\000)p Fv(\021)r(;\030)r Fm(\000)p Fv(\020)t Ft(\))1691 3791 y Fu(@)1740 3754 y Fv(\016)1770 3762 y Fo(2)1735 3816 y Fv(k)1807 3791 y Fu(@)1856 3757 y Fv(\016)1886 3765 y Fo(3)1851 3811 y Fv(x)1936 3724 y Fq(\002)1970 3791 y Fu(<)23 b(\021)j(>)2190 3757 y Fm(\000)p Fv(m)2328 3791 y Fu(f)9 b Fs(\()p Fu(\021)s Fs(\))2486 3724 y Fq(\003)2535 3791 y Fu(y)2579 3757 y Fv(\016)2609 3765 y Fo(4)2659 3699 y Fq(h)2698 3791 y Fu(@)2747 3754 y Fv(\016)2777 3762 y Fo(1)2742 3816 y Fv(l)2814 3791 y Fu(@)2863 3757 y Fv(\016)2893 3765 y Fo(6)2858 3811 y Fv(y)2943 3724 y Fq(\000)2981 3791 y Fu(l)3008 3757 y Fv(\016)3038 3765 y Fo(5)3074 3791 y Fu(g)3117 3724 y Fq(\001)3155 3699 y(i)3208 3791 y Fs(\()p Fu(\020)d Fs(\))p Fu(:)6 4045 y Fx(Step)32 b(7.)46 b Fz(Ob)n(viously)-7 b(,)29 b(this)i(in)n(tegral)e(is)h(dominated)g(for)f(an)n(y)h Fu(\030)k Fz(b)n(y)c Fl(k)d Fu(@)2279 4009 y Fv(\016)2309 4017 y Fo(2)2274 4070 y Fv(k)2345 4045 y Fu(@)2394 4015 y Fv(\016)2424 4023 y Fo(3)2389 4066 y Fv(x)2475 4045 y Fs([)p Fu(<)22 b Fl(\001)i Fu(>)2697 4015 y Fm(\000)p Fv(m)2834 4045 y Fu(f)9 b Fs(])27 b Fl(k)2976 4057 y Ft(1)3040 4045 y Fl(k)g Fu(y)3153 4015 y Fv(\016)3183 4023 y Fo(4)3234 3953 y Fq(h)3273 4045 y Fu(@)3322 4009 y Fv(\016)3352 4017 y Fo(1)3317 4070 y Fv(l)3388 4045 y Fu(@)3437 4015 y Fv(\016)3467 4023 y Fo(6)3432 4066 y Fv(y)3517 3978 y Fq(\000)3555 4045 y Fu(l)3582 4015 y Fv(\016)3612 4023 y Fo(5)3649 4045 y Fu(g)3692 3978 y Fq(\001)3729 3953 y(i)3796 4045 y Fl(k)3838 4057 y Ft(1)3875 4045 y Fz(.)-118 4170 y(The)35 b(\034rst)f(factor)g(is)h(part)f(of)h(the)g(norm)g Fu(r)1277 4140 y Fv(m)1275 4190 y(n)1340 4170 y Fs(\()p Fu(f)9 b Fs(\))35 b Fz(for)g(some)f(large)f Fu(n)i Fz(and)g(the)g(second)f(is)h(one)f(of)h(the) g(seminorms)f(of)h Fu(g)i Fz(in)-118 4270 y Fl(S)6 b Fz(.)43 b(The)30 b(Prop)r(osition)f(is)h(pro)n(v)n(ed)e(and,)i(as)e(a)i(b)r(on)n(us,) g(w)n(e)f(found)h(out)f(that)h Fs(\()p Fu(f)t(;)14 b(g)s Fs(\))27 b Fl(7!)f Fu(f)j Fl(\016)2739 4239 y Fv(B)2815 4270 y Fu(g)j Fz(extends)e(to)f(a)h(bilinear)f Fx(jointly)-118 4369 y(c)l(ontinuous)e Fz(mapping)g Fs(:)c Fu(R)749 4339 y Fv(m)748 4390 y Ft(0)830 4369 y Fl(\002)18 b(S)30 b(!)23 b(S)6 b Fz(.)2655 b Fe(\004)6 4528 y Fz(The)26 b(class)f Fu(C)433 4498 y Fm(1)427 4551 y Fd(p)r(ol,u)577 4528 y Fs(\(\004\))h Fz(is)g(indeed)g(con)n(v)n(enien)n(t.)35 b(It)26 b(has)f(a)g(v)n(ery)f(explicit)i(de\034nition)g(and)f(it)h(con)n (tains)f(all)g(the)h(p)r(olynomials)-118 4644 y(in)31 b Fu(x)h Fz(and)f Fu(p)p Fz(.)48 b(It)32 b(also)e(con)n(tains)g(the)i(classical)d(sym) n(b)r(ol)i(spaces)f Fu(S)2003 4614 y Fv(m)2066 4644 y Fs(\(\004\))g(:=)f Fl(f)p Fu(f)37 b Fl(2)30 b Fu(C)2602 4614 y Fm(1)2673 4644 y Fs(\(\004\))g Fl(j)f(j)p Fs(\()p Fu(@)2978 4614 y Fv(a)3018 4644 y Fu(f)9 b Fs(\)\()p Fu(\030)t Fs(\))p Fl(j)30 b(\024)f Fu(C)3410 4656 y Fv(a)3480 4644 y Fu(<)f(\030)34 b(>)3708 4614 y Fv(m)p Fm(\000j)p Fv(a)p Fm(j)-118 4744 y Fu(;)j Fl(8)p Fu(a)22 b Fl(2)h Fw(N)187 4714 y Ft(2)p Fv(N)289 4744 y Fl(g)28 b Fz(for)f(all)g Fu(m)p Fz(.)6 4844 y(The)e(magnetic)f(Mo)n(y)n(al)g(algebra)e(is)j(large)e (indeed,)j(but)f(man)n(y)f(distributions,)h(ev)n(en)g(with)g(a)f(go)r(o)r(d)g (b)r(eha)n(viour)g(at)g(in\034nit)n(y)-7 b(,)-118 4943 y(are)30 b(not)h(inside.)46 b(The)31 b(one-rank)e(pro)5 b(jection)30 b Fl(j)p Fu(u)e(><)g(u)p Fl(j)j Fz(is)f(in)h Fl(L)p Fs(\()p Fl(S)6 b Fs(\))33 b Fz(if)e(and)g(only)f(if)i Fu(u)c Fl(2)h(S)6 b Fz(.)47 b(Th)n(us,)31 b(b)n(y)g(Prop)r(osition)f(4.21,)-118 5043 y(there)d(are)g(plen)n(t)n(y)g(of)h(elemen)n(ts)f(in)h Fu(L)1072 5013 y Ft(2)1109 5043 y Fs(\(\004\))g Fz(not)g(b)r(elonging)f(to)g Fl(M)p Fs(\(\004\))p Fz(.)6 5142 y FA(A)m(c)m(kno)m(wledgemen)m(ts:)63 b Fz(Our)40 b(understanding)g(of)h(the)h(pseudo)r(di\033eren)n(tial)e(theory) h(without)g(magnetic)g(\034elds)g(w)n(as)-118 5242 y(in\035uenced)23 b(b)n(y)f(unpublished)g(lecture)h(notes)e(of)i(Vladimir)f(Georgescu)e(and)i (a)g(b)r(o)r(ok)g(in)h(preparation)d(of)j(Gruia)e(Arsu.)35 b(W)-7 b(e)23 b(are)-118 5342 y(grateful)h(to)h(George)e(Nenciu)i(and)g (Mihai)f(P)n(ascu)h(for)f(useful)h(discussions.)35 b(P)n(art)24 b(of)h(this)g(w)n(ork)f(w)n(as)f(done)i(while)g(the)g(authors)-118 5441 y(w)n(ere)31 b(visiting)h(the)h(Departmen)n(t)f(of)g(Theoretical)f(Ph)n (ysics)h(of)h(the)f(Univ)n(ersit)n(y)g(of)g(Genev)-5 b(a.)51 b(W)-7 b(e)32 b(are)g(greatful)f(to)h(W)-7 b(erner)-118 5541 y(Amrein)38 b(for)f(his)h(kind)g(hospitalit)n(y)-7 b(.)67 b(W)-7 b(e)38 b(also)e(ac)n(kno)n(wledge)g(partial)h(supp)r(ort)g(from)h(the)g(EURR) n(OMMA)-7 b(T)38 b(P)n(ogramme)-118 5641 y(\(con)n(tract)31 b(no.)50 b(ICA1-CT-2000-70022\),)28 b(the)33 b(CERES)g(Programme)d(\(con)n (tract)i(no.)50 b(38/2002\))29 b(and)j(the)g(CNCSIS)h(gran)n(t)-118 5740 y(no.)j(33536/2003.)1848 5989 y(19)p eop %%Page: 20 20 bop -118 154 a Fy(References)-118 336 y Fz([AHS])42 b(J.)24 b(A)-9 b(vron,)24 b(I.)g(Herbst)g(and)g(B.)g(Simon:)35 b Fx(Schr)l(\366)l (dinger)28 b(Op)l(er)l(ators)e(with)h(Magnetic)h(Fields.)g(I.)f(Gener)l(al)g (The)l(ory)p Fz(,)g(Duk)n(e)94 435 y(Math.)h(J.)f FA(45)p Fz(,)g(847-883,)e (\(1978\).)-118 601 y([An1])41 b(R.)28 b(F.)g(Anderson:)36 b Fx(The)31 b(W)-6 b(eyl)30 b(F)-6 b(unctional)30 b(Calculus)p Fz(,)e(J.)f(F)-7 b(unct.)28 b(Anal.)g FA(4)p Fz(,)g(240-267,)c(\(1969\).)-118 767 y([An2])41 b(R.)28 b(F.)g(Anderson:)36 b Fx(The)31 b(Multiplic)l(ative)h (W)-6 b(eyl)30 b(F)-6 b(unctional)29 b(Calculus)p Fz(,)f(J.)g(F)-7 b(unct.)28 b(Anal.)g FA(9)p Fz(,)f(423-440,)e(\(1972\).)-118 932 y([An)n(t])42 b(M.)28 b(A.)g(An)n(tonets:)36 b Fx(The)31 b(Classic)l(al)h(Limit)e(for)g(the)g(W)-6 b(eyl)30 b(Quantization)p Fz(,)e(Lett.)g(Math.)f(Ph)n(ys.)h FA(2)p Fz(,)f(241-245,)e(\(1978\).)-118 1098 y([Be1])41 b(J.)23 b(Bellissard:)33 b Fu(C)669 1068 y Fm(\003)707 1098 y Fx(-A)n(lgebr)l(as)26 b(in)g(Solid)h(State)e(Physics:)38 b Fs(2)p Fu(D)r Fx(-Ele)l(ctr)l(ons)25 b(in)h(a)g(Uniform)g(Magnetic)h(Field) 48 b Fz(in)23 b Fx(Op)l(er)l(ator)94 1198 y(A)n(lgebr)l(as)30 b(and)g(Applic)l(ations)p Fz(,)g(v)n(ol.)c(2,)i(D.)g(E.)g(Ev)-5 b(ans)28 b(and)f(M.)h(T)-7 b(ak)n(esaki)26 b(Eds.,)i(Cam)n(bridge)e(Univ)n (ersit)n(y)g(Press,)i(1988.)-118 1363 y([Be2])41 b(J.)35 b(Bellissard:)52 b Fx(Gap)39 b(L)l(abb)l(eling)f(The)l(or)l(ems)h(for)f(Schr)l(\366)l(dinger)g (Op)l(er)l(ators)60 b Fz(in)36 b Fx(Numb)l(er)h(The)l(ory)i(and)f(Physics)p Fz(,)h(J.-)94 1463 y(M.)c(Luc)n(k,)g(P)-7 b(.)35 b(Moussa)f(and)g(M.)h(W)-7 b(aldsc)n(hmidt)34 b(Eds.,)j(Springer)c(Preceedings)h(in)h(Ph)n(ysics,)g FA(47)p Fz(,)h(Springer,)f(Berlin,)94 1562 y(Heidelb)r(erg,)27 b(New)h(Y)-7 b(ork,)27 b(1993.)-118 1728 y([BMGH])42 b(L.)29 b(Boutet)g(de)h(Mon)n(v)n(el,)e(A.)i(Grigis)e(and)h(B.)h(Hel\033er:)40 b Fx(Par)l(ametrixes)32 b(d'op)l(\351r)l(ateurs)g(pseudo-di\033\351r)l (entiels)h(\340)f(c)l(ar-)94 1828 y(act\351ristiques)i(multiples)p Fz(,)g(Journ\351es:)43 b(\311quations)32 b(aux)f(D\351riv)n(\351es)g(P)n (artielles)g(de)h(Rennes,)h(93\025121.)c(Asterisque,)j(No.)94 1927 y(34-35,)25 b(So)r(c.)j(Math.)g(F)-7 b(rance,)27 b(P)n(aris,)f(1976.) -118 2093 y([EGBV])42 b(R.)31 b(Estrada,)f(J.)g(M.)h(Gracia-Bond\355a)c(and)j (J.)h(C.)f(V\341rilly:)42 b Fx(On)31 b(Asymptotic)i(Exp)l(ansions)g(of)g (Twiste)l(d)g(Pr)l(o)l(ducts)p Fz(,)94 2192 y(J.)27 b(Math.)h(Ph)n(ys.)g FA(30)p Fz(\(12\),)e(2789-2796,)e(\(1989\).)-118 2358 y([F)-7 b(o])77 b(G.)33 b(B.)g(F)-7 b(olland:)48 b Fx(Harmonic)36 b(A)n(nalysis)f(in) g(Phase)h(Sp)l(ac)l(e)p Fz(,)f(Princeton)e(Univ)n(ersit)n(y)f(Press,)i (Princeton,)h(New)e(Jersey)-7 b(,)94 2458 y(1989.)-118 2623 y([GI])71 b(V.)28 b(Georgescu)f(and)h(A.)g(Iftimo)n(vici:)38 b Fx(C*-A)n(lgebr)l(as)31 b(of)g(Ener)l(gy)f(Observables:)41 b(II.)30 b(Gr)l(ade)l(d)i(Symple)l(ctic)f(A)n(lgebr)l(as)f(and)94 2723 y(Magnetic)h(Hamiltonians)p Fz(,)d(Preprin)n(t)g(01-99)d(at)i(h)n (ttp://www.ma.utexas.edu/mp-arc/.)-118 2889 y([GBV1])41 b(J.)35 b(M.)h(Gracia-Bond\355a)c(and)j(J.)g(C.)g(V\341rilly:)52 b Fx(A)n(lgebr)l(a)37 b(of)g(Distributions)g(Suitable)g(for)g(Phase-Sp)l(ac)l (e)h(Quantum)94 2988 y(Me)l(chanics.)32 b(I)p Fz(,)c(J.)f(Math.)h(Ph)n(ys.)f FA(29)p Fz(\(4\),)g(869-879,)e(\(1988\).)-118 3154 y([GBV2])41 b(J.)35 b(M.)h(Gracia-Bond\355a)c(and)j(J.)g(C.)g(V\341rilly:)52 b Fx(A)n(lgebr)l(a)37 b(of)g(Distributions)g(Suitable)g(for)g(Phase-Sp)l(ac)l (e)h(Quantum)94 3253 y(Me)l(chanics.)32 b(II.)e(T)-6 b(op)l(olo)l(gies)32 b(on)e(the)f(Moyal)j(A)n(lgebr)l(a)p Fz(,)c(J.)g(Math.)f(Ph)n(ys.)h FA(29)p Fz(\(4\),)f(880-887,)d(\(1988\).)-118 3419 y([GLS])42 b(A.)32 b(Grossman,)f(G.)h(Lupias,)h(E.)f(M.)g(Stein:)46 b Fx(A)n(n)33 b(A)n(lgebr)l(a)h(of)g(Pseudo-di\033er)l(ential)i(Op)l(er)l (ators)d(and)h(Quantum)e(Me-)94 3519 y(chanics)f(in)f(Phase)h(Sp)l(ac)l(e)p Fz(,)d(Ann.)g(Inst.)g(F)-7 b(ourier,)26 b FA(18)p Fz(\(2\),)i(343-368,)c (\(1969\).)-118 3684 y([GMS])42 b(C.)34 b(G\351rard,)h(A.)f(Martinez)g(and)g (J.)g(Sj\366strand:)49 b Fx(A)36 b(Mathematic)l(al)h(Appr)l(o)l(ach)h(to)d (the)h(E\033e)l(ctive)h(Hamiltonian)g(in)94 3784 y(Perturb)l(e)l(d)30 b(Perio)l(dic)h(Pr)l(oblems)p Fz(,)e(Comm)n(un.)e(Math.)h(Ph)n(ys.)g FA(142)p Fz(,)f(217-244,)d(\(1991\).)-118 3950 y([HS1])41 b(B.)28 b(Hel\033er)g(and)g(J.)f(Sj\366strand:)37 b(Equation)27 b(de)h(Sc)n(hr\366)r (dinger)e(a)n(v)n(ec)h(c)n(hamp)g(magnetique)g(et)h(\351quation)g(de)f(Harp)r (er,)g(in)94 4049 y(Springer)f(Lecture)i(Notes)f(in)h(Ph)n(ysics,)f FA(345)p Fz(,)g(118-197,)d(\(1989\).)-118 4215 y([HS2])41 b(B.)25 b(Hel\033er)g(and)g(J.)g(Sj\366strand:)35 b Fx(On)26 b(Diamagnetism)i(and)g (de)g(Haas-van)f(A)n(lphen)h(E\033e)l(ct)p Fz(,)e(Ann.)f(I.H.P)-7 b(.,)27 b FA(52)p Fz(,)e(303-375,)94 4314 y(\(1990\).)-118 4480 y([Hor1])40 b(L.)28 b(H\366rmander:)35 b Fx(The)c(W)-6 b(eyl)29 b(Calculus)h(of)h(Pseudo-di\033er)l(ential)g(Op)l(er)l(ators)p Fz(,)d(Comm.)f(Pure)h(Appl.)g(Math.)f FA(32)p Fz(,)g(359-)94 4580 y(443,)f(\(1979\).)-118 4745 y([Hor2])40 b(L.)e(H\366rmander:)55 b Fx(The)40 b(A)n(nalysis)f(of)h(Partial)g(Di\033er)l(ential)g(Op)l(er)l (ators,)h(vol.)g(III)p Fz(,)60 b(Springer-V)-7 b(erlag,)38 b(New)f(Y)-7 b(ork,)94 4845 y(1985.)-118 5011 y([Le])77 b(H.)34 b(Leinfelder:)48 b Fx(Gauge)35 b(Invarianc)l(e)h(of)g(Schr)l(\366)l(dinger)g (Op)l(er)l(ators)f(and)g(R)l(elate)l(d)g(Sp)l(e)l(ctr)l(al)g(Pr)l(op)l (erties)p Fz(,)g(J.)e(Op)r(erator)94 5110 y(Th.)28 b FA(9)p Fz(,)f(163-179,)e(\(1983\).)-118 5276 y([Lu])68 b(J.)30 b(M.)g(Luttinger:)42 b Fx(The)33 b(E\033e)l(ct)f(of)h(the)f(Magnetic)h(Field)h(on)e(Ele)l(ctr)l (ons)g(in)g(a)g(Perio)l(dic)j(Potential)p Fz(,)c(Ph)n(ys.)f(Rev.,)h FA(84)p Fz(,)94 5375 y(814-817,)24 b(\(1951\).)-118 5541 y([MP])42 b(M.)34 b(M\240n)n(toiu)f(and)h(R.)g(Purice:)50 b Fx(The)36 b(A)n(lgebr)l(a)g(of)h(Observables)f(in)g(a)g(Magnetic)g(Field)p Fz(,)h(Mathematical)d(Results)f(in)94 5641 y(Quan)n(tum)25 b(Mec)n(hanics)g(\(T)-7 b(axco,)25 b(2001\),)f(Con)n(temp)r(orary)f (Mathematics)i FA(307)p Fz(,)h(Amer.)f(Math.)h(So)r(c.,)g(Pro)n(vidence,)f (RI,)94 5740 y(239-245,)f(\(2002\).)1848 5989 y(20)p eop %%Page: 21 21 bop -118 154 a Fz([MR])42 b(A.)23 b(Mohamed)f(and)g(G.)h(D.)g(Ra)-9 b(\010)-32 b(\031k)n(o)n(v:)32 b Fx(On)24 b(the)i(Sp)l(e)l(ctr)l(al)f(The)l (ory)h(of)g(the)f(Schr)l(\366)l(dinger)i(Op)l(er)l(ator)e(with)h(Ele)l(ctr)l (omagnetic)94 254 y(Potential)p Fz(,)43 b(Pseudo-di\033eren)n(tial)37 b(Calculus)i(and)f(Mathematical)h(Ph)n(ysics,)i(298\025390,)d(Math.)h(T)-7 b(op.,)41 b(5,)h(Ak)-5 b(ademie)94 353 y(V)e(erlag,)27 b(Berlin,)g(1994.)-118 519 y([Mo])48 b(J.)27 b(E.)i(Mo)n(y)n(al:)35 b Fx(Quantum)28 b(Me)l(chanics)j(as)f(a)g(Statistic)l(al)g(The)l(ory)p Fz(,)f(Pro)r(c.)f(Cam) n(b.)f(Ph)n(yl.)h(So)r(c.)f FA(45)p Fz(,)h(99-124,)d(\(1949\).)-118 685 y([Ne])67 b(G.)33 b(Nenciu:)48 b Fx(Dynamics)36 b(of)f(Band)h(Ele)l(ctr)l (ons)f(in)f(Ele)l(ctric)i(and)f(Magnetic)h(Fields:)50 b(R)n(igor)l(ous)35 b(Justi\034c)l(ation)f(of)h(the)94 785 y(E\033e)l(ctive)30 b(Hamiltonians)p Fz(,)f(Rev.)e(Mo)r(d.)h(Ph)n(ys,)g FA(63)p Fz(,)f(91-128,)e(\(1991\).)-118 951 y([RS])59 b(M.)38 b(Reed)f(and)g(B.)h (Simon:)56 b 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b(J.)27 b(Zak:)36 b Fx(Magnetic)31 b(T)-6 b(r)l(anslation)30 b(Gr)l(oups)p Fz(,)e(Ph)n(ys.)g(Rev.)g(A,)f FA(134)p Fz(,)g(1602-1607)d(and)j(1607-1611,)d (\(1965\).)1848 5989 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0311130735488--