This is a multi-part message in MIME format. ---------------0004141134716 Content-Type: text/plain; name="00-183.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-183.keywords" Quantum Statistical Mechanics, Renormalization Group, Fermi systems ---------------0004141134716 Content-Type: application/postscript; name="heisuno.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="heisuno.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.83 (MiKTeX 1.20b) Copyright 1998 Radical Eye Software %%Title: C:\GB\FERMI\Heis\heisuno.dvi %%CreationDate: Fri Apr 14 18:23:39 2000 %%Pages: 60 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: c:\texmf\miktex\bin\dvips.exe %+ C:\GB\FERMI\Heis\heisuno %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.04.14:1823 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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b Ft(can)0 4169 y(b)r(e)f (expressed)f(in)h(terms)g(of)g(the)g Fe(Sc)n(h)n(winger)f(functions)p Ft(,)i(that)f(is)g(the)g(truncated)g(exp)r(ectations)g(in)g(the)0 4319 y(Grand)d(Canonical)g(Ensem)n(ble)g(of)h(the)g(time)g(order)f(pro) r(duct)g(of)h(the)g(\014eld)g Fq(a)2416 4289 y Fu(\006)2416 4339 y Fk(x)2498 4319 y Ft(at)g(di\013eren)n(t)g(space-time)0 4468 y(p)r(oin)n(ts.)42 b(There)29 b(is)g(of)h(course)e(a)h(relation)f (b)r(et)n(w)n(een)h(these)h(functions)g(and)f(the)g(exp)r(ectations)g (of)h(some)0 4618 y(suitable)35 b(observ)-5 b(ables)33 b(in)i(the)g(spin)g(system.)58 b(Ho)n(w)n(ev)n(er,)34 b(b)n(y)h(lo)r(oking)f(at)g(\(1.4\),)i(one)f(sees)f(that)h(this)0 4767 y(relation)c(is)h(simple)g(enough)f(only)g(in)i(the)f(case)f(of)h (the)g(truncated)g(exp)r(ectations)f(of)h(the)g(time)h(order)0 4916 y(pro)r(duct)c(of)h(the)g(fermionic)f Fe(densit)n(y)g(op)r(erator) f Fq(\032)1587 4928 y Fk(x)1657 4916 y Ft(=)e Fq(a)1792 4886 y Fp(+)1792 4937 y Fk(x)1847 4916 y Fq(a)1891 4886 y Fu(\000)1891 4937 y Fk(x)1976 4916 y Ft(at)k(di\013eren)n(t)f (space-time)g(p)r(oin)n(ts,)h(whic)n(h)0 5066 y(w)n(e)24 b(shall)h(call)f(the)i Fe(densit)n(y)e(Sc)n(h)n(winger)g(functions)p Ft(;)i(they)f(coincide)f(with)i(the)f(truncated)g(exp)r(ectations)0 5215 y(of)g(the)h(time)g(order)e(pro)r(duct)h(of)g(the)h(op)r(erator)e Fq(S)1566 5185 y Fp(3)1561 5236 y Fk(x)1628 5215 y Ft(=)e Fq(e)1754 5185 y Fn(x)1792 5193 y Fj(0)1824 5185 y Fn(H)1887 5215 y Fq(S)1943 5185 y Fp(3)1938 5236 y Fn(x)1980 5215 y Fq(e)2019 5185 y Fu(\000)p Fn(H)t(x)2167 5193 y Fj(0)2229 5215 y Ft(at)j(di\013eren)n(t)g(space-time)g(p)r(oin)n(ts.)71 5370 y(As)32 b(it)g(is)g(w)n(ell)f(kno)n(wn,)h(the)g(Sc)n(h)n(winger)f (functions)h(can)f(b)r(e)h(written)g(as)f(p)r(o)n(w)n(er)g(series)g(in) h Fq(\025)g Ft(and)f Fq(u)p Ft(,)0 5520 y(con)n(v)n(ergen)n(t)25 b(for)h Fm(j)p Fq(\025)p Fm(j)p Fq(;)14 b Fm(j)p Fq(u)p Fm(j)24 b(\024)e Fq(")916 5532 y Fn(\014)961 5520 y Ft(,)27 b(for)g(some)f(constan)n(t)g Fq(")1718 5532 y Fn(\014)1790 5520 y Ft(\(the)i(only)e(trivial)h(b)r(ound)g(of)g Fq(")2781 5532 y Fn(\014)2853 5520 y Ft(go)r(es)f(to)h(zero,)0 5669 y(as)33 b Fq(\014)38 b Fm(!)c(1)p Ft(\).)56 b(This)33 b(p)r(o)n(w)n(er)g(expansion)g(is)h(constructed)f(in)h(the)g(usual)g(w) n(a)n(y)f(in)h(terms)f(of)h(F)-7 b(eynman)0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(12)p eop %%Page: 13 13 13 12 bop 0 83 a Ft(graphs,)26 b(b)n(y)i(using)f(as)g Fe(free)g(propagator)e Ft(the)j(function)108 331 y Fq(g)151 297 y Fn(L;\014)261 331 y Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\))24 b(=)649 267 y(T)-7 b(r)749 200 y Fl(\002)783 267 y Fq(e)822 237 y Fu(\000)p Fn(\014)s(H)969 245 y Fj(0)1005 267 y Fo(T)p Ft(\()p Fq(a)1147 237 y Fu(\000)1147 288 y Fk(x)1204 267 y Fq(a)1248 237 y Fp(+)1248 288 y Fk(y)1303 267 y Ft(\))1335 200 y Fl(\003)p 649 312 721 4 v 833 388 a Ft(T)g(r)o([)p Fq(e)980 364 y Fu(\000)p Fn(\014)s(H)1127 372 y Fj(0)1164 388 y Ft(])1403 331 y(=)552 572 y(=)657 516 y(1)p 649 553 57 4 v 649 629 a 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Fn(L;\014)1598 1629 y Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\))24 b(=)f(lim)2092 1641 y Fn(M)6 b Fu(!1)2312 1629 y Fq(g)2355 1599 y Fn(L;\014)s(;M)2554 1629 y Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\),)29 b(where)731 1877 y Fq(g)774 1843 y Fn(L;\014)s(;M)973 1877 y Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\))24 b(=)1395 1821 y(1)p 1361 1858 108 4 v 1361 1934 a Fq(L\014)1550 1798 y Fl(X)1493 1977 y Fk(k)p Fu(2D)1630 1986 y Fh(L;\014)1955 1821 y Fq(e)1994 1791 y Fu(\000)p Fn(i)p Fk(k)p Fu(\001)p Fp(\()p Fk(x)p Fu(\000)p Fk(y)q Fp(\))p 1751 1858 770 4 v 1751 1934 a Fm(\000)p Fq(ik)1888 1946 y Fp(0)1943 1934 y Ft(+)18 b(cos)13 b Fq(p)2193 1946 y Fn(F)2266 1934 y Fm(\000)18 b Ft(cos)13 b Fq(k)2553 1877 y(;)560 b Ft(\(2)p Fq(:)p Ft(7\))0 2162 y Fo(k)25 b Ft(=)f(\()p Fq(k)s(;)14 b(k)322 2174 y Fp(0)359 2162 y Ft(\),)29 b Fo(k)19 b Fm(\001)g Fo(x)25 b Ft(=)f Fq(k)761 2174 y Fp(0)799 2162 y Fq(x)846 2174 y Fp(0)902 2162 y Ft(+)19 b Fq(k)s(x)p Ft(,)29 b Fm(D)1195 2174 y Fn(L;\014)1329 2162 y 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Fq(da)p Ft(\))q(^)-43 b Fq(a)801 2813 y Fu(\000)801 2874 y Fk(k)841 2882 y Fj(1)879 2849 y Ft(^)g Fq(a)922 2813 y Fp(+)922 2874 y Fk(k)962 2882 y Fj(2)1021 2849 y Ft(=)23 b Fq(L\014)t(\016)1254 2861 y Fk(k)1294 2869 y Fj(1)1325 2861 y Fn(;)p Fk(k)1385 2869 y Fj(2)1424 2849 y Ft(^)-45 b Fq(g)s Ft(\()p Fo(k)1546 2861 y Fp(1)1584 2849 y Ft(\))23 b Fq(;)100 b Ft(^)-45 b Fq(g)s Ft(\()p Fo(k)p Ft(\))24 b(=)2401 2793 y(1)p 2037 2830 V 2037 2906 a Fm(\000)p Fq(ik)2174 2918 y Fp(0)2229 2906 y Ft(+)18 b(cos)13 b Fq(p)2479 2918 y Fn(F)2553 2906 y Fm(\000)18 b Ft(cos)13 b Fq(k)2840 2849 y(;)273 b Ft(\(2)p Fq(:)p Ft(8\))0 3087 y(w)n(e)27 b(ha)n(v)n(e)445 3326 y(lim)402 3380 y Fn(M)6 b Fu(!1)661 3270 y Ft(1)p 627 3307 108 4 v 627 3383 a Fq(L\014)816 3247 y Fl(X)759 3426 y Fk(k)p Fu(2D)896 3435 y Fh(L;\014)1021 3326 y Fq(e)1060 3292 y Fu(\000)p Fn(i)p Fk(k)p Fu(\001)p Fp(\()p Fk(x)p Fu(\000)p Fk(y)q Fp(\))1399 3326 y Ft(^)-45 b Fq(g)s Ft(\()p Fo(k)p Ft(\))24 b(=)66 b(lim)1664 3380 y Fn(M)6 b 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Ft(\))i(=)f(1,)h(if)0 662 y Fm(j)p Fo(k)73 632 y Fu(0)97 662 y Fm(j)23 b Ft(=)g Fq(t)261 674 y Fp(0)298 662 y Fq(\015)346 632 y Fn(h)388 662 y Ft(,)28 b(so)f(that)942 812 y Fq(f)983 824 y Fn(h)1022 832 y Fj(1)1058 812 y Ft(\()p Fo(k)1140 777 y Fu(0)1164 812 y Ft(\))p Fq(f)1237 824 y Fn(h)1276 832 y Fj(2)1312 812 y Ft(\()p Fo(k)1394 777 y Fu(0)1418 812 y Ft(\))c(=)g(0)g Fq(;)97 b Ft(if)55 b Fm(j)p Fq(h)1920 824 y Fp(1)1976 812 y Fm(\000)18 b Fq(h)2107 824 y Fp(2)2144 812 y Fm(j)23 b Fq(>)g Ft(1)f Fq(:)730 b Ft(\(2)p Fq(:)p Ft(38\))71 1015 y(W)-7 b(e)28 b(\014nally)f(de\014ne,)h(for)f(an)n(y)g Fq(h)c Fm(\024)g Ft(0:)933 1232 y(^)915 1254 y Fq(f)956 1266 y Fn(h)999 1254 y Ft(\()p Fo(k)p Ft(\))h(=)f Fq(f)1266 1266 y Fn(h)1308 1254 y Ft(\()p Fq(k)f Fm(\000)c Fq(p)1530 1266 y Fn(F)1585 1254 y Fq(;)c(k)1665 1266 y Fp(0)1702 1254 y Ft(\))19 b(+)f Fq(f)1877 1266 y Fn(h)1920 1254 y Ft(\()p Fq(k)j Ft(+)d Fq(p)2141 1266 y Fn(F)2196 1254 y Fq(;)c(k)2276 1266 y Fp(0)2313 1254 y Ft(\))24 b(;)703 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Ft(\()p Fq(db)1488 352 y Fp(\(1\))1577 382 y Ft(\))p Fq(b)1645 339 y Fp(\(1\))p Fu(\000)1645 392 y Fk(x)p Fn(;!)1747 400 y Fj(1)1786 382 y Fq(b)1822 339 y Fp(\()p Fn(h)p Fp(\)+)1822 392 y Fk(y)q Fn(;!)1925 400 y Fj(2)1998 382 y Ft(is)30 b(b)r(ounded)h(for)g Fq(M)36 b Fm(!)28 b(1)p Ft(,)k(a)e(prop)r(ert)n(y)0 531 y(whic)n(h)e(should)f (otherwise)g(b)r(e)h(true)f(only)g(for)1468 469 y Fl(P)1556 556 y Fn(!)1598 564 y Fj(1)1630 556 y Fn(;!)1692 564 y Fj(2)1742 464 y Fl(R)1811 531 y Fq(P)12 b Ft(\()p Fq(db)1987 501 y Fp(\(1\))2077 531 y Ft(\))p Fq(b)2145 488 y Fp(\(1\))p Fn(\033)2268 496 y Fj(1)2300 488 y Fn(!)2342 496 y Fj(1)2145 542 y Fk(x)p Fn(;!)2247 550 y Fj(1)2379 531 y Fq(b)2415 488 y Fp(\()p Fn(h)p Fp(\))p Fn(\033)2544 496 y Fj(2)2576 488 y Fn(!)2618 496 y Fj(2)2415 542 y Fk(y)q Fn(;!)2518 550 y Fj(2)2654 531 y Ft(.)37 b(In)28 b(the)g(follo)n(wing,)0 681 y(w)n(e)f(shall)g(use)h(this)g(prop)r(ert)n(y)e(in)i(order)e(to)i 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y Fq(a)1502 53 y Fp(\()p Fn(h)p Fp(\))1597 83 y Ft(\),)28 b(b)n(y)f(using)g(\(2.51\).)0 453 y Fo(Remark.)73 b Ft(Note)28 b(that)i(the)f(sum)g(o)n(v)n(er)e Fq(k)1358 465 y Fp(0)1424 453 y Ft(in)i(\(2.49\))g(can)f(b)r(e)h (though)n(t)g(as)f(a)h(\014nite)g(sum)g(for)f(an)n(y)g Fq(M)9 b Ft(,)0 602 y(if)36 b Fq(h)g Fm(\024)f Ft(0,)i(b)r(ecause)e(of) g(the)h(supp)r(ort)f(prop)r(erties)g(of)1771 580 y(^)1754 602 y Fq( )1811 559 y Fp(\()p Fn(h)p Fp(\))p Fn(\033)1808 627 y Fk(k)1848 611 y Fg(0)1870 627 y Fn(;!)1946 602 y Ft(.)60 b(Hence,)38 b(all)d(quan)n(tities)g(that)h(w)n(e)f(shall)0 752 y(calculate)27 b(will)h(dep)r(end)g(on)f Fq(M)37 b Ft(only)27 b(trough)g(the)h(propagator)f(~)-45 b Fq(g)2084 722 y Fp(\(1\))2173 752 y Ft(\()p Fo(k)2255 722 y Fu(0)2279 752 y Ft(\),)28 b(if)g Fq(M)36 b Ft(is)28 b(large)e(enough.)0 972 y Fo(2.4)98 b Ft(If)28 b(w)n(e)f(apply)g(\(2.56\))g(to)g Fm(Z)1055 984 y Fn(L;\014)1193 972 y Ft(and)h(w)n(e)f(use)g(\(2.29\))g (and)h(\(2.22\),)f(w)n(e)g(get)832 1169 y 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y Fn(Z)573 344 y Fh(h)p Fg(\000)p Fj(1)685 335 y Fn(;\033)743 344 y Fh(h)p Fg(\000)p Fj(1)855 335 y Fn(;C)923 344 y Fh(h)p Fg(\000)p Fj(1)1038 323 y Ft(\()p Fq(d )1170 289 y Fp(\()p Fu(\024)p Fn(h)p Fu(\000)p Fp(1\))1403 323 y Ft(\))1449 210 y Fl(Z)1546 323 y Fq(P)1599 349 y Fn(Z)1644 358 y Fh(h)p Fg(\000)p Fj(1)1756 349 y Fn(;\033)1814 358 y Fh(h)p Fg(\000)p Fj(1)1926 349 y Fn(;)1959 334 y Fp(~)1946 349 y Fn(f)1985 322 y Fg(\000)p Fj(1)1978 371 y Fh(h)2066 323 y Ft(\()p Fq(d )2198 289 y Fp(\()p Fn(h)p Fp(\))2293 323 y Ft(\))14 b Fq(e)2378 289 y Fu(\000)2439 274 y Fp(~)2430 289 y Fu(V)2477 264 y Fj(\()p Fh(h)p Fj(\))2560 289 y Fp(\()2586 243 y Fu(p)p 2641 243 84 3 v 46 x Fn(Z)2686 298 y Fh(h)2724 289 y Fn( )2770 264 y Fj(\()p Fg(\024)p Fh(h)p Fj(\))2899 289 y Fp(\))2952 323 y Fq(;)120 b Ft(\(2)p Fq(:)p Ft(88\))0 563 y(where)499 712 y Fq(Z)556 724 y Fn(h)p Fu(\000)p Fp(1)706 712 y Ft(=)23 b Fq(Z)851 724 y Fn(h)894 712 y Ft(\(1)18 b(+)g Fq(z)1108 724 y Fn(h)1151 712 y Ft(\))23 b Fq(;)1427 690 y Ft(~)1409 712 y Fq(f)1450 724 y Fn(h)1493 712 y Ft(\()p Fo(k)1575 678 y Fu(0)1599 712 y Ft(\))g(=)g Fq(Z)1799 724 y Fn(h)p Fu(\000)p Fp(1)1926 712 y Ft([)1974 655 y Fq(C)2039 619 y Fu(\000)p Fp(1)2033 680 y Fn(h)2129 655 y Ft(\()p Fo(k)2211 625 y Fu(0)2235 655 y Ft(\))p 1959 693 323 4 v 1977 759 a(~)1959 780 y Fq(Z)2016 792 y Fn(h)p Fu(\000)p Fp(1)2144 780 y Ft(\()p Fo(k)2226 756 y Fu(0)2250 780 y Ft(\))2311 712 y Fm(\000)2404 648 y Fq(C)2469 612 y Fu(\000)p Fp(1)2463 673 y Fn(h)p Fu(\000)p Fp(1)2591 648 y Ft(\()p Fo(k)2673 618 y Fu(0)2697 648 y Ft(\))p 2404 693 326 4 v 2474 769 a Fq(Z)2531 781 y Fn(h)p Fu(\000)p Fp(1)2739 712 y Ft(])g Fq(:)287 b Ft(\(2)p Fq(:)p Ft(89\))0 940 y(Note)31 b(that)405 918 y(~)388 940 y Fq(f)429 952 y Fn(h)471 940 y Ft(\()p Fo(k)553 910 y Fu(0)577 940 y Ft(\))h(has)e(the)i(same)e(supp)r(ort)h(of)g Fq(f)1600 952 y Fn(h)1643 940 y Ft(\()p Fo(k)1725 910 y Fu(0)1749 940 y Ft(\);)i(in)f(fact,)g(b)n(y)f(using)f(\(2.38\),)i(it) 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Fk(k)1608 5678 y Fg(0)1741 5538 y Fq(e)1780 5508 y Fu(\000)p Fn(i)p Fk(k)1895 5483 y Fg(0)1917 5508 y Fp(\()p Fk(x)p Fu(\000)p Fk(y)q Fp(\))p 1683 5575 481 4 v 1683 5651 a Fm(\000)p Fq(ik)1820 5663 y Fp(0)1875 5651 y Ft(+)18 b Fq(!)s(v)2056 5623 y Fu(\003)2053 5673 y Fp(0)2094 5651 y Fq(k)2140 5627 y Fu(0)2191 5572 y Ft(~)2173 5594 y Fq(f)2214 5606 y Fn(h)2257 5594 y Ft(\()p Fo(k)2339 5560 y Fu(0)2363 5594 y Ft(\))23 b Fq(:)612 b Ft(\(2)p Fq(:)p Ft(100\))0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(23)p eop %%Page: 24 24 24 23 bop 0 83 a Fr(Mor)l(e)l(over,)26 b(given)e(the)g(p)l(ositive)h (inte)l(gers)e Fq(N)t(;)14 b(n)1480 95 y Fp(0)1517 83 y Fq(;)g(n)1604 95 y Fp(1)1664 83 y Fr(and)24 b(putting)f Fq(n)g Ft(=)f Fq(n)2305 95 y Fp(0)2347 83 y Ft(+)t Fq(n)2466 95 y Fp(1)2503 83 y Fr(,)j(ther)l(e)e(exist)g(a)h(c)l(onstant)0 232 y Fq(C)59 244 y Fn(N)s(;n)209 232 y Fr(such)30 b(that)662 472 y Fm(j)p Fq(@)734 437 y Fn(n)775 445 y Fj(0)729 492 y Fn(x)767 500 y Fj(0)822 450 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b(0)0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(25)p eop %%Page: 26 26 26 25 bop 0 83 a Ft(b)n(y)28 b(the)g(remark)f(follo)n(wing)f(\(2.62\),) i(w)n(e)f(ha)n(v)n(e)g(to)h(consider)f(only)h(t)n(w)n(o)f(kinds)h(of)g (bilinear)f(terms:)38 b(those)0 232 y(with)f Fq(!)250 244 y Fp(1)324 232 y Ft(=)h Fq(!)479 244 y Fp(2)552 232 y Ft(and)e(those)g(with)h Fq(!)1198 244 y Fp(1)1272 232 y Ft(=)h Fm(\000)p Fq(!)1492 244 y Fp(2)1528 232 y Ft(.)63 b(It)37 b(turns)f(out)g(that,)j(for)d(the)h(bilinear)f(terms)g(with)0 382 y Fq(!)52 394 y Fp(1)119 382 y Ft(=)31 b Fm(\000)p Fq(!)332 394 y Fp(2)368 382 y Ft(,)i(a)f(T)-7 b(a)n(ylor)30 b(expansion)h(up)i(to)f(order)e(0)i(is)g(su\016cien)n(t;)i(the)f (reason)d(is)i(that)h(the)f(F)-7 b(eynman)0 531 y(graphs)35 b(con)n(tributing)h(to)g(suc)n(h)g(terms)g(con)n(tain)f(at)h(least)g (one)g(non)g(diagonal)f(propagator)e(and,)38 b(b)n(y)0 681 y(lemma)31 b(2.6,)g(suc)n(h)g(propagators)d(are)j(smaller)f(than)h 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Fq(\015)1521 5618 y Fn(h)1560 5593 y Fg(\003)1626 5649 y Ft(it)g(is)f(a)h(\\massiv)n(e")d(theory)i(with)h(mass)f Fq(O)r Ft(\()p Fq(\015)3049 5618 y Fn(h)3088 5593 y Fg(\003)3127 5649 y Ft(\).)0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)h(18:23)1046 b Ft(27)p eop %%Page: 28 28 28 27 bop 696 83 a Fv(3.)50 b(Analyticit)m(y)34 b(of)k(the)f (e\013ectiv)m(e)g(p)s(oten)m(tial)0 374 y Fo(3.1)107 b Ft(W)-7 b(e)37 b(w)n(an)n(t)e(to)i(study)g(the)g(expansion)e(of)i (the)g(e\013ectiv)n(e)f(p)r(oten)n(tial,)j(whic)n(h)e(follo)n(ws)f (from)g(the)0 524 y(renormalization)e(pro)r(cedure)g(discussed)i(in)g Fm(x)p Ft(2.)61 b(In)36 b(order)e(to)i(do)f(that,)k(w)n(e)c(\014nd)h (it)h(con)n(v)n(enien)n(t)d(to)0 673 y(write)d Fm(V)274 643 y Fp(\()p Fn(h)p Fp(\))368 673 y Ft(,)h Fq(h)c Fm(\024)g Ft(1,)j(in)g(terms)f(of)h(the)g(v)-5 b(ariables)30 b Fq( )1671 630 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fn(\033)1668 684 y Fk(x)p Fn(;!)1858 673 y Ft(.)46 b(The)31 b(t)n(w)n(o)f(con)n (tributions)g(to)h Fm(V)2933 643 y 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Fu(2D)1756 5418 y Fg(0)1754 5460 y Fh(L;\014)1865 5336 y Fq(e)1904 5301 y Fn(i\033)r Fk(k)2007 5276 y Fg(0)2030 5301 y Fk(x)2097 5336 y Fq(:)975 b Ft(\(3)p Fq(:)p Ft(15\))0 5669 y(Similar)27 b(expressions)f(are)h(obtained,)g (if)h(the)g(lo)r(calization)f(p)r(oin)n(t)g(is)h(c)n(hosen)f(in)g(a)h (di\013eren)n(t)f(w)n(a)n(y)-7 b(.)0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(30)p eop %%Page: 31 31 31 30 bop 71 83 a Ft(In)34 b(the)h(new)f(represen)n(tation,)h(the)g (action)e(of)i Fm(R)f Ft(is)h(seen)f(as)f(the)i(decomp)r(osition)f(of)g (the)h(original)0 232 y(term)25 b(in)h(the)g(sum)f(of)g(three)g(terms,) h(whic)n(h)f(are)g(still)g(of)h(the)f(form)g(\(3.3\),)h(but)g(with)g(a) f(di\013eren)n(t)g(k)n(ernel,)0 382 y(con)n(taining)i(suitable)g(delta) h(functions.)71 609 y(2\)If)c(2)p Fq(n)f Ft(=)f(2)i(and,)h(p)r(ossibly) f(after)g(a)g(suitable)g(p)r(erm)n(utation)g(of)g(the)h(\014elds,)g Fq(\033)p 2491 622 51 4 v 27 w Ft(=)d(\(+)p Fq(;)14 b Fm(\000)p Ft(\),)25 b Fq(!)2983 621 y Fp(1)3043 609 y Ft(=)e Fq(!)3183 621 y Fp(2)3243 609 y Ft(=)0 758 y Fq(!)s Ft(,)k(b)n(y)h(\(2.74\),)832 921 y Fm(L)903 808 y Fl(Z)1000 921 y Fq(d)p Fo(x)1093 933 y Fp(1)1131 921 y Fq(d)p Fo(x)1224 933 y Fp(2)1262 921 y Fq(W)1352 878 y Fp(\()p Fn(h)p Fp(\))1340 943 y(2)p Fn(;\033)p 1393 956 41 4 v 3 w(;!)p 1454 956 44 4 v 1501 921 a Ft(\()p Fo(x)1583 933 y Fp(1)1640 921 y Fm(\000)18 b Fo(x)1773 933 y Fp(2)1810 921 y Ft(\))p Fq( )1899 887 y Fp(\()p Fu(\024)p Fn(h)p Fp(\)+)1896 942 y Fk(x)1936 950 y Fj(1)1969 942 y Fn(;!)2097 921 y Fq( )2154 887 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)2151 942 y Fk(x)2191 950 y Fj(2)2224 942 y Fn(;!)2376 921 y Ft(=)855 1139 y(=)943 1026 y Fl(Z)1040 1139 y Fq(d)p Fo(x)1133 1151 y Fp(1)1170 1139 y Fq(d)p Fo(x)1263 1151 y Fp(2)1301 1139 y Fq(W)1391 1096 y Fp(\()p Fn(h)p Fp(\))1379 1161 y(2)p Fn(;\033)p 1432 1174 41 4 v 3 w(;!)p 1493 1174 44 4 v 1541 1139 a Ft(\()p Fo(x)1623 1151 y Fp(1)1679 1139 y Fm(\000)g Fo(x)1812 1151 y Fp(2)1850 1139 y Ft(\))p 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2479 y Ft(\))24 b(=)e(cos\()p Fq(\031)s(x)2251 2491 y Fp(0)2289 2479 y Fq(\014)2340 2444 y Fu(\000)p Fp(1)2429 2479 y Ft(\))i Fq(:)587 b Ft(\(3)p Fq(:)p Ft(18\))71 2755 y(As)30 b(in)g(the)g(item)h(1\),)f(w)n(e)f(de\014ne)i(the)f(lo)r (calization)f(p)r(oin)n(t)h(as)f(the)h Fo(x)g Ft(co)r(ordinate)f(of)h (the)g(\014eld)g(whic)n(h)0 2904 y(is)h(left)g(unc)n(hanged)g Fm(L)p Ft(.)47 b(W)-7 b(e)31 b(are)f(free)h(to)g(c)n(ho)r(ose)e(it)j (equal)e(to)h Fo(x)2065 2916 y Fp(1)2133 2904 y Ft(or)f Fo(x)2288 2916 y Fp(2)2326 2904 y Ft(.)47 b(This)31 b(freedom)f (a\013ects)h(also)0 3054 y(the)d(action)f(of)h Fm(R)p Ft(,)g(whic)n(h)f(can)g(b)r(e)h(written)g(as)827 3312 y Fm(R)911 3199 y Fl(Z)1008 3312 y Fq(d)p Fo(x)1101 3324 y Fp(1)1139 3312 y Fq(d)p Fo(x)1232 3324 y Fp(2)1269 3312 y Fq(W)1359 3268 y Fp(\()p Fn(h)p Fp(\))1347 3334 y(2)p Fn(;\033)p 1400 3347 41 4 v 3 w(;!)p 1461 3347 44 4 v 1509 3312 a Ft(\()p Fo(x)1591 3324 y Fp(1)1647 3312 y Fm(\000)18 b Fo(x)1780 3324 y Fp(2)1818 3312 y Ft(\))p Fq( 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b(to)g(pro)n(v)n(e)e (that)j(the)g(\\zeros")d Fq(d)1111 3564 y Fn(L)1160 3552 y Ft(\()p Fq(y)15 b Fm(\000)d Fq(x)p Ft(\))25 b(and)f Fq(d)1630 3564 y Fn(\014)1675 3552 y Ft(\()p Fq(y)1748 3564 y Fp(0)1797 3552 y Fm(\000)12 b Fq(x)1921 3564 y Fp(0)1958 3552 y Ft(\))24 b(giv)n(e)g(a)f(con)n(tribution)h(to)g(the)h (b)r(ounds)f(of)0 3701 y(order)g Fq(\015)263 3671 y Fu(\000)p Fn(h)354 3646 y Fg(0)380 3701 y Ft(,)h(with)h Fq(h)663 3671 y Fu(0)709 3701 y Fm(\025)d Fq(h)p Ft(,)i(if)h Fq(h)f Ft(is)g(the)g(scale)g(at)g(whic)n(h)g(the)g(zero)f(w)n(as)g(pro)r (duced)h(b)n(y)g(the)g(action)g(of)g Fm(R)p Ft(.)0 3851 y(In)e Fm(x)p Ft(3.7)f(w)n(e)g(shall)g(realize)g(this)h(task)f(b)n(y)g (\\distributing")g(the)h(zeros)e(along)h(a)g(path)h(connecting)f(a)g (family)0 4000 y(of)29 b(space-time)f(p)r(oin)n(ts)h(asso)r(ciated)f (with)i(a)e(subset)h(of)g(\014eld)h(v)-5 b(ariables.)40 b(Let)29 b Fo(x)2532 4012 y Fp(0)2595 4000 y Ft(=)c Fo(x)p Fq(;)14 b Fo(x)2822 4012 y Fp(1)2860 4000 y Fq(;)g(:)g(:)g(:)f(;)h Fo(x)3094 4012 y Fn(n)3165 4000 y Ft(=)25 b Fo(y)0 4150 y Ft(b)r(e)j(the)g(family)g(of)f(p)r(oin)n(ts)h(connected)f(b)n(y)g (the)h(path;)g(it)g(is)g(easy)e(to)i(sho)n(w)f(that)703 4394 y Fq(d)746 4406 y Fn(L)796 4394 y Ft(\()p Fq(y)21 b Fm(\000)d Fq(x)p Ft(\))24 b(=)1203 4290 y Fn(n)1164 4315 y Fl(X)1165 4491 y Fn(r)r Fp(=1)1297 4394 y Fq(d)1340 4406 y Fn(L)1390 4394 y Ft(\()p Fq(x)1469 4406 y Fn(r)1525 4394 y Fm(\000)18 b Fq(x)1655 4406 y Fn(r)r Fu(\000)p Fp(1)1778 4394 y Ft(\))p Fq(e)1849 4359 y Fu(\000)p Fn(i)1936 4337 y Fh(\031)p 1934 4346 40 4 v 1934 4379 a(L)1983 4359 y Fp(\()p Fn(x)2047 4367 y Fh(r)2080 4359 y Fp(+)p Fn(x)2169 4367 y Fh(r)q Fg(\000)p Fj(1)2276 4359 y Fu(\000)p Fn(x)2366 4367 y Fh(n)2406 4359 y Fu(\000)p Fn(x)2496 4367 y Fj(0)2528 4359 y Fp(\))2581 4394 y Fq(:)491 b Ft(\(3)p Fq(:)p Ft(55\))0 4652 y(A)28 b(similar)f(expression)f(is)h(v) -5 b(alid)28 b(for)f Fq(d)1217 4664 y Fn(\014)1262 4652 y Ft(\()p Fq(y)1335 4664 y Fp(0)1391 4652 y Fm(\000)18 b Fq(x)1521 4664 y Fp(0)1558 4652 y Ft(\).)71 4801 y(It)35 b(can)f(happ)r(en)g(that)h(one)f(of)g(the)h(terms)f(in)h(the)g(r.h.s.) 57 b(of)34 b(\(3.55\))g(or)f(the)i(analogous)d(expansion)0 4951 y(for)g Fq(d)175 4963 y Fn(\014)220 4951 y Ft(\()p Fq(y)293 4963 y Fp(0)351 4951 y Fm(\000)21 b Fq(x)484 4963 y Fp(0)522 4951 y Ft(\))32 b(dep)r(ends)h(on)f(the)g(same)g (space-time)f(p)r(oin)n(ts)h(as)f(the)i(in)n(tegration)e(v)-5 b(ariables)30 b(in)j(the)0 5100 y(r.h.s.)41 b(of)29 b(a)f(term)h(lik)n (e)g(\(3.21\))f(or)g(\(3.26\).)41 b(W)-7 b(e)29 b(w)n(an)n(t)f(to)h (study)h(the)f(e\013ect)g(of)g(this)h(ev)n(en)n(t.)40 b(Let)30 b(us)f(call)0 5250 y Fq(W)12 b Ft(\()p Fo(x)g Fm(\000)g Fo(y)q Ft(\))26 b(the)e(k)n(ernel)g(app)r(earing)f(in)i(the)g (l.h.s.)35 b(of)25 b(\(3.21\))e(or)h(\(3.26\),)g Fq(W)2338 5262 y Fn(R)2393 5250 y Ft(\()p Fo(x)12 b Fm(\000)g Fo(y)q Ft(\))26 b(its)e(regularization,)0 5399 y(that)k(is)f(the)h(quan)n(tit) n(y)f(app)r(earing)g(in)h(braces)e(in)i(the)g(corresp)r(onding)e (r.h.s.,)h(and)g(let)h(us)g(de\014ne)42 5631 y Fq(I)78 5643 y Fn(n)119 5651 y Fj(1)151 5643 y Fn(;n)212 5651 y Fj(2)272 5631 y Ft(=)359 5518 y Fl(Z)456 5631 y Fq(d)p Fo(x)p Fq(d)p Fo(y)q Fq( )700 5597 y Fp(\()p Fu(\024)p Fn(h)p Fp(\)+)697 5652 y Fk(x)p Fn(;!)900 5631 y Fq( )957 5597 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)954 5652 y Fk(y)q Fn(;!)1155 5631 y Fq(W)1233 5643 y Fn(R)1288 5631 y Ft(\()p Fo(x)15 b Fm(\000)g Fo(y)q Ft(\)[)p Fq(e)1610 5597 y Fu(\000)p Fn(i\031)1740 5569 y Fh(y)p 1736 5584 V 1736 5617 a(L)1790 5631 y Fq(d)1833 5643 y Fn(L)1883 5631 y Ft(\()p Fq(y)j Fm(\000)c Fq(x)p Ft(\)])2155 5597 y Fn(n)2196 5605 y Fj(1)2233 5631 y Ft([)p Fq(e)2295 5594 y Fu(\000)p Fn(i\031)2421 5562 y Fh(y)2451 5574 y Fj(0)p 2421 5581 63 4 v 2435 5614 a Fh(\014)2498 5631 y Fq(d)2541 5643 y Fn(\014)2586 5631 y Ft(\()p Fq(y)2659 5643 y Fp(0)2711 5631 y Fm(\000)g Fq(x)2837 5643 y Fp(0)2874 5631 y Ft(\)])2929 5597 y Fn(n)2970 5605 y Fj(2)3030 5631 y Fq(:)42 b Ft(\(3)p Fq(:)p Ft(56\))0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(41)p eop %%Page: 42 42 42 41 bop 0 83 a Ft(In)38 b(the)h(follo)n(wing)e(w)n(e)h(shall)g(meet)h (suc)n(h)f(expressions)e(for)i(v)-5 b(alues)38 b(of)g Fq(n)2373 95 y Fp(1)2448 83 y Ft(and)h Fq(n)2671 95 y Fp(2)2708 83 y Ft(,)i(suc)n(h)d(that)g(1)j Fm(\024)0 232 y Fq(n)50 244 y Fp(1)106 232 y Ft(+)18 b Fq(n)239 244 y Fp(2)299 232 y Fm(\024)k Ft(2.)71 382 y(If)d Fq(W)12 b Ft(\()p Fo(x)r Fm(\000)r Fo(y)q Ft(\))20 b(is)f(the)h(k)n(ernel)e (app)r(earing)g(in)i(the)f(l.h.s.)35 b(of)19 b(\(3.26\),)h(it)g(is)f (easy)f(to)h(see)g(that,)i(if)f Fq(n)2940 394 y Fp(1)2978 382 y Ft(+)r Fq(n)3095 394 y Fp(2)3155 382 y Fm(\025)j Ft(1,)51 632 y Fq(I)87 644 y Fn(n)128 652 y Fj(1)161 644 y Fn(;n)222 652 y Fj(2)281 632 y Ft(=)369 519 y Fl(Z)466 632 y Fq(d)p Fo(x)p Fq(d)p Fo(y)q Fq( )710 598 y Fp(\()p Fu(\024)p Fn(h)p Fp(\)+)707 653 y Fk(x)p Fn(;!)909 632 y Fq( )966 598 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)963 653 y Fk(y)q Fn(;!)1165 632 y Fq(W)12 b Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\)[)p Fq(e)1584 598 y Fu(\000)p Fn(i\031)1715 571 y Fh(y)p 1711 586 40 4 v 1711 618 a(L)1765 632 y Fq(d)1808 644 y Fn(L)1858 632 y Ft(\()p Fq(y)j Fm(\000)d Fq(x)p Ft(\)])2137 598 y Fn(n)2178 606 y Fj(1)2216 632 y Ft([)p Fq(e)2278 595 y Fu(\000)p Fn(i\031)2404 563 y Fh(y)2434 575 y Fj(0)p 2404 583 63 4 v 2417 615 a Fh(\014)2480 632 y Fq(d)2523 644 y Fn(\014)2568 632 y Ft(\()p Fq(y)2641 644 y Fp(0)2697 632 y Fm(\000)g Fq(x)2827 644 y Fp(0)2865 632 y Ft(\)])2920 598 y Fn(n)2961 606 y Fj(2)3021 632 y Fq(;)51 b Ft(\(3)p Fq(:)p Ft(57\))0 883 y(that)28 b(is)f(the)h(presence)f(of)h(the)g(zeros)e(simply)i (erases)e(the)i(e\013ect)g(of)f(the)h(regularization.)71 1032 y(Let)36 b(us)g(no)n(w)g(supp)r(ose)g(that)g Fq(W)12 b Ft(\()p Fo(x)25 b Fm(\000)f Fo(y)q Ft(\))37 b(is)f(the)h(k)n(ernel)e (app)r(earing)g(in)h(the)h(l.h.s.)63 b(of)36 b(\(3.21\))f(and)0 1182 y Fq(W)78 1194 y Fn(R)133 1182 y Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\))29 b(its)e(regularization.)35 b(W)-7 b(e)28 b(ha)n(v)n(e)309 1425 y Fq(I)345 1437 y Fp(1)p Fn(;)p Fp(0)458 1425 y Ft(=)546 1312 y Fl(Z)643 1425 y Fq(d)p Fo(x)p Fq(d)p Fo(y)q Fq( )887 1390 y Fp(\()p Fu(\024)p Fn(h)p Fp(\)+)884 1445 y Fk(x)p Fn(;!)1086 1425 y Fq(W)12 b Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\))p Fq(d)1486 1437 y Fn(L)1537 1425 y Ft(\()p Fq(y)k Fm(\000)c Fq(x)p Ft(\))1794 1332 y Fl(n)1850 1425 y Fq(D)1921 1390 y Fp(1)p Fn(;)p Fp(3\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)1919 1445 y Fk(y)q Fn(;)p Fk(x)p Fn(;!)2205 1425 y Fm(\000)454 1640 y(\000)g Fq(c)573 1652 y Fn(\014)617 1640 y Ft(\()p Fq(y)690 1652 y Fp(0)746 1640 y Fm(\000)g Fq(x)876 1652 y Fp(0)914 1640 y Ft(\)[)979 1584 y(1)p 979 1621 42 4 v 979 1697 a(2)1041 1618 y(\026)1031 1640 y Fq(@)1080 1606 y Fp(2)1075 1661 y(1)1117 1640 y Ft(\()p Fq(e)1188 1606 y Fu(\000)p Fn(i\031)1317 1584 y Fh(x)p 1314 1593 40 4 v 1314 1626 a(L)1368 1640 y Fq( )1425 1606 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)1422 1661 y Fk(x)p Fn(;!)1623 1640 y Ft(\))h(+)1767 1584 y Fq(i)14 b Ft(cos)e Fq(p)1976 1596 y Fn(F)p 1767 1621 265 4 v 1860 1697 a Fq(v)1900 1709 y Fp(0)2052 1618 y Ft(\026)2041 1640 y Fq(@)2085 1652 y Fp(1)2123 1640 y Ft(\()p Fq(e)2194 1606 y Fu(\000)p Fn(i\031)2323 1584 y Fh(x)p 2320 1593 40 4 v 2320 1626 a(L)2374 1640 y Fq( )2431 1606 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)2428 1661 y Fk(x)p Fn(;!)2629 1640 y Ft(\)])2684 1548 y Fl(o)2763 1640 y Fq(;)3095 1531 y Ft(\(3)p Fq(:)p Ft(58\))677 1970 y Fq(I)713 1982 y Fp(0)p Fn(;)p Fp(1)826 1970 y Ft(=)914 1857 y Fl(Z)1011 1970 y Fq(d)p Fo(x)p Fq(d)p Fo(y)q Fq( )1255 1935 y Fp(\()p Fu(\024)p Fn(h)p Fp(\)+)1252 1990 y Fk(x)p Fn(;!)1454 1970 y Fq(W)g Ft(\()p Fo(x)19 b Fm(\000)f Fo(y)q Ft(\))p Fq(d)1854 1982 y Fn(\014)1900 1970 y Ft(\()p Fq(y)1973 1982 y Fp(0)2029 1970 y Fm(\000)g Fq(x)2159 1982 y Fp(0)2196 1970 y Ft(\))p Fq(D)2299 1935 y Fp(1)p Fn(;)p Fp(4\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)2297 1990 y Fk(y)q Fn(;)p Fk(x)p Fn(;!)2607 1970 y Fq(;)465 b Ft(\(3)p Fq(:)p Ft(59\))0 2178 y(where)667 2329 y Fq(D)738 2294 y Fp(1)p Fn(;)p Fp(3\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)736 2349 y Fk(y)q Fn(;)p Fk(x)p Fn(;!)1046 2329 y Ft(=)23 b Fq(e)1173 2294 y Fu(\000)p Fn(i\031)1303 2267 y Fh(y)p 1299 2282 V 1299 2314 a(L)1352 2329 y Fq( )1409 2294 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)1406 2349 y Fk(y)q Fn(;!)1627 2329 y Fm(\000)18 b Fq(c)1746 2341 y Fn(\014)1790 2329 y Ft(\()p Fq(y)1863 2341 y Fp(0)1919 2329 y Fm(\000)g Fq(x)2049 2341 y Fp(0)2087 2329 y Ft(\))p Fq(e)2158 2294 y Fu(\000)p Fn(i\031)2287 2272 y Fh(x)p 2284 2281 V 2284 2314 a(L)2338 2329 y Fq( )2395 2294 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)2392 2349 y Fk(x)p Fn(;!)2617 2329 y Fq(:)455 b Ft(\(3)p Fq(:)p Ft(60\))676 2538 y Fq(D)747 2504 y Fp(1)p Fn(;)p Fp(4\()p Fu(\024)p Fn(h)p Fp(\))p Fu(\000)745 2559 y Fk(y)q Fn(;)p Fk(x)p Fn(;!)1055 2538 y Ft(=)23 b Fq(e)1182 2501 y Fu(\000)p Fn(i\031)1308 2469 y Fh(y)1338 2481 y Fj(0)p 1308 2489 63 4 v 1322 2521 a Fh(\014)1384 2538 y Fq( 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Fh(=apr)q(ile=)p Fj(2000;)e(18:23)1046 b Ft(42)p eop %%Page: 43 43 43 42 bop 0 83 a Ft(op)r(eration,)30 b(b)r(efore)g(p)r(erforming)f(the) i("decomp)r(osition)e(of)h(the)h(zeros";)e(one)h(should)g(get)g(in)h (this)f(case)0 232 y(a)e(zero)f(of)h(order)f(four)h(and)g(the)h (iteration)f(of)g(the)g(pro)r(cedure)g(of)g(decomp)r(osition)g(of)g (the)h(zeros)e(w)n(ould)0 382 y(pro)r(duce)35 b(zeros)f(of)h(arbitrary) e(order)h(and,)j(as)d(a)h(consequence,)h(bad)f(com)n(binatorial)e (factors)h(in)i(the)0 531 y(b)r(ounds.)0 755 y Fo(3.7)104 b Ft(W)-7 b(e)35 b(are)e(no)n(w)g(ready)g(to)h(describ)r(e)g(in)g(more) f(detail)i(our)e(expansion.)55 b(First)34 b(of)g(all,)i(w)n(e)d(insert) 0 904 y(the)h(decomp)r(osition)f(\(3.14\))g(of)h Fq(V)1116 874 y Fp(\()p Fn(h)p Fp(\))1210 904 y Ft(\()p Fq(\034)5 b(;)14 b( )1377 874 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))1525 904 y Ft(\))34 b(in)f(the)i(v)n(ertices)d(with)i Fm(j)p Fq(P)2420 916 y Fn(v)2460 904 y Fm(j)f Ft(=)g(4,)i(b)n(y)e(follo)n (wing)g(the)0 1054 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Fu(2T)1707 3012 y Fh(h;n)1833 2882 y Ft(~)1814 2903 y Fq(E)1875 2915 y Fn(h)p Fp(+1)2002 2903 y Ft(\()p Fq(\034)i Ft(\))25 b Fq(;)936 b Ft(\(3)p Fq(:)p Ft(72\))0 3189 y(where)917 3317 y(~)898 3338 y Fq(E)959 3350 y Fn(h)p Fp(+1)1087 3338 y Ft(\()p Fq(\034)9 b Ft(\))24 b(=)1351 3282 y(1)p 1317 3319 108 4 v 1317 3395 a Fq(L\014)1502 3259 y Fl(X)1459 3426 y Fb(P)p Fg(2P)1581 3434 y Fh(\034)1619 3426 y(;)p Fb(r)1474 3475 y Fh(P)1510 3483 y(v)1540 3495 y Fj(0)1576 3475 y(=)p Fg(;)1688 3225 y Fl(Z)1785 3338 y Fq(d)p Fo(x)1878 3350 y Fn(v)1911 3358 y Fj(0)1948 3338 y Fq(K)2025 3295 y Fp(\()p Fn(h)p Fp(\))2019 3363 y Fn(\034)s(;)p Fk(P)p Fn(;)p Fk(r)2178 3338 y Ft(\()p Fo(x)2260 3350 y Fn(v)2293 3358 y Fj(0)2330 3338 y Ft(\))g Fq(:)686 b Ft(\(3)p Fq(:)p Ft(73\))0 3702 y Fo(3.8)97 b Ft(W)-7 b(e)28 b(w)n(an)n(t)f(no)n(w)g(to)g(iterate)g (the)h(previous)f(pro)r(cedure,)f(b)n(y)i(using)f(equation)g(\(3.38\),) f(in)i(order)e(to)0 3851 y(suitably)c(tak)n(e)g(in)n(to)h(accoun)n(t)e (the)i(non)g(trivial)f Fm(R)h Ft(op)r(erations)e(in)i(the)g(v)n (ertices)e Fq(v)27 b Fm(6)p Ft(=)22 b Fq(v)2699 3863 y Fp(0)2737 3851 y Ft(.)35 b(W)-7 b(e)23 b(shall)f(fo)r(cus)0 4001 y(our)e(discussion)g(on)h Fm(R)p Fq(V)769 3971 y Fp(\()p Fn(h)p Fp(\))864 4001 y Ft(\()p Fq(\034)5 b(;)14 b Fo(P)p Fq(;)g Fo(r)p Ft(\),)23 b(but)f(the)f(follo)n(wing)f(analysis) f(applies)i(also)f(to)g Fm(L)p Fq(V)2773 3971 y Fp(\()p Fn(h)p Fp(\))2868 4001 y Ft(\()p Fq(\034)5 b(;)14 b Fo(P)p Fq(;)g Fo(r)p Ft(\))22 b(and)19 4129 y(~)0 4150 y Fq(E)61 4162 y Fn(h)p Fp(+1)188 4150 y Ft(\()p Fq(\034)9 b Ft(\).)71 4300 y(Let)33 b(us)g(consider)e(the)i(truncated)g(exp)r(ectation)g(in)g (the)g(r.h.s.)52 b(of)33 b(\(3.38\))f(and)g(let)i(us)e(put)i Fq(s)d Ft(=)h Fq(s)3245 4312 y Fn(v)3284 4300 y Ft(,)0 4449 y Fq(P)53 4461 y Fn(i)109 4449 y Fm(\021)c Fq(P)255 4461 y Fn(v)288 4469 y Fh(i)319 4449 y Fm(n)p Fq(Q)427 4461 y Fn(v)460 4469 y Fh(i)490 4449 y Ft(.)46 b(Moreo)n(v)n(er)28 b(w)n(e)i(order)f(in)i(an)g(arbitrary)d(w)n(a)n(y)i(the)h(sets)f Fq(P)2395 4414 y Fu(\006)2383 4472 y Fn(i)2479 4449 y Fm(\021)e(f)p Fq(f)36 b Fm(2)29 b Fq(P)2828 4461 y Fn(i)2856 4449 y Fq(;)14 b(\033)s Ft(\()p Fq(f)9 b Ft(\))28 b(=)g Fm(\006g)p Ft(,)0 4598 y(w)n(e)23 b(call)h Fq(f)317 4563 y Fu(\006)308 4622 y Fn(ij)396 4598 y Ft(their)g(elemen)n(ts)g(and)f(w) n(e)h(de\014ne)g Fo(x)1489 4568 y Fp(\()p Fn(i)p Fp(\))1592 4598 y Ft(=)f Fm([)1735 4624 y Fn(f)7 b Fu(2)p Fn(P)1870 4598 y Fg(\000)1861 4645 y Fh(i)1923 4598 y Fo(x)p Ft(\()p Fq(f)i Ft(\),)25 b Fo(y)2186 4568 y Fp(\()p Fn(i)p Fp(\))2290 4598 y Ft(=)d Fm([)2432 4624 y Fn(f)7 b Fu(2)p Fn(P)2567 4598 y Fj(+)2558 4645 y Fh(i)2618 4598 y Fo(x)p Ft(\()p Fq(f)i Ft(\),)26 b Fo(x)2881 4610 y Fn(ij)2963 4598 y Ft(=)c Fo(x)p Ft(\()p Fq(f)3182 4563 y Fu(\000)3173 4622 y Fn(i;j)3252 4598 y Ft(\),)0 4748 y Fo(y)50 4760 y Fn(ij)134 4748 y Ft(=)i Fo(x)p Ft(\()p Fq(f)355 4712 y Fp(+)346 4771 y Fn(i;j)425 4748 y Ft(\).)40 b(Note)29 b(that)903 4686 y Fl(P)991 4706 y Fn(s)991 4773 y(i)p Fp(=1)1116 4748 y Fm(j)p Fq(P)1204 4712 y Fu(\000)1192 4771 y Fn(i)1260 4748 y Fm(j)c Ft(=)1397 4686 y Fl(P)1485 4706 y Fn(s)1485 4773 y(i)p Fp(=1)1611 4748 y Fm(j)p Fq(P)1699 4712 y Fp(+)1687 4771 y Fn(i)1754 4748 y Fm(j)g(\021)f Fq(n)p Ft(,)29 b(otherwise)f(the)h(truncated)f(exp)r(ectation)0 4897 y(v)-5 b(anishes.)57 b(A)34 b(couple)g Fq(l)i Fm(\021)e Ft(\()p Fq(f)982 4862 y Fu(\000)973 4921 y Fn(ij)1038 4897 y Fq(;)14 b(f)1125 4862 y Fp(+)1116 4922 y Fn(i)1139 4905 y Fg(0)1161 4922 y Fn(j)1191 4905 y Fg(0)1219 4897 y Ft(\))34 b Fm(\021)g Ft(\()p Fq(f)1466 4862 y Fu(\000)1457 4922 y Fn(l)1522 4897 y Fq(;)14 b(f)1609 4862 y Fp(+)1600 4922 y Fn(l)1664 4897 y Ft(\))35 b(will)f(b)r(e)h(called)f(a)g(line)g (joining)h(the)f(\014elds)h(with)0 5047 y(lab)r(els)26 b Fq(f)282 5011 y Fu(\000)273 5070 y Fn(ij)337 5047 y Fq(;)14 b(f)424 5011 y Fp(+)415 5071 y Fn(i)438 5055 y Fg(0)461 5071 y Fn(j)491 5055 y Fg(0)544 5047 y Ft(and)26 b Fq(!)j Ft(indices)d Fq(!)1111 5011 y Fu(\000)1108 5072 y Fn(l)1166 5047 y Fq(;)14 b(!)1258 5011 y Fp(+)1255 5072 y Fn(l)1339 5047 y Ft(and)26 b(connecting)f(the)h(p)r(oin)n(ts)g Fo(x)2351 5059 y Fn(l)2400 5047 y Fm(\021)d Fo(x)2538 5059 y Fn(i;j)2642 5047 y Ft(and)j Fo(y)2852 5059 y Fn(l)2901 5047 y Fm(\021)d Fo(y)3039 5059 y Fn(i)3062 5043 y Fg(0)3085 5059 y Fn(j)3115 5043 y Fg(0)3142 5047 y Ft(,)k(the)0 5196 y Fr(endp)l(oints)34 b Ft(of)e Fq(l)r Ft(;)j(moreo)n(v)n(er)c(w)n (e)h(shall)g(put)i Fq(m)1480 5208 y Fn(l)1537 5196 y Fm(\021)e Fq(m)p Ft(\()p Fq(f)1789 5161 y Fu(\000)1780 5221 y Fn(l)1844 5196 y Ft(\))23 b(+)e Fq(m)p Ft(\()p Fq(f)2140 5161 y Fp(+)2131 5221 y Fn(l)2195 5196 y Ft(\).)53 b(Then,)34 b(it)g(is)e(w)n(ell)h(kno)n(wn)f(\(see)0 5346 y([Le],)c([BGPS],)f(for)g(example\))h(that,)g(up)f(to)h(a)f(sign,)g(if) h Fq(s)23 b(>)g Ft(1,)149 5569 y(~)135 5590 y Fm(E)186 5556 y Fn(T)179 5610 y(h)238 5590 y Ft(\()288 5568 y(~)270 5590 y Fq( )327 5556 y Fp(\()p Fn(h)p Fp(\))423 5590 y Ft(\()p Fq(P)508 5602 y Fp(1)546 5590 y Ft(\))p Fq(;)14 b(:::;)738 5568 y Ft(~)721 5590 y Fq( )778 5556 y Fp(\()p Fn(h)p Fp(\))873 5590 y Ft(\()p Fq(P)958 5602 y Fn(s)994 5590 y Ft(\)\))24 b(=)1169 5511 y Fl(X)1205 5689 y Fn(T)1307 5511 y Fl(Y)1303 5690 y Fn(l)p Fu(2)p Fn(T)1441 5568 y Ft(\026)1431 5590 y Fq(@)1480 5553 y Fn(m)1539 5562 y Fh(l)1475 5612 y Fp(1)1567 5590 y Fq(g)1610 5547 y Fp(\()p Fn(h)p Fp(\))1607 5629 y Fn(!)1651 5602 y Fg(\000)1649 5651 y Fh(l)1699 5629 y Fn(;!)1763 5602 y Fj(+)1761 5651 y Fh(l)1814 5590 y Ft(\()p Fo(x)1896 5602 y Fn(l)1941 5590 y Fm(\000)18 b Fo(y)2074 5602 y Fn(l)2100 5590 y Ft(\))2146 5477 y Fl(Z)2243 5590 y Fq(dP)2339 5602 y Fn(T)2391 5590 y Ft(\()p Fo(t)p Ft(\))c(det)h Fq(G)2701 5556 y Fn(h;T)2812 5590 y Ft(\()p Fo(t)p Ft(\))24 b Fq(;)135 b Ft(\(3)p Fq(:)p Ft(74\))0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(45)p eop %%Page: 46 46 46 45 bop 0 83 a Ft(where)38 b Fq(T)50 b Ft(is)39 b(a)f(set)h(of)g (lines)f(forming)g(an)h Fr(anchor)l(e)l(d)i(tr)l(e)l(e)f(gr)l(aph)g Ft(b)r(et)n(w)n(een)e(the)h(clusters)g(of)f(p)r(oin)n(ts)0 232 y Fo(x)50 202 y Fp(\()p Fn(i)p Fp(\))152 232 y Fm([)23 b Fo(y)281 202 y Fp(\()p Fn(i)p Fp(\))361 232 y Ft(,)35 b(that)f(is)f Fq(T)45 b Ft(is)33 b(a)g(set)h(of)f(lines,)i(whic)n(h)e (b)r(ecomes)h(a)f(tree)g(graph)f(if)i(one)f(iden)n(ti\014es)h(all)f (the)0 382 y(p)r(oin)n(ts)26 b(in)h(the)f(same)g(cluster.)36 b(Moreo)n(v)n(er)24 b Fo(t)f Ft(=)g Fm(f)p Fq(t)1576 394 y Fn(i;i)1642 378 y Fg(0)1691 382 y Fm(2)h Ft([0)p Fq(;)14 b Ft(1])p Fq(;)g Ft(1)21 b Fm(\024)i Fq(i;)14 b(i)2220 352 y Fu(0)2265 382 y Fm(\024)23 b Fq(s)p Fm(g)p Ft(,)j Fq(dP)2579 394 y Fn(T)2632 382 y Ft(\()p Fo(t)p Ft(\))h(is)f(a)g(probabilit)n(y)0 531 y(measure)21 b(with)i(supp)r(ort) g(on)f(a)g(set)h(of)f Fo(t)h Ft(suc)n(h)f(that)h Fq(t)1640 543 y Fn(i;i)1706 527 y Fg(0)1756 531 y Ft(=)g Fo(u)1897 543 y Fn(i)1933 531 y Fm(\001)8 b Fo(u)2017 543 y Fn(i)2040 527 y Fg(0)2090 531 y Ft(for)22 b(some)g(family)g(of)h(v)n(ectors)d Fo(u)3082 543 y Fn(i)3133 531 y Fm(2)k Ff(R)3272 495 y Fn(s)0 681 y Ft(of)h(unit)h(norm.)35 b(Finally)25 b Fq(G)858 651 y Fn(h;T)969 681 y Ft(\()p Fo(t)p Ft(\))h(is)e(a)h(\()p Fq(n)13 b Fm(\000)g Fq(s)g Ft(+)g(1\))g Fm(\002)g Ft(\()p Fq(n)g Fm(\000)g Fq(s)g Ft(+)g(1\))24 b(matrix,)h(whose)f(elemen)n(ts)h (are)f(giv)n(en)0 830 y(b)n(y)j Fq(G)180 790 y Fn(h;T)180 855 y(ij;i)273 838 y Fg(0)297 855 y Fn(j)327 838 y Fg(0)377 830 y Ft(=)c Fq(t)495 842 y Fn(i;i)561 826 y Fg(0)599 808 y Ft(\026)588 830 y Fq(@)637 765 y Fn(m)p Fp(\()p Fn(f)761 738 y Fg(\000)754 786 y Fh(ij)810 765 y Fp(\)+)p Fn(m)p Fp(\()p Fn(f)1011 738 y Fj(+)1004 794 y Fh(i)1026 782 y Fg(0)1049 794 y Fh(j)1075 782 y Fg(0)1102 765 y Fp(\))632 852 y(1)1132 830 y Fq(g)1175 787 y Fp(\()p Fn(h)p Fp(\))1172 870 y Fn(!)1216 843 y Fg(\000)1214 892 y Fh(l)1265 870 y Fn(;!)1329 843 y Fj(+)1327 892 y Fh(l)1380 830 y Ft(\()p Fo(x)1462 842 y Fn(ij)1539 830 y Fm(\000)18 b Fo(y)1672 842 y Fn(i)1695 826 y Fg(0)1718 842 y Fn(j)1748 826 y Fg(0)1776 830 y Ft(\))28 b(with)g(\()p Fq(f)2107 795 y Fu(\000)2098 853 y Fn(ij)2163 830 y Fq(;)14 b(f)2250 795 y Fp(+)2241 855 y Fn(i)2264 838 y Fg(0)2286 855 y Fn(j)2316 838 y Fg(0)2344 830 y Ft(\))28 b(not)f(b)r(elonging)g (to)h Fq(T)12 b Ft(.)71 980 y(If)31 b Fq(s)c Ft(=)g(1,)k(the)g(sum)f(o) n(v)n(er)f Fq(T)41 b Ft(is)31 b(empt)n(y)-7 b(,)31 b(but)g(w)n(e)f (shall)g(still)h(use)f(equation)g(\(3.74\),)g(b)n(y)g(in)n(terpreting)0 1129 y(the)e(r.h.s.)36 b(as)27 b(1,)h(if)g Fq(P)684 1141 y Fp(1)749 1129 y Ft(is)f(empt)n(y)h(\(whic)n(h)g(is)f(p)r(ossible,)h (for)f Fq(s)c Ft(=)f(1\),)28 b(and)f(as)g(det)14 b Fq(G)2633 1099 y Fn(h)2677 1129 y Ft(\()p Fo(1)p Ft(\))28 b(otherwise.)71 1349 y(Inserting)f(\(3.74\))g(in)g(the)h(r.h.s.)37 b(of)27 b(\(3.38\))g(\(with)i Fq(v)d Ft(=)d Fq(v)1867 1361 y Fp(0)1904 1349 y Ft(\))28 b(w)n(e)f(obtain,)h(up)g(to)f(a)g(sign,)215 1585 y Fm(R)p Fq(V)352 1551 y Fp(\()p Fn(h)p Fp(\))447 1585 y Ft(\()p Fq(\034)5 b(;)14 b Fo(P)p Fq(;)g Fo(r)p Ft(\))24 b(=)897 1529 y(1)p 852 1566 132 4 v 852 1642 a Fq(s)891 1654 y Fn(v)924 1662 y Fj(0)961 1642 y Ft(!)994 1508 y Fl(p)p 1077 1508 100 4 v 77 x Fq(Z)1134 1597 y Fn(h)1176 1522 y Fu(j)p Fn(P)1238 1530 y Fh(v)1268 1542 y Fj(0)1305 1522 y Fu(j)1342 1506 y Fl(X)1350 1684 y Fn(T)1389 1692 y Fh(v)1419 1704 y Fj(0)1476 1472 y Fl(Z)1573 1585 y Fq(d)p Fo(x)1666 1597 y Fn(v)1699 1605 y Fj(0)1750 1472 y Fl(Z)1847 1585 y Fq(dP)1943 1597 y Fn(T)1982 1605 y Fh(v)2012 1617 y Fj(0)2054 1585 y Ft(\()p Fo(t)p Ft(\)[)p Fm(R)2266 1563 y Ft(~)2248 1585 y Fq( )2305 1551 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))2453 1585 y Ft(\()p Fq(P)2538 1597 y Fn(v)2571 1605 y Fj(0)2608 1585 y Ft(\)])f Fm(\001)233 1902 y(\001)298 1810 y Fl(h)384 1823 y(Y)351 2002 y Fn(l)p Fu(2)p Fn(T)456 2010 y Fh(v)486 2022 y Fj(0)547 1880 y Ft(\026)537 1902 y Fq(@)586 1864 y Fn(m)645 1873 y Fh(l)581 1924 y Fp(1)673 1902 y Fq(g)716 1859 y Fp(\()p Fn(h)p Fp(+1\))713 1941 y Fn(!)757 1914 y Fg(\000)755 1963 y Fh(l)805 1941 y Fn(;!)869 1914 y Fj(+)867 1963 y Fh(l)920 1902 y Ft(\()p Fo(x)1002 1914 y Fn(l)1046 1902 y Fm(\000)18 b Fo(y)1179 1914 y Fn(l)1205 1902 y Ft(\))1237 1810 y Fl(i)1291 1902 y Ft(det)c Fq(G)1485 1868 y Fn(h)p Fp(+1)p Fn(;T)1667 1876 y Fh(v)1697 1888 y Fj(0)1738 1902 y Ft(\()p Fo(t)p Ft(\))1839 1776 y Fl(r)p 1922 1776 204 4 v 1932 1846 a Fq(Z)1989 1858 y Fn(h)p Fp(+1)p 1932 1883 184 4 v 1974 1959 a Fq(Z)2031 1971 y Fn(h)2126 1790 y Fu(j)p Fn(P)2188 1798 y Fh(v)2218 1810 y Fj(0)2255 1790 y Fu(j)2297 1785 y Fn(s)2328 1793 y Fh(v)2358 1805 y Fj(0)2293 1823 y Fl(Y)2292 2000 y Fn(i)p Fp(=1)2400 1902 y Ft([)p Fq(K)2500 1868 y Fp(\()p Fn(h)p Fp(+2\))2494 1922 y Fn(v)2527 1930 y Fh(i)2678 1902 y Ft(\()p Fo(x)2760 1914 y Fn(v)2793 1922 y Fh(i)2825 1902 y Ft(\)])3095 1772 y(\(3)p Fq(:)p Ft(75\))71 2195 y(Let)35 b(us)g(no)n(w)g(consider)f(the)i(con)n(tribution)e(to)h(the)h (r.h.s.)59 b(of)35 b(\(3.75\))g(of)g(one)g(of)g(the)g(terms)g(in)h(the) 0 2344 y(represen)n(tation)i(\(3.69\))i(of)g Fm(R)1002 2322 y Ft(~)985 2344 y Fq( )1042 2314 y Fp(\()p Fu(\024)p Fn(h)p Fp(\))1189 2344 y Ft(\()p Fq(P)1274 2356 y Fn(v)1307 2364 y Fj(0)1344 2344 y Ft(\))h(with)g Fq(n)1669 2356 y Fp(1)1706 2344 y Ft(\()p Fq(\013)p Ft(\))27 b(+)g Fq(n)1992 2356 y Fp(2)2029 2344 y Ft(\()p Fq(\013)p Ft(\))45 b Fq(>)e Ft(0.)74 b(F)-7 b(or)40 b(eac)n(h)f(c)n(hoice)h(of)g Fq(T)3215 2356 y Fn(v)3248 2364 y Fj(0)3284 2344 y Ft(,)0 2494 y(w)n(e)31 b(decomp)r(ose)f(the)i(factors)e Fq(d)1011 2451 y Fn(n)1052 2459 y Fj(1)1085 2451 y Fp(\()p Fn(\013)p Fp(\))1011 2518 y Fn(L)1184 2494 y Ft(\()p Fq(y)24 b Fm(\000)c Fq(x)p Ft(\))32 b(and)g Fq(d)1686 2451 y Fn(n)1727 2459 y Fj(2)1759 2451 y Fp(\()p Fn(\013)p Fp(\))1686 2519 y Fn(\014)1858 2494 y Ft(\()p Fq(y)1931 2506 y Fp(0)1990 2494 y Fm(\000)20 b Fq(x)2122 2506 y Fp(0)2160 2494 y Ft(\),)32 b(b)n(y)f(using)g(equation)g(\(3.55\))f(and)0 2643 y(the)37 b(analogous)e(equation)h(for)h Fq(d)1073 2655 y Fn(\014)1118 2643 y Ft(\()p Fq(y)1191 2655 y Fp(0)1253 2643 y Fm(\000)24 b Fq(x)1389 2655 y Fp(0)1427 2643 y Ft(\),)39 b(with)f Fo(x)1770 2655 y Fp(0)1846 2643 y Ft(=)g Fo(x)p Ft(,)i Fo(x)2112 2655 y Fn(n)2196 2643 y Ft(=)f Fo(y)f Ft(and)f(the)h(other)e(p)r(oin)n(ts)h Fo(x)3247 2655 y Fn(r)3284 2643 y Ft(,)0 2793 y Fq(r)26 b Ft(=)c(1)p Fq(;)14 b(:)g(:)g(:)g(;)g(n)k Fm(\000)g Ft(1,)27 b(c)n(hosen)g(in)h(the)g(follo)n(wing)e(w)n(a)n(y)-7 b(.)71 2942 y(Let)38 b(us)g(consider)f(the)i(unique)f(subset)g(\()p Fq(l)1437 2954 y Fp(1)1475 2942 y Fq(;)14 b(:)g(:)g(:)f(;)h(l)1684 2954 y Fn(m)1747 2942 y Ft(\))39 b(of)f Fq(T)1972 2954 y Fn(v)2005 2962 y Fj(0)2041 2942 y Ft(,)j(whic)n(h)d(selects)g(a)g (path)g(joining)g(the)0 3091 y(cluster)32 b(con)n(taining)f Fo(x)731 3103 y Fp(0)801 3091 y Ft(with)i(the)g(cluster)f(con)n (taining)f Fo(x)1874 3103 y Fn(n)1920 3091 y Ft(,)j(if)f(one)f(iden)n (ti\014es)g(all)g(the)h(p)r(oin)n(ts)f(in)h(the)0 3241 y(same)h(cluster.)59 b(Let)35 b(\()s(\026)-45 b Fq(v)766 3253 y Fn(i)p Fu(\000)p Fp(1)879 3241 y Fq(;)17 b Ft(\026)-45 b 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y Fq(C)1749 1930 y Fl(P)1837 1951 y Fh(s)1865 1959 y(v)1837 2018 y(i)p Fj(=1)1945 1986 y Fu(j)p Fn(P)2007 1994 y Fh(v)2037 2007 y(i)2068 1986 y Fu(j\000j)p Fn(P)2202 1994 y Fh(v)2237 1986 y Fu(j\000)p Fp(2\()p Fn(s)2399 1994 y Fh(v)2435 1986 y Fu(\000)p Fp(1\))2573 2027 y Fm(\001)529 2235 y(\001)42 b Fq(\015)651 2167 y Fh(h)685 2175 y(v)p 651 2182 70 4 v 672 2215 a Fj(2)731 2201 y Ft(\()764 2139 y Fl(P)851 2160 y Fh(s)879 2168 y(v)851 2226 y(i)p Fj(=1)960 2195 y Fu(j)p Fn(P)1022 2203 y Fh(v)1052 2216 y(i)1082 2195 y Fu(j\000j)p Fn(P)1216 2203 y Fh(v)1252 2195 y Fu(j\000)p Fp(2\()p Fn(s)1414 2203 y Fh(v)1449 2195 y Fu(\000)p Fp(1\))1560 2201 y Ft(\))1596 2235 y Fq(\015)1644 2195 y Fn(h)1683 2203 y Fh(v)1730 2139 y Fl(P)1818 2160 y Fh(s)1846 2168 y(v)1818 2226 y(i)p Fj(=1)1915 2201 y Ft([)1938 2195 y Fn(q)1968 2203 y Fh(\013)2010 2195 y Fp(\()p Fn(P)2078 2203 y Fh(v)2108 2216 y(i)2139 2195 y Fu(n)p Fn(Q)2225 2203 y Fh(v)2255 2216 y(i)2286 2195 y Fp(\)+)p Fn(m)p Fp(\()p Fn(P)2490 2203 y Fh(v)2520 2216 y(i)2550 2195 y Fu(n)p Fn(Q)2636 2203 y Fh(v)2666 2216 y(i)2697 2195 y Fp(\))2723 2201 y Ft(])2773 2235 y Fm(\001)529 2410 y(\001)g Fq(\015)642 2361 y Fu(\000)p Fn(h)733 2369 y Fh(v)779 2305 y Fl(P)867 2392 y Fh(l)p Fg(2)p Fh(T)961 2400 y(v)1000 2367 y Ft([)1023 2361 y Fn(q)1053 2369 y Fh(\013)1096 2361 y Fp(\()p Fn(f)1161 2334 y Fj(+)1154 2383 y Fh(l)1208 2361 y Fp(\)+)p Fn(q)1315 2369 y Fh(\013)1357 2361 y Fp(\()p Fn(f)1422 2334 y Fg(\000)1415 2383 y Fh(l)1471 2361 y Fp(\)+)p Fn(m)p Fp(\()p Fn(f)1672 2334 y Fj(+)1665 2383 y Fh(l)1718 2361 y Fp(\)+)p Fn(m)p Fp(\()p Fn(f)1919 2334 y Fg(\000)1912 2383 y Fh(l)1968 2361 y Fp(\))1994 2367 y Ft(])2044 2410 y Fq(:)3095 2182 y Ft(\(3)p Fq(:)p Ft(94\))71 2534 y(In)33 b(order)e(to)i(pro)n(v)n(e)e (this)i(inequalit)n(y)-7 b(,)33 b(w)n(e)g(shall)f(supp)r(ose,)h(for)g (simplicit)n(y)-7 b(,)34 b(that)f(all)f(the)h(op)r(erators)11 2662 y(^)0 2684 y Fq(@)49 2641 y Fn(q)r Fp(\()p Fn(f)7 b Fp(\))44 2712 y Fn(j)s Fp(\()p Fn(f)g Fp(\))205 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Fq(!)e Ft(=)c Fm(\000)p Ft(1.)3095 5428 y(\(3)p Fq(:)p Ft(98\))0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(52)p eop %%Page: 53 53 53 52 bop 71 83 a Ft(Let)31 b(us)f(no)n(w)g(de\014ne)h Fq(n)801 95 y Fp(+)884 83 y Ft(=)c Fm(j)1018 62 y Ft(~)999 83 y Fq(P)1064 53 y Fu(\000)p Fn(;)p Fp(+)1191 83 y Fm(j)p Ft(,)32 b Fq(m)1342 95 y Fp(+)1425 83 y Ft(=)27 b Fm(j)1559 62 y Ft(~)1540 83 y Fq(P)1605 53 y Fp(+)p Fn(;)p Fp(+)1731 83 y Fm(j)p Ft(,)k Fq(m)d Ft(=)g Fm(j)2043 62 y Ft(~)2025 83 y Fq(P)2090 53 y Fu(\000)p Fn(;)p Fp(+)2216 83 y Fm(j)21 b Ft(+)f Fm(j)2387 62 y Ft(~)2368 83 y Fq(P)2433 53 y Fu(\000)p Fn(;)p Fu(\000)2560 83 y Fm(j)29 b Ft(=)e Fm(j)2746 62 y Ft(~)2727 83 y Fq(P)2792 53 y Fp(+)p Fn(;)p Fp(+)2918 83 y Fm(j)20 b Ft(+)g Fm(j)3088 62 y Ft(~)3069 83 y Fq(P)3134 53 y Fp(+)p Fn(;)p Fu(\000)3261 83 y Fm(j)p Ft(;)0 232 y(b)n(y)26 b(using)h(\(3.95\))f(and)g(\(3.98\),)h(it)g(is)g(easy)e(to)i (see,)g(b)n(y)f(pro)r(ceeding)g(as)g(in)h Fm(x)p Ft(2.7,)f(that,)h(if)g 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545 y Fu(\000)p Fn(n)2940 553 y Fj(+)3028 644 y Fq(:)44 b Ft(\(3)p Fq(:)p Ft(99\))0 894 y(Since)20 b(2)p Fq(m)i Ft(=)434 832 y Fl(P)521 853 y Fn(s)552 861 y Fh(v)521 919 y Fn(i)p Fp(=1)647 894 y Fm(j)p Fq(P)723 906 y Fn(v)756 914 y Fh(i)787 894 y Fm(j)r(\000)r(j)p Fq(P)955 906 y Fn(v)995 894 y Fm(j)r(\000)r Ft(2\()p Fq(s)1200 906 y Fn(v)1242 894 y Fm(\000)r Ft(1\))e(and)1556 832 y Fl(P)1644 853 y Fn(s)1675 861 y Fh(v)1644 919 y Fn(i)p Fp(=1)1769 894 y Fq(q)1806 906 y Fn(\013)1854 894 y Ft(\()p Fq(P)1939 906 y Fn(v)1972 914 y Fh(i)2003 894 y Fq(=Q)2111 906 y Fn(v)2144 914 y Fh(i)2174 894 y Ft(\))r Fm(\000)2275 832 y Fl(P)2363 919 y Fn(l)p Fu(2)p Fn(T)2468 927 y Fh(v)2508 894 y Ft([)p Fq(q)2568 906 y Fn(\013)2616 894 y Ft(\()p Fq(f)2698 859 y Fp(+)2689 919 y Fn(l)2753 894 y Ft(\))r(+)r Fq(q)2891 906 y Fn(\013)2939 894 y Ft(\()p Fq(f)3021 859 y Fu(\000)3012 919 y Fn(l)3076 894 y Ft(\)])k(=)f(0,)0 1044 y(w)n(e)d(get)g(the)g(inequalit)n(y)g (\(3.94\),)h(if)g Fq(m)1158 1056 y Fp(+)1236 1044 y Fm(\025)i Fq(n)1374 1056 y Fp(+)1429 1044 y Ft(,)e(b)n(y)f(using)g(\(2.116\).)33 b(The)21 b(case)e Fq(m)2507 1056 y Fp(+)2585 1044 y Fq(<)k(n)2723 1056 y Fp(+)2798 1044 y Ft(can)d(b)r(e)h(treated)0 1193 y(in)28 b(a)f(similar)g(w)n(a)n(y)-7 b(,)26 b(b)n(y)i(exc)n(hanging)e (the)i(de\014nitions)f(of)h Fq(A)1869 1150 y Fp(\()p Fn(h)p Fp(\))1869 1204 y Fk(x)p Fn(;!)1977 1193 y Ft(\()p Fo(k)2059 1163 y Fu(0)2083 1193 y Ft(\))g(and)f Fq(B)2371 1150 y Fp(\()p Fn(h)p Fp(\))2367 1204 y Fk(x)p Fn(;!)2474 1193 y Ft(\()p Fo(k)2556 1163 y Fu(0)2580 1193 y Ft(\).)0 1414 y Fo(3.14)97 b Ft(Pro)r(of)27 b(of)g(Theorem)g(3.12.)71 1563 y(By)g(using)g(\(3.81\))g(and)h(\(3.94\))f(w)n(e)g(get)169 1689 y Fl(Z)266 1802 y Fq(d)p Fo(x)359 1814 y Fn(v)392 1822 y Fj(0)430 1802 y Fm(j)p Fq(W)531 1814 y Fn(\034)s(;)p Fk(P)p Fn(;)p Fk(r)p Fn(;T)5 b(;\013)818 1802 y Ft(\()p Fo(x)900 1814 y Fn(v)933 1822 y Fj(0)970 1802 y Ft(\))p Fm(j)24 b(\024)e Fq(C)1201 1768 y Fn(n)1247 1802 y Fq(J)1293 1814 y Fn(\034)s(;)p Fk(P)p Fn(;)p Fk(r)p Fn(;T)5 b(;\013)1681 1723 y Fl(Y)1593 1902 y Fn(v)14 b Fi(not)24 b(e.p.)1889 1710 y Fl(n\020)1994 1802 y Fq(Z)2051 1814 y Fn(h)2090 1822 y Fh(v)2129 1802 y Fq(=)-5 b(Z)2223 1814 y Fn(h)2262 1822 y Fh(v)2297 1814 y Fu(\000)p Fp(1)2386 1710 y Fl(\021)2436 1727 y Fu(j)p Fn(P)2498 1735 y Fh(v)2533 1727 y Fu(j)p Fn(=)p Fp(2)2624 1802 y Fm(\001)174 2100 y(\001)18 b Fq(C)280 2004 y Fl(P)368 2025 y Fh(s)396 2033 y(v)368 2091 y(i)p Fj(=1)477 2060 y Fu(j)p Fn(P)539 2068 y Fh(v)569 2081 y(i)599 2060 y Fu(j\000j)p Fn(P)733 2068 y Fh(v)769 2060 y Fu(j\000)p Fp(2\()p Fn(s)931 2068 y Fh(v)966 2060 y Fu(\000)p Fp(1\))1095 1983 y Fl(\022)1166 2044 y Fm(j)p Fq(\033)1236 2056 y Fn(h)1275 2064 y Fh(v)1315 2044 y Fm(j)p 1166 2081 V 1189 2157 a Fq(\015)1237 2133 y Fn(h)1276 2141 y Fh(v)1348 1983 y Fl(\023)1409 2000 y Fn(\032)p Fp(\()p Fn(T)1508 2008 y Fh(v)1545 2000 y Fp(\))1589 2100 y Fq(\015)1646 2032 y Fh(h)1680 2040 y(v)p 1646 2047 70 4 v 1667 2080 a Fj(2)1726 2066 y Ft(\()1758 2004 y 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3816 y Fh(v)1270 3808 y Fp(\()p Fn(s)1327 3816 y Fh(v)1363 3808 y Fu(\000)p Fp(1\))1478 3843 y Fq(\015)1526 3794 y Fn(h)1565 3802 y Fh(v)1612 3738 y Fl(P)1699 3825 y Fh(l)p Fg(2)p Fh(T)1793 3833 y(v)1833 3800 y Ft([)1856 3794 y Fn(q)1886 3802 y Fh(\013)1928 3794 y Fp(\()p Fn(f)1993 3767 y Fj(+)1986 3816 y Fh(l)2040 3794 y Fp(\)+)p Fn(q)2147 3802 y Fh(\013)2189 3794 y Fp(\()p Fn(f)2254 3767 y Fg(\000)2247 3816 y Fh(l)2303 3794 y Fp(\)+)p Fn(m)p Fp(\()p Fn(f)2504 3767 y Fj(+)2497 3816 y Fh(l)2551 3794 y Fp(\)+)p Fn(m)p Fp(\()p Fn(f)2752 3767 y Fg(\000)2745 3816 y Fh(l)2801 3794 y Fp(\))2827 3800 y Ft(])2854 3750 y Fl(i)2916 3843 y Fq(;)3053 3705 y Ft(\(3)p Fq(:)p Ft(102\))0 4064 y(where)25 b Fq(n)288 4076 y Fn(\027)329 4064 y Ft(\()p Fq(v)s Ft(\))h(is)f(the)h(n)n(um)n(b) r(er)f(of)g(v)n(ertices)g(of)g(t)n(yp)r(e)g Fq(\027)31 b Ft(with)26 b(scale)e Fq(h)2154 4076 y Fn(v)2207 4064 y Ft(+)14 b(1)25 b(and)2528 4043 y(\026)2512 4064 y Fq(T)2561 4076 y Fn(v)2625 4064 y Ft(is)h(the)f(subset)h(of)f(the)0 4214 y(lines)j(of)g Fq(T)334 4226 y Fn(v)401 4214 y Ft(corresp)r (onding)e(to)i Fr(non)i(diagonal)g Ft(propagators,)25 b(that)j(is)g(propagators)d(with)j(di\013eren)n(t)g Fq(!)0 4363 y Ft(indices.)71 4513 y(It)g(is)f(easy)g(to)g(see)h(that)670 4687 y Fl(X)581 4867 y Fn(v)i Fi(not)25 b(e.p.)893 4766 y Fq(h)941 4778 y Fn(v)1021 4662 y(s)1052 4670 y Fh(v)994 4687 y Fl(X)1001 4864 y Fn(i)p Fp(=1)1128 4766 y Fq(q)1165 4778 y Fn(\013)1213 4766 y Ft(\()p Fq(P)1298 4778 y Fn(v)1331 4786 y Fh(i)1362 4766 y Fm(n)p Fq(Q)1470 4778 y Fn(v)1503 4786 y Fh(i)1533 4766 y Ft(\))18 b(+)g Fq(h)28 b(q)1779 4778 y Fn(\013)1826 4766 y Ft(\()p Fq(P)1911 4778 y Fn(v)1944 4786 y Fj(0)1982 4766 y Ft(\))23 b(=)2154 4687 y Fl(X)2125 4866 y Fn(f)7 b Fu(2)p Fn(I)2238 4874 y Fh(v)2268 4886 y Fj(0)2318 4766 y Fq(h)2366 4778 y Fn(\013)2413 4766 y Ft(\()p Fq(f)i Ft(\))p Fq(q)2564 4778 y Fn(\013)2612 4766 y Ft(\()p Fq(f)g Ft(\))327 b(\(3)p Fq(:)p Ft(103\))0 5057 y(and,)27 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Ft(,)h(there)f(are)f(essen)n(tially)g(t)n(w)n(o)h(di\013eren)n(t)g(p)r (ossibilities)g(for)g(the)h(op)r(erators)3192 61 y(~)3182 83 y Fq(@)3231 43 y Fn(q)3261 51 y Fh(\013)3226 106 y Fn(j)3253 114 y Fh(\013)0 232 y Ft(and)34 b(the)h(space-time)f(p)r(oin) n(ts)h Fo(x)1043 202 y Fu(0)1043 256 y Fn(l)1069 232 y Ft(\()p Fq(t)1131 244 y Fn(l)1157 232 y Ft(\),)h Fo(y)1299 202 y Fu(0)1298 256 y Fn(l)1324 232 y Ft(\()p Fq(s)1395 244 y Fn(l)1421 232 y Ft(\).)58 b(Let)35 b(us)g(consider,)g(for)f (example,)i Fq(f)2700 197 y Fu(\000)2691 258 y Fn(l)2756 232 y Ft(;)i(then)d(the)g(\014rst)0 382 y(p)r(ossibilit)n(y)27 b(is)h(that)669 360 y(~)659 382 y Fq(@)708 342 y Fn(q)738 350 y Fh(\013)703 405 y Fn(j)730 413 y Fh(\013)812 382 y Ft(is)f(a)g(deriv)-5 b(ativ)n(e)27 b(of)h(order)e Fq(q)1694 394 y Fn(\013)1769 382 y Ft(and)851 610 y Fo(x)901 575 y Fu(0)901 630 y Fn(l)927 610 y Ft(\()p Fq(t)989 622 y Fn(l)1014 610 y Ft(\))e(=)1162 609 y(~)1157 610 y Fo(x)1207 622 y Fn(l)1252 610 y Ft(+)18 b Fq(t)1365 622 y Fn(l)1390 610 y Ft(\()1426 609 y(\026)1422 610 y Fo(x)1472 622 y Fn(l)1517 610 y Fm(\000)1604 609 y Ft(~)1600 610 y Fo(x)1650 622 y Fn(l)1676 610 y Ft(\))23 b Fq(;)14 b Ft(for)27 b(some)2107 609 y(~)2103 610 y Fo(x)2153 622 y Fn(l)2202 610 y Fm(2)c Fo(x)2330 575 y Fp(\()p Fn(i)p Fp(\))2433 610 y Fq(;)597 b Ft(\(3)p Fq(:)p Ft(115\))4 836 y(\026)0 837 y Fo(x)50 849 y Fn(l)108 837 y Ft(b)r(eing)31 b(de\014ned)h(in)g(terms)f(of)h Fo(x)1110 849 y Fn(l)1168 837 y Ft(as)k(\026)-48 b Fq(y)35 b Ft(is)c(de\014ned)h(in)g(terms)f(of) h Fq(y)i Ft(in)e Fm(x)p Ft(3.5)f(\(that)h(is)2820 836 y(\026)2816 837 y Fo(x)2866 849 y Fn(l)2923 837 y Ft(and)g Fo(x)3139 849 y Fn(l)3196 837 y Ft(are)0 987 y(equiv)-5 b(alen)n(t)23 b(represen)n(tation)f(of)i(the)g(same)f(p)r(oin)n(t)h(on) f(the)h(space-time)f(torus\).)35 b(The)24 b(second)e(p)r(ossibilit)n(y) 0 1136 y(is)31 b(that)281 1114 y(~)270 1136 y Fq(@)319 1096 y Fn(q)349 1104 y Fh(\013)314 1159 y Fn(j)341 1167 y Fh(\013)426 1136 y Ft(is)g(a)g(lo)r(cal)f(op)r(erator)f(of)i(the)g (form)g Fq(L)1624 1106 y Fu(\000)p Fn(n)1717 1114 y Fj(1)1753 1136 y Fq(\014)1804 1106 y Fu(\000)p Fn(n)1897 1114 y Fj(2)19 b Ft(\026)1934 1136 y Fq(@)1983 1099 y Fn(n)2024 1107 y Fj(3)1978 1158 y Fp(1)2060 1136 y Fq(@)2109 1099 y Fn(n)2150 1107 y Fj(4)2104 1158 y Fp(0)2186 1136 y Ft(,)32 b(with)g Fq(q)2471 1148 y Fn(\013)2547 1136 y Fm(\024)2640 1074 y Fl(P)2727 1095 y Fp(4)2727 1161 y Fn(i)p Fp(=1)2853 1136 y Fq(n)2903 1148 y Fn(i)2959 1136 y Fm(\024)c Fq(q)3089 1148 y Fn(\013)3157 1136 y Ft(+)21 b(1,)0 1286 y(and)27 b Fo(x)211 1256 y Fu(0)211 1309 y Fn(l)237 1286 y Ft(\()p Fq(t)299 1298 y Fn(l)325 1286 y Ft(\))c(=)472 1285 y(~)468 1286 y Fo(x)518 1298 y Fn(l)567 1286 y Fm(2)g Fo(x)695 1256 y Fp(\()p Fn(i)p Fp(\))775 1286 y Ft(.)37 b(Note)28 b(that,)g(b)n(y)f(\(2.40\),)g Fq(L)1674 1256 y Fu(\000)p Fn(n)1767 1264 y Fj(1)1803 1286 y Fq(\014)1854 1256 y Fu(\000)p Fn(n)1947 1264 y Fj(2)2007 1286 y Fm(\024)c Fq(\015)2143 1256 y Fn(h)2182 1265 y Fh(L;\014)2279 1256 y Fp(\()p Fn(n)2346 1264 y Fj(1)2378 1256 y Fp(+)p Fn(n)2470 1264 y Fj(2)2503 1256 y Fp(\))2556 1286 y Fm(\024)f Fq(\015)2691 1256 y Fn(h)2730 1264 y Fh(v)2765 1256 y Fp(\()p Fn(n)2832 1264 y Fj(1)2865 1256 y Fp(+)p Fn(n)2957 1264 y Fj(2)2989 1256 y Fp(\))3019 1286 y Ft(.)71 1435 y(By)33 b(pro)r(ceeding)f(as)h(in)h(the)f(pro)r(of) g(of)g(lemma)h(\(2.6\))f(and)g(using)g(\(2.105\))f(it)i(is)f(v)n(ery)f (easy)h(to)g(sho)n(w)0 1585 y(that,)28 b(for)f(an)n(y)g Fq(N)32 b(>)22 b Ft(1,)353 1687 y Fl(\014)353 1737 y(\014)353 1787 y(\014)353 1837 y(\014)391 1786 y Ft(~)381 1808 y Fq(@)430 1757 y Fn(q)460 1765 y Fh(\013)502 1757 y Fp(\()p Fn(f)567 1730 y Fg(\000)560 1779 y Fh(l)616 1757 y Fp(\))425 1847 y Fn(j)452 1855 y Fh(\013)494 1847 y Fp(\()p Fn(f)559 1820 y Fg(\000)552 1869 y Fh(l)608 1847 y Fp(\))656 1786 y Ft(~)646 1808 y Fq(@)695 1757 y Fn(q)725 1765 y Fh(\013)767 1757 y Fp(\()p Fn(f)832 1730 y Fj(+)825 1779 y Fh(l)879 1757 y Fp(\))690 1847 y Fn(j)717 1855 y Fh(\013)760 1847 y Fp(\()p Fn(f)825 1820 y Fj(+)818 1869 y Fh(l)871 1847 y Fp(\))909 1808 y Ft([)p Fq(d)975 1765 y Fn(b)1004 1773 y Fh(\013)1046 1765 y Fp(\()p Fn(l)p Fp(\))975 1836 y Fn(j)1002 1844 y Fh(\013)1045 1836 y Fp(\()p Fn(l)p Fp(\))1124 1808 y Ft(\()p Fo(x)1206 1773 y Fu(0)1206 1828 y Fn(l)1232 1808 y Ft(\()p Fq(t)1294 1820 y Fn(l)1320 1808 y Ft(\))p Fq(;)14 b Fo(y)1440 1773 y Fu(0)1439 1828 y Fn(l)1465 1808 y Ft(\()p Fq(s)1536 1820 y Fn(l)1561 1808 y Ft(\)\))1636 1786 y(\026)1625 1808 y Fq(@)1674 1770 y Fn(m)1733 1779 y Fh(l)1669 1830 y Fp(1)1762 1808 y Fq(g)1805 1765 y Fp(\()p Fn(h)1870 1773 y Fh(v)1905 1765 y Fp(\))1802 1847 y Fn(!)1846 1820 y Fg(\000)1844 1869 y Fh(l)1894 1847 y Fn(;!)1958 1820 y Fj(+)1956 1869 y Fh(l)2009 1808 y Ft(\()p Fo(x)2091 1773 y Fu(0)2091 1828 y Fn(l)2117 1808 y Ft(\()p Fq(t)2179 1820 y Fn(l)2205 1808 y Ft(\))19 b Fm(\000)f Fo(y)2390 1773 y Fu(0)2389 1828 y Fn(l)2415 1808 y Ft(\()p Fq(s)2486 1820 y Fn(l)2511 1808 y Ft(\)\)])2598 1687 y Fl(\014)2598 1737 y(\014)2598 1787 y(\014)2598 1837 y(\014)2650 1808 y Fm(\024)362 2062 y(\024)23 b Fq(C)525 2005 y(\015)573 1974 y Fn(h)612 1982 y Fh(v)647 1974 y Fp([1+)p Fn(q)780 1982 y Fh(\013)823 1974 y Fp(\()p Fn(f)888 1948 y Fj(+)881 1997 y Fh(l)934 1974 y Fp(\)+)p Fn(q)1041 1982 y Fh(\013)1084 1974 y Fp(\()p Fn(f)1149 1948 y Fg(\000)1142 1997 y Fh(l)1198 1974 y Fp(\)+)p Fn(m)p Fp(\()p Fn(f)1399 1948 y Fg(\000)1392 1997 y Fh(l)1447 1974 y Fp(\)+)p Fn(m)p Fp(\()p Fn(f)1648 1948 y Fj(+)1641 1997 y Fh(l)1695 1974 y Fp(\))p Fu(\000)p Fn(b)1802 1982 y Fh(\013)1844 1974 y Fp(\()p Fn(l)p Fp(\)])p 525 2043 1415 4 v 710 2119 a Ft(1)18 b(+)g([)p Fq(\015)924 2095 y Fn(h)963 2103 y Fh(v)1003 2119 y Fm(j)p Fo(d)p Ft(\()p Fo(x)1161 2090 y Fu(0)1161 2144 y Fn(l)1187 2119 y Ft(\()p Fq(t)1249 2131 y Fn(l)1275 2119 y Ft(\))g Fm(\000)h Fo(y)1460 2090 y Fu(0)1459 2144 y Fn(l)1484 2119 y Ft(\()p Fq(s)1555 2131 y Fn(l)1581 2119 y Ft(\)\))p Fm(j)p Ft(])1691 2095 y Fn(N)1950 1969 y Fl(\020)2010 2005 y Fm(j)p Fq(\033)2080 2017 y Fn(h)2119 2025 y Fh(v)2159 2005 y Fm(j)p 2010 2043 173 4 v 2033 2119 a Fq(\015)2081 2095 y Fn(h)2120 2103 y Fh(v)2192 1969 y Fl(\021)2241 1987 y Fn(\032)2275 1996 y Fh(l)2327 2062 y Fq(;)3053 1936 y Ft(\(3)p Fq(:)p Ft(116\))0 2302 y(where)32 b Fo(d)p Ft(\()p Fo(x)p Ft(\))h(is)f (de\014ned)g(in)h(\(2.97\))e(and)h Fq(\032)1378 2314 y Fn(l)1434 2302 y Ft(=)e(1)i(if)g Fq(!)s Ft(\()p Fq(f)1820 2266 y Fu(\000)1811 2327 y Fn(l)1876 2302 y Ft(\))f Fm(6)p Ft(=)f Fq(!)s Ft(\()p Fq(f)2171 2266 y Fp(+)2162 2327 y Fn(l)2226 2302 y Ft(\),)j Fq(\032)2357 2314 y Fn(l)2413 2302 y Ft(=)d(0)i(otherwise.)50 b(W)-7 b(e)32 b(used)0 2451 y(here)38 b(the)h(fact)g(that,)j(if)e Fq(h)873 2463 y Fn(v)954 2451 y Ft(=)h(+1,)g(then)e Fq(q)1468 2463 y Fn(\013)1516 2451 y Ft(\()p Fq(f)1598 2416 y Fu(\000)1589 2476 y Fn(l)1653 2451 y Ft(\))j(=)g Fq(q)1871 2463 y Fn(\013)1918 2451 y Ft(\()p Fq(f)2000 2416 y Fp(+)1991 2476 y Fn(l)2055 2451 y Ft(\))g(=)f(0,)h(whic)n(h)c(allo)n(ws)g(to)g(a) n(v)n(oid)g(the)0 2601 y(problems)25 b(connected)h(with)g(the)h (singularit)n(y)d(of)i(the)g(time)h(deriv)-5 b(ativ)n(es)24 b(of)i(the)h(scale)e(1)g(propagator)e(at)0 2750 y Fq(x)47 2720 y Fu(0)47 2774 y Fn(l;)p Fp(0)126 2750 y Ft(\()p Fq(t)188 2762 y Fn(l)213 2750 y Ft(\))c Fm(\000)f Fq(y)391 2720 y Fu(0)388 2774 y Fn(l;)p Fp(0)466 2750 y Ft(\()p Fq(s)537 2762 y Fn(l)563 2750 y Ft(\))23 b(=)g(0.)71 2900 y(Let)31 b(us)g(no)n(w)g(consider)f(the)h(con)n(tribution)f(of)i (the)f(endp)r(oin)n(ts)g(to)g(the)h(r.h.s.)46 b(of)31 b(\(3.101\))f(and)h(recall)0 3049 y(\(see)h Fm(x)p Ft(3.10\))g(that)g Fq(T)654 3061 y Fn(v)689 3041 y Fg(\003)687 3082 y Fh(i)761 3049 y Ft(is)g(empt)n(y)-7 b(,)34 b(if)f Fm(j)p Fo(x)1279 3061 y Fn(v)1314 3041 y Fg(\003)1312 3082 y Fh(i)1354 3049 y Fm(j)e Ft(=)g(1,)i(hence)f Fq(b)1873 3061 y Fn(\013)1920 3049 y Ft(\()p Fq(v)1995 3019 y Fu(\003)1992 3071 y Fn(i)2034 3049 y Ft(\))g(=)e(0,)k(while,)g(if)f Fo(x)2669 3061 y Fn(v)2704 3041 y Fg(\003)2702 3082 y Fh(i)2774 3049 y Ft(=)e(\()p Fo(x)2952 3061 y Fn(i)2980 3049 y Fq(;)14 b Fo(y)3067 3061 y Fn(i)3095 3049 y Ft(\),)34 b Fq(T)3233 3061 y Fn(v)3268 3041 y Fg(\003)3266 3082 y Fh(i)0 3198 y Ft(con)n(tains)28 b(the)h(line)g Fq(l)654 3210 y Fn(i)710 3198 y Ft(connecting)f Fo(x)1176 3210 y Fn(i)1233 3198 y Ft(with)h Fo(y)1473 3210 y Fn(i)1530 3198 y Ft(and)f Fq(h)1740 3210 y Fn(v)1775 3191 y Fg(\003)1773 3232 y Fh(i)1839 3198 y Ft(=)c(2.)40 b(By)28 b(using)g(\(3.33\))g(and)h (\(3.39\),)f(w)n(e)g(get,)0 3348 y(if)g Fq(h)124 3360 y Fn(i)175 3348 y Fm(\021)22 b Fq(h)310 3360 y Fn(v)345 3340 y Fg(\003)343 3381 y Fh(i)412 3348 y Ft(and)27 b Fq(S)624 3360 y Fn(\027)689 3348 y Fm(\021)22 b(f)p Fq(i)g Ft(:)i Fq(v)959 3318 y Fu(\003)956 3369 y Fn(i)1025 3348 y Ft(is)j(of)h(t)n(yp)r(e)f Fq(\027)5 b Fm(g)p Ft(,)780 3468 y Fl(\014)780 3518 y(\014)780 3567 y(\014)780 3617 y(\014)780 3667 y(\014)808 3521 y(h)894 3509 y Fn(n)861 3534 y Fl(Y)861 3711 y Fn(i)p Fp(=1)982 3613 y Fq(d)1025 3568 y Fn(b)1054 3576 y Fh(\013)1096 3568 y Fp(\()p Fn(v)1157 3543 y Fg(\003)1155 3585 y Fh(i)1192 3568 y Fp(\))1025 3641 y Fn(j)1052 3649 y Fh(\013)1095 3641 y Fp(\()p Fn(v)1156 3621 y Fg(\003)1154 3662 y Fh(i)1191 3641 y Fp(\))1222 3613 y Ft(\()p Fo(x)1304 3625 y Fn(i)1333 3613 y Fq(;)14 b Fo(y)1420 3625 y Fn(i)1448 3613 y Ft(\))p Fq(K)1557 3576 y Fn(h)1596 3584 y Fh(i)1551 3637 y Fn(v)1586 3617 y Fg(\003)1584 3658 y Fh(i)1626 3613 y Ft(\()p Fo(x)1708 3625 y Fn(v)1743 3606 y Fg(\003)1741 3646 y Fh(i)1783 3613 y Ft(\))1815 3521 y Fl(i)1854 3468 y(\014)1854 3518 y(\014)1854 3567 y(\014)1854 3617 y(\014)1854 3667 y(\014)1905 3613 y Fm(\024)789 3862 y(\024)23 b Fq(C)942 3828 y Fn(n)987 3862 y Fq(")1026 3828 y Fn(n)1026 3883 y(h)1170 3783 y Fl(Y)1085 3965 y Fn(i)p Fp(:)p Fu(j)p Fk(x)1187 3979 y Fh(v)1218 3965 y Fg(\003)1217 4006 y Fh(i)1257 3965 y Fu(j)p Fp(=2)1700 3806 y Ft(1)p 1385 3843 674 4 v 1385 3919 a([1)18 b(+)g Fm(j)p Fo(d)p Ft(\()p Fo(x)1709 3931 y Fn(i)1755 3919 y Fm(\000)g Fo(y)1888 3931 y Fn(i)1916 3919 y Ft(\))p Fm(j)p Ft(])1994 3895 y Fn(N)2101 3783 y Fl(Y)2082 3961 y Fn(i)p Fu(2)p Fn(S)2191 3969 y Fh(\027)2241 3862 y Fq(\015)2289 3828 y Fp(\()p Fn(h)2354 3836 y Fh(i)2380 3828 y Fu(\000)p Fp(1\))2518 3862 y Fq(:)3053 3762 y Ft(\(3)p Fq(:)p Ft(117\))71 4163 y(Let)26 b(us)f(no)n(w)g(remark)f (that,)j(after)e(the)h(insertion)f(of)h(the)g(b)r(ounds)f(\(3.116\))g (and)g(\(3.117\))g(in)h(the)g(r.h.s.)0 4312 y(of)33 b(\(3.101\),)g(b)n (y)g(p)r(ossibly)f(c)n(hanging)g(the)h(constan)n(t)g Fq(C)6 b Ft(,)34 b(w)n(e)f(can)g(substitute)2500 4245 y Fl(R)2570 4312 y Fq(d)p Fo(x)2663 4324 y Fn(v)2696 4332 y Fj(0)2733 4312 y Ft(,)i(whic)n(h)d(is)h(there)0 4462 y(a)f(shorthand)f(for)603 4399 y Fl(Q)681 4487 y Fk(x)p Fu(2)p Fk(x)806 4495 y Fh(v)836 4507 y Fj(0)890 4399 y Fl(P)977 4487 y Fn(x)p Fu(2)p Fp(\003)1123 4395 y Fl(R)1192 4462 y Fq(dx)1282 4474 y Fp(0)1320 4462 y Ft(,)i(with)g(the)g(real)e(in)n(tegral)g(o)n(v)n(er)f(\()p Ff(T)2461 4474 y Fn(L;\014)2571 4462 y Ft(\))2603 4432 y Fu(j)p Fk(x)2663 4440 y Fh(v)2693 4452 y Fj(0)2729 4432 y Fu(j)2753 4462 y Ft(,)j(where)f Ff(T)3109 4474 y Fn(L;\014)3251 4462 y Ft(is)0 4611 y(the)24 b(space-time)g(torus)f([) p Fm(\000)p Fq(L=)p Ft(2)p Fq(;)14 b(L=)p Ft(2])d Fm(\002)g Ft([)p Fm(\000)p Fq(\014)t(=)p Ft(2)p Fq(;)j(\014)t(=)p Ft(2].)30 b(Moreo)n(v)n(er,)22 b(equation)i(\(3.115\))e(can)i(b)r(e)g (though)n(t,)0 4761 y(and)29 b(w)n(e)g(shall)f(do)h(that,)h(as)f (de\014ning)g(an)g(in)n(terv)-5 b(al)28 b(on)h Ff(T)1813 4773 y Fn(L;\014)1924 4761 y Ft(,)g(when)h Fq(t)2225 4773 y Fn(l)2279 4761 y Ft(spans)f(the)g(in)n(terv)-5 b(al)29 b([0)p Fq(;)14 b Ft(1];)29 b(this)0 4910 y(is)e(p)r(ossible)h (thanks)f(to)g(the)h(in)n(tro)r(duction)g(of)f(the)h(partition)f (\(3.42\))g(in)h Fm(x)p Ft(3.5.)71 5059 y(Hence,)40 b(in)d(order)f(to)h (complete)g(the)g(pro)r(of)f(of)h(\(3.102\),)i(w)n(e)d(ha)n(v)n(e)g(to) h(sho)n(w)f(that,)k(\014xed)d(a)g(p)r(oin)n(t)4 5208 y(\026)0 5209 y Fo(x)25 b Fm(2)g Fo(x)205 5221 y Fn(v)238 5229 y Fj(0)275 5209 y Ft(,)j(the)h(in)n(terp)r(olation)f(parameters)e (asso)r(ciated)h(with)i(the)g(regularization)d(op)r(erations)h(and)h (an)0 5358 y(in)n(teger)f Fq(N)k Fm(\025)23 b Ft(3,)263 5473 y Fl(Z)309 5662 y Fp(\004)371 5586 y Fq(d)p Ft(\()p Fo(x)496 5598 y Fn(v)529 5606 y Fj(0)567 5586 y Fm(n)613 5585 y Ft(\026)609 5586 y Fo(x)p Ft(\))710 5507 y Fl(Y)705 5682 y Fn(v)r Fu(2)p Fn(\034)853 5507 y Fl(Y)836 5686 y Fn(l)p Fu(2)p Fn(T)941 5694 y Fh(v)1502 5530 y Ft(1)p 1001 5567 1045 4 v 1001 5643 a(1)18 b(+)g([)p Fq(\015)1215 5619 y Fn(h)1254 5627 y Fh(v)1293 5643 y Fm(j)p Fo(d)p Ft(\()p Fo(x)1451 5614 y Fu(0)1451 5668 y Fn(l)1477 5643 y Ft(\()p Fq(t)1539 5655 y Fn(l)1565 5643 y Ft(\))h Fm(\000)f Fo(y)1750 5614 y Fu(0)1749 5668 y Fn(l)1775 5643 y Ft(\()p Fq(s)1846 5655 y Fn(l)1871 5643 y Ft(\)\))p Fm(j)p Ft(])1981 5619 y Fn(N)2078 5586 y Fm(\024)2171 5507 y Fl(Y)2166 5682 y Fn(v)r Fu(2)p Fn(\034)2297 5586 y Fq(C)6 b(\015)2410 5552 y Fu(\000)p Fn(h)2501 5560 y Fh(v)2536 5552 y Fp(\()p Fn(s)2593 5560 y Fh(v)2629 5552 y Fu(\000)p Fp(1\))2767 5586 y Fq(;)263 b Ft(\(3)p Fq(:)p Ft(118\))0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(56)p eop %%Page: 57 57 57 56 bop 0 83 a Ft(where)24 b(\004)h(denotes)f(the)h(subset)f(of)h(\() p Ff(T)1184 95 y Fn(L;\014)1294 83 y Ft(\))1326 53 y Fu(j)p Fk(x)1386 61 y Fh(v)1416 73 y Fj(0)1453 53 y Fu(n)1490 52 y Fp(\026)1487 53 y Fk(x)o Fu(j)1575 83 y Ft(satisfying)f(all)g(the) h(constrain)n(ts)e(asso)r(ciated)g(with)i(the)0 232 y(in)n(terp)r (olated)i(p)r(oin)n(ts)h(of)f(the)h(form)f(\(3.115\).)71 382 y(Let)32 b(us)g(call)508 361 y(~)492 382 y Fq(T)41 b Ft(=)31 b Fm([)733 394 y Fn(v)789 361 y Ft(~)772 382 y Fq(T)821 394 y Fn(v)860 382 y Ft(,)j(where)1177 361 y(~)1161 382 y Fq(T)1210 394 y Fn(v)1281 382 y Ft(is)e(the)h(set)f(of)g (lines)g(connecting)g Fo(x)2414 352 y Fu(0)2414 405 y Fn(l)2439 382 y Ft(\()p Fq(t)2501 394 y Fn(l)2527 382 y Ft(\))h(with)f Fo(y)2836 352 y Fu(0)2835 405 y Fn(l)2861 382 y Ft(\()p Fq(s)2932 394 y Fn(l)2958 382 y Ft(\),)h(for)f(an)n(y)0 531 y Fq(l)j Fm(2)e Fq(T)197 543 y Fn(v)236 531 y Ft(.)331 510 y(~)314 531 y Fq(T)45 b Ft(is)34 b(not)f(a)h(tree)f(in)h(general;)h (ho)n(w)n(ev)n(er,)e(for)h(an)n(y)f Fq(v)s Ft(,)2083 510 y(~)2067 531 y Fq(T)2116 543 y Fn(v)2189 531 y Ft(is)g(still)h(an)g (anc)n(hored)e(tree)h(graph)0 681 y(b)r(et)n(w)n(een)22 b(the)h(clusters)e(of)i(p)r(oin)n(ts)f Fo(x)1132 651 y Fp(\()p Fn(i)p Fp(\))1212 681 y Ft(,)h Fq(i)g Ft(=)f(1)p Fq(;)14 b(:)g(:)g(:)f(;)h(s)1662 693 y Fn(v)1702 681 y Ft(.)35 b(Hence,)23 b(the)g(pro)r(of)f(of)g(\(3.118\))f(b)r(ecomes)h (trivial,)0 830 y(if)28 b(w)n(e)f(can)h(sho)n(w)e(that)1297 980 y Fq(d)p Ft(\()p Fo(x)1422 992 y Fn(v)1455 1000 y Fj(0)1493 980 y Fm(n)1539 979 y Ft(\026)1535 980 y Fo(x)p Ft(\))d(=)1732 901 y Fl(Y)1728 1093 y Fn(l)p Fu(2)1807 1078 y Fp(~)1794 1093 y Fn(T)1856 980 y Fq(d)p Fo(r)1938 992 y Fn(l)1987 980 y Fq(;)1043 b Ft(\(3)p Fq(:)p Ft(119\))0 1224 y(where)27 b Fo(r)279 1236 y Fn(l)328 1224 y Ft(=)c Fo(x)466 1194 y Fu(0)466 1247 y Fn(l)491 1224 y Ft(\()p Fq(t)553 1236 y Fn(l)579 1224 y Ft(\))c Fm(\000)f Fo(y)764 1194 y Fu(0)763 1247 y Fn(l)789 1224 y Ft(\()p Fq(s)860 1236 y Fn(l)886 1224 y Ft(\).)71 1373 y(In)28 b(order)e(to)i(pro)n(v)n (e)e(\(3.119\),)g(w)n(e)i(can)f(pro)r(ceed,)g(for)g(example,)g(as)h(in) f([BM1].)37 b(Let)28 b(us)g(consider)e(\014rst)0 1523 y(a)e(v)n(ertex)f Fq(v)28 b Ft(with)d Fm(j)p Fq(T)640 1535 y Fn(v)679 1523 y Fm(j)e Fq(>)f Ft(0,)j(whic)n(h)f(is)h(maximal)e (with)i(resp)r(ect)f(to)g(the)h(tree)f(order;)g(hence)h(either)f Fq(v)j Ft(is)e(a)0 1672 y(non)j(lo)r(cal)f(endp)r(oin)n(t)h(with)g Fq(h)941 1684 y Fn(v)1003 1672 y Ft(=)23 b(2)k(or)g(it)h(is)g(a)f(non)h (trivial)f(v)n(ertex)f(with)j(no)e(v)n(ertex)g Fq(v)2758 1642 y Fu(0)2809 1672 y Ft(with)h Fm(j)p Fq(T)3070 1684 y Fn(v)3105 1668 y Fg(0)3132 1672 y Fm(j)23 b Fq(>)g Ft(0)0 1821 y(follo)n(wing)28 b(it.)42 b(In)29 b(this)g(case)934 1800 y(~)917 1821 y Fq(T)966 1833 y Fn(v)1031 1821 y Ft(=)c Fq(T)1170 1833 y Fn(v)1209 1821 y Ft(,)30 b(that)f(is)g(no)g (line)g(dep)r(ends)g(on)g(the)h(in)n(terp)r(olation)e(parameters,)0 1971 y(and)178 1950 y(~)161 1971 y Fq(T)210 1983 y Fn(v)277 1971 y Ft(is)g(a)f(tree)g(on)g(the)h(set)g Fo(x)1034 1983 y Fn(v)1074 1971 y Ft(,)f(so)g(that)h(w)n(e)g(get)f(immediately)h (the)g(iden)n(tit)n(y)1278 2196 y Fq(d)p Fo(x)1371 2208 y Fn(v)1434 2196 y Ft(=)23 b Fq(d)1569 2195 y Ft(\026)1565 2196 y Fo(x)1615 2161 y Fp(\()p Fn(v)r Fp(\))1738 2117 y Fl(Y)1720 2309 y Fn(l)p Fu(2)1799 2294 y Fp(~)1786 2309 y Fn(T)1825 2317 y Fh(v)1875 2196 y Fq(d)p Fo(r)1957 2208 y Fn(l)2006 2196 y Fq(;)1024 b Ft(\(3)p Fq(:)p Ft(120\))0 2483 y(where)249 2482 y(\026)245 2483 y Fo(x)295 2453 y Fp(\()p Fn(v)r Fp(\))419 2483 y Ft(is)32 b(an)g(arbitrary)e(p)r(oin)n (t)j(of)f Fo(x)1356 2495 y Fn(v)1396 2483 y Ft(.)51 b(If)32 b(w)n(e)g(use)g(\(3.120\))f(for)h(the)h(family)f Fq(S)2707 2495 y Fp(0)2777 2483 y Ft(of)g(all)g(maximal)0 2632 y(v)n(ertices)26 b(with)j Fm(j)p Fq(T)562 2644 y Fn(v)601 2632 y Fm(j)23 b Fq(>)f Ft(0,)28 b(w)n(e)f(get)1140 2857 y Fq(d)p Fo(x)1233 2869 y Fn(v)1266 2877 y Fj(0)1326 2857 y Ft(=)1438 2778 y Fl(Y)1414 2956 y Fn(v)r Fu(2)p Fn(S)1535 2964 y Fj(0)1581 2765 y Fl(h)1620 2857 y Fq(d)1667 2856 y Ft(\026)1663 2857 y Fo(x)1713 2822 y Fp(\()p Fn(v)r Fp(\))1836 2778 y Fl(Y)1819 2970 y Fn(l)p Fu(2)1898 2955 y Fp(~)1885 2970 y Fn(T)1924 2978 y Fh(v)1974 2857 y Fq(d)p Fo(r)2056 2869 y Fn(l)2082 2765 y Fl(i)2144 2857 y Fq(:)886 b Ft(\(3)p Fq(:)p Ft(121\))0 3147 y(Let)31 b(us)h(no)n(w)e(consider)g(a)h(line)999 3125 y(\026)1000 3147 y Fq(l)f Fm(2)1156 3126 y Ft(~)1140 3147 y Fq(T)11 b Ft(,)32 b(whic)n(h)f(connects)g(t)n(w)n(o)g(clusters)f(of)h(p)r(oin)n (ts)h Fo(x)2704 3159 y Fn(v)2737 3167 y Fj(1)2805 3147 y Ft(and)f Fo(x)3020 3159 y Fn(v)3053 3167 y Fj(2)3090 3147 y Ft(,)i(with)0 3296 y Fq(v)40 3308 y Fn(i)91 3296 y Fm(2)23 b Fq(S)220 3308 y Fp(0)258 3296 y Ft(,)k Fq(i)c Ft(=)g(1)p Fq(;)14 b Ft(2.)35 b(By)28 b(\(3.115\))797 3521 y Fo(r)835 3526 y Fp(\026)836 3541 y Fn(l)885 3521 y Ft(=)23 b Fo(x)1023 3486 y Fu(0)1022 3533 y Fp(\026)1023 3548 y Fn(l)1049 3521 y Ft(\()p Fq(t)1110 3526 y Fp(\026)1111 3541 y Fn(l)1136 3521 y Ft(\))c Fm(\000)f Fo(y)1321 3486 y Fu(0)1320 3541 y Fn(l)1346 3521 y Ft(\()p Fq(s)1416 3526 y Fp(\026)1417 3541 y Fn(l)1443 3521 y Ft(\))23 b(=)g Fq(t)1615 3526 y Fp(\026)1616 3541 y Fn(l)1641 3521 y Fo(x)1690 3526 y Fp(\026)1691 3541 y Fn(l)1735 3521 y Ft(+)18 b(\(1)h Fm(\000)f Fq(t)2023 3526 y Fp(\026)2024 3541 y Fn(l)2049 3521 y Ft(\))2085 3520 y(\026)2081 3521 y Fo(x)2130 3526 y Fp(\026)2131 3541 y Fn(l)2176 3521 y Fm(\000)g Fo(y)2310 3486 y Fu(0)2308 3533 y Fp(\026)2309 3548 y Fn(l)2335 3521 y Ft(\()p Fq(s)2405 3526 y Fp(\026)2406 3541 y Fn(l)2431 3521 y Ft(\))24 b Fq(;)543 b Ft(\(3)p Fq(:)p Ft(122\))0 3745 y(implying)28 b(that)199 3969 y(\026)194 3970 y Fo(x)244 3936 y Fp(\()p Fn(v)303 3944 y Fj(1)336 3936 y Fp(\))390 3970 y Ft(=)22 b Fo(r)515 3975 y Fp(\026)516 3990 y Fn(l)560 3970 y Ft(+)648 3969 y(\026)643 3970 y Fo(x)693 3936 y Fp(\()p Fn(v)752 3944 y Fj(1)785 3936 y Fp(\))834 3970 y Fm(\000)c Fo(r)955 3975 y Fp(\026)956 3990 y Fn(l)1005 3970 y Ft(=)k Fo(r)1130 3975 y Fp(\026)1131 3990 y Fn(l)1176 3970 y Ft(+)c Fq(t)1288 3975 y Fp(\026)1289 3990 y Fn(l)1314 3970 y Ft(\()1350 3969 y(\026)1346 3970 y Fo(x)1396 3936 y Fp(\()p Fn(v)1455 3944 y Fj(1)1488 3936 y Fp(\))1537 3970 y Fm(\000)g Fo(x)1669 3975 y Fp(\026)1670 3990 y Fn(l)1696 3970 y Ft(\))h(+)f(\(1)g Fm(\000)g Fq(t)2034 3975 y Fp(\026)2035 3990 y Fn(l)2060 3970 y Ft(\)\()2128 3969 y(\026)2124 3970 y Fo(x)2174 3936 y Fp(\()p Fn(v)2233 3944 y Fj(1)2267 3936 y Fp(\))2315 3970 y Fm(\000)2403 3969 y Ft(\026)2398 3970 y Fo(x)2447 3975 y Fp(\026)2448 3990 y Fn(l)2474 3970 y Ft(\))h(+)f Fo(y)2659 3936 y Fu(0)2657 3982 y Fp(\026)2658 3998 y Fn(l)2684 3970 y Ft(\()p Fq(s)2754 3975 y Fp(\026)2755 3990 y Fn(l)2781 3970 y Ft(\))23 b Fq(:)194 b Ft(\(3)p Fq(:)p Ft(123\))0 4195 y(Since)25 b Fo(y)265 4165 y Fu(0)263 4211 y Fp(\026)264 4227 y Fn(l)290 4195 y Ft(\()p Fq(s)360 4200 y Fp(\026)361 4215 y Fn(l)387 4195 y Ft(\))g(dep)r(ends)g(only)f (on)h(the)g(v)-5 b(ariables)24 b Fo(x)1586 4207 y Fn(v)1619 4215 y Fj(2)1681 4195 y Ft(and)g(\()1875 4194 y(\026)1871 4195 y Fo(x)1921 4165 y Fp(\()p Fn(v)1980 4173 y Fj(1)2013 4165 y Fp(\))2056 4195 y Fm(\000)13 b Fo(x)2183 4200 y Fp(\026)2184 4215 y Fn(l)2210 4195 y Ft(\))25 b(and)g(\()2462 4194 y(\026)2458 4195 y Fo(x)2508 4165 y Fp(\()p Fn(v)2567 4173 y Fj(1)2600 4165 y Fp(\))2643 4195 y Fm(\000)2725 4194 y Ft(\026)2721 4195 y Fo(x)2770 4200 y Fp(\026)2771 4215 y Fn(l)2796 4195 y Ft(\))g(b)r(oth)g(dep)r(end)0 4344 y(only)i(on)g Fm(f)p Fo(r)378 4356 y Fn(l)404 4344 y Fq(;)14 b(l)24 b Fm(2)585 4323 y Ft(~)569 4344 y Fq(T)618 4356 y Fn(v)651 4364 y Fj(1)687 4344 y Fm(g)p Ft(,)j(w)n(e)g(get)884 4494 y Fp(2)848 4519 y Fl(Y)847 4696 y Fn(i)p Fp(=1)968 4506 y Fl(h)1008 4598 y Fq(d)1055 4597 y Ft(\026)1051 4598 y Fo(x)1101 4564 y Fp(\()p Fn(v)1160 4572 y Fh(i)1187 4564 y Fp(\))1260 4519 y Fl(Y)1231 4711 y Fn(l)p Fu(2)1309 4696 y Fp(~)1297 4711 y Fn(T)1336 4719 y Fh(v)1366 4732 y(i)1410 4598 y Fq(d)p Fo(r)1492 4610 y Fn(l)1518 4506 y Fl(i)1580 4598 y Ft(=)c Fq(d)p Fo(r)1749 4603 y Fp(\026)1750 4618 y Fn(l)1776 4598 y Fq(d)1823 4597 y Ft(\026)1819 4598 y Fo(x)1869 4564 y Fp(\()p Fn(v)1928 4572 y Fj(2)1961 4564 y Fp(\))2042 4494 y(2)2006 4519 y Fl(Y)2005 4696 y Fn(i)p Fp(=1)2156 4519 y Fl(Y)2126 4711 y Fn(l)p Fu(2)2205 4696 y Fp(~)2192 4711 y Fn(T)2231 4719 y Fh(v)2261 4732 y(i)2306 4598 y Fq(d)p Fo(r)2388 4610 y Fn(l)2437 4598 y Fq(:)593 b Ft(\(3)p Fq(:)p Ft(124\))71 4887 y(By)27 b(iterating)g(this)h(pro)r(cedure,)f(one)g(gets)g(\(3.119\).)0 5107 y Fo(3.16)94 b Ft(As)24 b(w)n(e)g(ha)n(v)n(e)f(discussed)h(in)h Fm(x)o Ft(2.13,)f(it)h(is)f(not)g(necessary)f(to)h(p)r(erform)g(the)g (scale)g(decomp)r(osition)0 5257 y(of)i(the)h(Grassmanian)d(in)n (tegration)h(up)i(to)f(the)h(last)f(scale)f Fq(h)1916 5269 y Fn(L;\014)2026 5257 y Ft(,)i(but)f(w)n(e)g(can)g(stop)g(it)h(to) f(the)h(scale)e Fq(h)3246 5227 y Fu(\003)3284 5257 y Ft(,)0 5406 y(de\014ned)j(in)g(\(2.116\).)35 b(Hence,)28 b(w)n(e)g(rede\014ne)1417 5385 y(~)1398 5406 y Fq(E)1459 5418 y Fn(h)1498 5402 y Fg(\003)1537 5406 y Ft(,)g(so)e(that)380 5631 y Fq(e)419 5597 y Fu(\000)p Fn(L\014)572 5582 y Fp(~)558 5597 y Fn(E)607 5609 y Fh(h)641 5597 y Fg(\003)706 5631 y Ft(=)794 5518 y Fl(Z)891 5631 y Fq(P)944 5643 y Fn(Z)989 5655 y Fh(h)1023 5643 y Fg(\003)1058 5655 y(\000)p Fj(1)1136 5643 y Fn(;\033)1194 5655 y Fh(h)1228 5643 y Fg(\003)1263 5655 y(\000)p Fj(1)1340 5643 y Fn(;C)1408 5655 y Fh(h)1442 5643 y Fg(\003)1485 5631 y Ft(\()p Fq(d )1617 5597 y Fp(\()p Fu(\024)p Fn(h)1734 5572 y Fg(\003)1769 5597 y Fp(\))1799 5631 y Ft(\))14 b Fq(e)1884 5595 y Fu(\000)1945 5580 y Fp(^)1936 5595 y Fu(V)1983 5570 y Fj(\()p Fh(h)2039 5553 y Fg(\003)2074 5570 y Fj(\))2101 5595 y Fp(\()2127 5542 y Fm(p)p 2196 5542 192 4 v 53 x Fn(Z)2241 5607 y Fh(h)2275 5595 y Fg(\003)2310 5607 y(\000)p Fj(1)2388 5595 y Fn( )2434 5570 y Fj(\()p Fg(\024)p Fh(h)2535 5553 y Fg(\003)2570 5570 y Fj(\))2597 5595 y Fp(\))2650 5631 y Fq(;)380 b Ft(\(3)p Fq(:)p Ft(125\))0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)28 b(18:23)1046 b Ft(57)p eop %%Page: 58 58 58 57 bop 0 83 a Ft(implying)28 b(that)1245 257 y Fq(E)1306 269 y Fn(L;\014)1440 257 y Ft(=)1592 153 y Fp(1)1549 178 y Fl(X)1527 357 y Fn(h)p Fp(=)p Fn(h)1656 340 y Fg(\003)1691 257 y Ft([)1733 236 y(~)1714 257 y Fq(E)1775 269 y Fn(h)1837 257 y Ft(+)18 b Fq(t)1950 269 y Fn(h)1993 257 y Ft(])23 b Fq(:)991 b Ft(\(3)p Fq(:)p Ft(126\))71 489 y(Thanks)35 b(to)i(Lemma)f(2.12,)h(w)n(e)f(can)f(pro)r(ceed)h(as)g(in)g(the)h(pro)r (of)f(of)g(Theorem)f(3.12)g(to)h(pro)n(v)n(e)f(the)0 639 y(follo)n(wing)27 b(Theorem.)0 859 y Fo(3.17)100 b Fs(Theorem.)46 b Fr(Ther)l(e)33 b(exists)f(a)h(c)l(onstant)k Ft(\026)-48 b Fq(")33 b Fr(such)f(that,)i(if)f Fq(")2180 871 y Fn(h)2219 855 y Fg(\003)2285 859 y Fm(\024)g Ft(\026)-47 b Fq(")32 b Fr(and,)i(for)f Fq(h)28 b Ft(=)f Fq(h)2990 829 y Fu(\003)3028 859 y Fr(,)33 b(\(2.98\))0 1008 y(holds)e(and)f(the) g(b)l(ounds)g(\(3.88\))h(ar)l(e)f(satis\014e)l(d,)h(then)1136 1179 y Fl(X)1035 1357 y Fn(\034)7 b Fu(2T)1156 1369 y Fh(h)1190 1357 y Fg(\003)1223 1369 y(\000)p Fj(1)p Fh(;n)1370 1257 y Fm(j)1413 1236 y Ft(~)1393 1257 y Fq(E)1454 1269 y Fn(h)1493 1253 y Fg(\003)1532 1257 y Ft(\()p Fq(\034)i Ft(\))p Fm(j)25 b(\024)d Fq(\015)1824 1223 y Fp(2)p Fn(h)1896 1198 y Fg(\003)1935 1257 y Ft(\()p Fq(C)6 b(")2071 1269 y Fn(h)2110 1253 y Fg(\003)2149 1257 y Ft(\))2181 1223 y Fn(n)2249 1257 y Fq(:)781 b Ft(\(3)p Fq(:)p Ft(127\))0 1727 y Fo(3.18)93 b Ft(Theorems)22 b(3.12)f(and)i(3.17,)g(together)f (with)i(\(3.126\))d(and)i(\(2.118\),)g(imply)g(that)h(the)f(expansion)0 1876 y(de\014ning)34 b Fq(E)381 1888 y Fn(L;\014)525 1876 y Ft(is)f(con)n(v)n(ergen)n(t,)g(uniformly)h(in)g Fq(L;)14 b(\014)t Ft(.)55 b(With)34 b(some)g(more)e(w)n(ork)h(\(essen)n (tially)g(trivial,)0 2026 y(but)28 b(cum)n(b)r(ersome)f(to)g(describ)r (e\))h(one)f(can)g(also)g(pro)n(v)n(e)f(that)i(lim)2073 2038 y Fn(L;\014)s Fu(!1)2329 2026 y Fq(E)2390 2038 y Fn(L;\014)2528 2026 y Ft(do)r(es)g(exist.)0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)g(18:23)1046 b Ft(58)p eop %%Page: 59 59 59 58 bop 1391 83 a Fv(References)0 303 y Ft([A])203 b(I.)35 b(A\017ec)n(k:)51 b(Field)35 b(theory)f(metho)r(ds)g(and)h (quan)n(tum)f(critical)g(phenomena.)58 b(Pro)r(c.)f(of)34 b(Les)311 453 y(Houc)n(hes)27 b(summer)h(sc)n(ho)r(ol)e(on)i(Critical)f (phenomena,)h(Random)f(Systems,)h(Gauge)f(theories,)311 602 y(North)h(Holland)f(\(1984\).)0 694 y([B])206 b(R.J.)20 b(Baxter:)31 b(Eigh)n(t-V)-7 b(ertex)19 b(Mo)r(del)g(in)h(Lattice)f (Statistics.)35 b Fr(Phys.)i(R)l(ev.)f(L)l(ett.)e Fo(26)p Ft(,)20 b(832{833)311 843 y(\(1971\).)0 934 y([BG])141 b(G.)28 b(Benfatto,)g(G.)g(Galla)n(v)n(otti:)36 b(P)n(erturbation)26 b(Theory)h(of)h(the)g(F)-7 b(ermi)28 b(Surface)f(in)h(Quan)n(tum)311 1084 y(Liquid.)36 b(A)24 b(General)e(Quasiparticle)g(F)-7 b(ormalism)23 b(and)g(One-Dimensional)g(Systems.)35 b Fr(J.)26 b(Stat.)311 1233 y(Phys.)39 b Fo(59)p Ft(,)27 b(541{664)d(\(1990\).)0 1324 y([BGM])65 b(G.)24 b(Benfatto,)g(G.)g (Galla)n(v)n(otti,)f(V.)h(Mastropietro:)33 b(Renormalization)22 b(Group)h(and)g(the)h(F)-7 b(ermi)311 1474 y(Surface)27 b(in)h(the)g(Luttinger)f(Mo)r(del.)37 b Fr(Phys.)j(R)l(ev.)e(B)29 b Fo(45)p Ft(,)e(5468{5480)c(\(1992\).)0 1565 y([BGPS])38 b(G.)21 b(Benfatto,)h(G.)f(Galla)n(v)n(otti,)g(A.)g(Pro)r(cacci,)f(B.)h (Scopp)r(ola:)33 b(Beta)20 b(F)-7 b(unctions)21 b(and)f(Sc)n(h)n (winger)311 1715 y(F)-7 b(unctions)34 b(for)e(a)h(Man)n(y)g(F)-7 b(ermions)32 b(System)h(in)h(One)f(Dimension.)54 b Fr(Comm.)g(Math.)h (Phys.)311 1864 y Fo(160)p Ft(,)27 b(93{171)e(\(1994\).)0 1955 y([BM1])88 b(F.)22 b(Bonetto,)h(V.)f(Mastropietro:)32 b(Beta)21 b(F)-7 b(unction)22 b(and)g(Anomaly)f(of)h(the)g(F)-7 b(ermi)22 b(Surface)f(for)g(a)311 2105 y Fq(d)i Ft(=)g(1)k(System)g(of) g(In)n(teracting)f(F)-7 b(ermions)26 b(in)h(a)g(P)n(erio)r(dic)f(P)n (oten)n(tial.)35 b Fr(Comm.)k(Math.)g(Phys.)311 2254 y Fo(172)p Ft(,)27 b(57{93)e(\(1995\).)0 2346 y([BM2])88 b(F.)42 b(Bonetto,)i(V.)e(Mastropietro:)63 b(Filled)42 b(Band)f(F)-7 b(ermi)42 b(Systems.)78 b Fr(Mat.)f(Phys.)h(Ele)l(ct.)311 2495 y(Journal)28 b Fo(2)p Ft(,)f(1{43)f(\(1996\).)0 2586 y([BeM])93 b(G.Benfatto,)34 b(V.)f(Mastropietro:)45 b(Renormalization)31 b(group,)i(hidden)g(symmetries)f(and)h(ap-)311 2736 y(pro)n(ximate)26 b(W)-7 b(ard)28 b(iden)n(tities)g(in)f(the)h Fq(X)7 b(Y)18 b(Z)34 b Ft(mo)r(del,)27 b(I)r(I.)h Fr(pr)l(eprint)h Ft(\(2000\).)0 2827 y([EFIK])59 b(F.Essler,)29 b(H.F)-7 b(rahm,)31 b(A.)f(Izergin,)g(V.Korepin:)40 b(Determinan)n(t)30 b(represen)n(tation)e(for)h(correla-)311 2977 y(tion)g(functions)f(of)h (spin-)1123 2944 y Fp(1)p 1122 2958 34 4 v 1122 3005 a(2)1194 2977 y Ft(XXX)g(and)f(XXZ)h(Heisen)n(b)r(erg)f(Magnets.)38 b Fr(Comm.)k(Math.)f(Phys.)311 3126 y Fo(174)p Ft(,)27 b(191{214)d(\(1995\).)0 3217 y([GS])154 b(G.)27 b(Gen)n(tile,)g(B.)f (Scopp)r(ola:)36 b(Renormalization)25 b(group)g(and)i(the)f(ultra)n (violet)g(problem)g(in)h(the)311 3367 y(Luttinger)h(mo)r(del,)f Fr(Comm.)40 b(Meth.)f(Phys.)f Fo(154)p Ft(,)27 b(153{179)d(\(1993\).)0 3458 y([JKM])81 b(J.D.)34 b(Johnson,)g(S.)g(Krinsky)-7 b(,)34 b(B.M.McCo)n(y:)48 b(V)-7 b(ertical-Arro)n(w)32 b(Correlation)f(Length)j(in)g(the)311 3608 y(Eigh)n(t-V)-7 b(ertex)19 b(Mo)r(del)i(and)g(the)g(Lo)n(w-Lying)e(Excitations)g(of)i (the)g Fq(X)7 b(Y)18 b(Z)26 b Ft(Hamiltonian.)34 b Fr(Phys.)311 3757 y(R)l(ev.)k(A)28 b Fo(8)p Ft(,)f(2526{2547)d(\(1973\).)0 3848 y([LP])156 b(A.)22 b(Luther,)h(I.P)n(esc)n(hel:)33 b(Calculation)21 b(of)g(critical)h(exp)r(onen)n(ts)f(in)h(t)n(w)n(o)f (dimensions)g(from)h(quan-)311 3998 y(tum)28 b(\014eld)g(theory)f(in)h (one)f(dimension.)37 b Fr(Phys.)j(R)l(ev.)e(B)28 b Fo(12)p Ft(,)f(9,)h(3908{3917)23 b(\(1975\).)0 4089 y([Le])176 b(A.)36 b(Lesniewski:)52 b(E\013ectiv)n(e)35 b(action)f(for)h(the)h(Y) -7 b(uk)i(a)n(w)n(a)35 b(2)g(quan)n(tum)g(\014eld)h(Theory)-7 b(.)59 b Fr(Comm.)311 4239 y(Math.)40 b(Phys.)e Fo(108)p Ft(,)27 b(437-467)d(\(1987\).)0 4330 y([LSM])91 b(E.)24 b(Lieb,)h(T.)f(Sc)n(h)n(ultz,)h(D.)f(Mattis:)35 b(Tw)n(o)24 b(Soluble)g(Mo)r(dels)g(of)g(an)f(An)n(tiferromagnetic)g(Chain.)311 4479 y Fr(A)n(nn.)38 b(of)30 b(Phys.)39 b Fo(16)p Ft(,)27 b(407{466)d(\(1961\).)0 4571 y([LSM1])49 b(E.)29 b(Lieb,)h(T.)g(Sc)n(h) n(ultz,)f(D.)h(Mattis:)41 b(Tw)n(o-dimensional)27 b(Ising)i(mo)r(del)h (as)f(a)g(soluble)g(problem)311 4720 y(of)f(man)n(y)f(fermions.)36 b Fr(Journ.)i(Math.)i(Phys.)e Ft(Rev.)f(Mo)r(dern)27 b(Ph)n(ys.)36 b(36,)27 b(856{871)d(\(1964\).)0 4811 y([M1])147 b(V.)38 b(Mastropietro:)55 b(Small)37 b(denominators)f(and)i(anomalous) e(b)r(eha)n(viour)g(in)h(the)h(Holstein-)311 4961 y(Hubbard)28 b(mo)r(del.)37 b Fr(Comm.)i(Math.)g(Phys)29 b Ft(201,)d(1,)i(81-115)d (\(1999\).)0 5052 y([M2])147 b(V.)20 b(Mastropietro:)32 b(Renormalization)18 b(group)h(for)g(the)h(XYZ)g(mo)r(del,)i (Renormalization)c(group)311 5202 y(for)27 b(the)h(XYZ)g(mo)r(del.)37 b Fr(L)l(etters)29 b(in)h(Mathematic)l(al)i(physics)p Ft(,)d(47,)e(339-352)d(\(1999\).)0 5293 y([Mc])152 b(B.M.)32 b(McCo)n(y:)45 b(Spin)33 b(Correlation)d(F)-7 b(unctions)33 b(of)f(the)g Fq(X)c Fm(\000)21 b Fq(Y)51 b Ft(Mo)r(del.)g Fr(Phys.)h(R)l(ev.)f Fo(173)p Ft(,)311 5442 y(531{541)24 b(\(1968\).)0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)k(18:23)1046 b Ft(59)p eop %%Page: 60 60 60 59 bop 0 83 a Ft([MD])126 b(W.Metzner,)32 b(C)e(Di)h(Castro:)42 b(Conserv)-5 b(ation)29 b(la)n(ws)g(and)i(correlation)d(functions)j(in) g(the)g(Lut-)311 232 y(tinger)c(liquids.)37 b Fr(Phys.)j(R)l(ev.)e(B)28 b Fo(47)p Ft(,)f(16107{16123)c(\(1993\).)0 324 y([NO])138 b(J.W.)35 b(Negele,)g(H.)g(Orland:)49 b(Quan)n(tum)34 b(man)n(y-particle)f(systems.)56 b(Addison-W)-7 b(esley)g(,)36 b(New)311 473 y(Y)-7 b(ork)27 b(\(1988\).)0 565 y([S])219 b(S.B.)29 b(Suterland:)39 b(Tw)n(o-Dimensional)27 b(Hydrogen)h(Bonded)g (Crystals.)39 b Fr(J.)31 b(Math.)43 b(Phys.)e Fo(11)p Ft(,)311 714 y(3183{3186)23 b(\(1970\).)0 805 y([Sp])173 b(H.Sp)r(ohn:)38 b(Bosonization,)26 b(vicinal)h(surfaces)f(and)i(Hydro) r(dynamic)f(\015uctuation)g(theory)-7 b(.)311 955 y Fr(c)l (ond-mat/9908381)31 b Ft(\(1999\).)0 1046 y([Sp)r(e])134 b(T.Sp)r(encer:)48 b(A)34 b(mathematical)e(approac)n(h)g(to)h(univ)n (ersalit)n(y)e(in)j(t)n(w)n(o)e(dimensions.)53 b Fr(pr)l(eprint)311 1196 y Ft(\(1999\).)0 1287 y([YY])141 b(C.N.)26 b(Y)-7 b(ang,)26 b(C.P)-7 b(.)26 b(Y)-7 b(ang:)36 b(One)25 b(dimensional)h(c)n (hain)f(of)h(anisotropic)e(spin-spin)i(in)n(teractions,)311 1436 y(I)i(and)f(I)r(I.)h Fr(Phys.)40 b(R)l(ev.)d Fo(150)p Ft(,)27 b(321{339)d(\(1966\).)0 5869 y Fj(14)p Fh(=apr)q(ile=)p Fj(2000;)k(18:23)1046 b Ft(60)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0004141134716--