MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY. e S(A)/ da, h N

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1 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 9 4. Matrx tgrals Lt h N b th spac of Hrmta matrcs of sz N. Th r product o h N s gv by (A, B) = Tr(AB). I ths scto w wll cosdr tgrals of th form Z N = N 2 /2 S(A)/ da, h N whr th Lbsgu masur da s ormalzd by th codto Tr(A2 )/2 da =, ad S(A) = Tr(A 2 )/2 m 0 g mtr(a m )/m s th acto fuctoal. 3 W wll b trstd th bhavor of th coffcts of th xpaso of Z N g for larg N. Th study of ths bhavor wll lad us to cosdrg ot smply Fyma graphs, but actually fat (or rbbo) graphs, whch ar fact 2-dmsoal surfacs. Thus, bfor w procd furthr, w d to do som 2-dmsoal combatoral topology. 4.. Fat graphs. Rcall from th proof of Fyma s thorm that gv a ft collcto of flowrs ad a parg σ o th st T of dpots of thr dgs, w ca obta a graph Γ σ by coctg (or glug) th pots whch fall th sam par. Now, gv a -flowr, lt us scrb t a closd dsk D (so that th ds of th dgs ar o th boudary) ad tak ts small tubular ghborhood D. Ths producs a rgo wth pcws smooth boudary. W wll qup ths rgo wth a ortato, ad call t a fat -valt flowr. Th boudary of a fat -valt flowr has th form P Q P 2 Q 2...P Q P,whr P,Q ar th agl pots, th trvals P j Q j ar arcs o D, ad Q j P j+ ar (smooth) arcs lyg sd D (s Fg. 0). 3-valt flowr fat 3-valt flowr Q P P 2 Q 2 Q 3 P 3 Fgur 0 Now, gv a collcto of usual flowrs ad a parg σ as abov, w ca cosdr th corrspodg fat flowrs, ad glu thm (rspctg th ortato) alog trvals P j Q j accordg to σ. Ths wll produc a compact ortd surfac wth boudary (th boudary s glud from trvals P j Q j+ ). W wll dot ths surfac by Γ σ, ad call t th fattg of Γ wth rspct to σ. A fattg of a graph wll b calld a fat (or rbbo) graph. Thus, a fat graph s ot just a ortd surfac wth boudary, but such a surfac togthr wth a partto of ths surfac to fat flowrs. Not that th sam graph Γ ca hav may dffrt fattgs, ad partcular th gus g of th fattg s ot dtrmd by Γ (s Fg. ) Matrx tgrals larg N lmt ad plaar graphs. Lt us ow rtur to th study of th tgral Z N. By th proof of Fyma s thorm, g (/2 ) l Z N = ( ) F σ,! whr th summato s tak ovr all pargs of T = T () that produc a coctd graph Γ σ,ad F σ dots th cotracto of th tsors Tr(A )usg σ. For a surfac Σ wth boudary, lt ν(σ) dot th umbr of coctd compots of th boudary. = N ν( Γ Proposto 4.. F σ ) σ. 3 Not that w dvd by m ad ot by m!. W wll s blow why such ormalzato wll b mor covt. σ

2 20 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY Γ g =0 Γ 2 g =0 Γ 3 g = Fgur. Glug a fat graph from fat flowrs Proof. Lt b th stadard bass of C N,ad th dual bass. Th th tsor Tr(A m )cab wrtt as N Tr(A m )= ( m,a m ).,..., m= O ca vsualz ach moomal ths sum as a lablg of th agl pots P,Q,...,P m,q m o th boudary of a fat m-valt flowr by, 2, 2, 3,..., m,. Now, th cotracto usg σ of som st of such moomals s ozro ff th subscrpt s costat alog ach boudary compot of Γ σ (s Fg. 2). Ths mpls th rsult. j j k k l m Cotracto ozro ff = r, j = p, j = m, k = r, k = p, = m, that s = r = k = p = j = m. Fgur 2. Cotracto dfd by a fat graph. Lt G c () s th st of somorphsm classs of coctd fat graphs wth -valt vrtcs. For Γ G c (), lt b( Γ) b th umbr of dgs mus th umbr of vrtcs of th udrlyg usual graph Γ. Corollary 4.2. l Z N = ( g ) Γ G c() N ν( Γ) b( Γ). Proof. Lt G fat () = S ( Z/Z). Ths group acts o T,sothat Γ σ = Γ gσ, for ay g G fat (sc cyclc prmutatos of dgs of a flowr xtd to ts fattg). Morovr, th group acts trastvly o th st of σ gvg a fxd fat graph Γ σ, ad th stablzr of ay σ s Aut( Γ σ ). Ths mpls th rsult. Now for ay compact surfac Σ wth boudary, lt g(σ) b th gus of Σ. Th for a coctd fat graph Γ, b( Γ) = 2g( Γ) 2+ν( Γ) (mus th Eulr charactrstc). Thus, dfg ẐN( ) =Z N ( /N ), w fd

3 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 2 Thorm 4.3. l Ẑ N = ( g ) Γ) b( Γ) N 2 2g(. Γ G c() whr G c ()[0] dots th st of plaar coctd fat graphs,.. thos whch hav gus zro. (2) Morovr, thr xsts a xpaso l Ẑ N /N 2 = g 0 a gn 2g,whr ad G c ()[g] dots th st of coctd fat graphs whch hav gus g Itgrato ovr ral symmtrc matrcs. O may also cosdr th matrx tgral ovr th spac s N of ral symmtrc matrcs of sz N. Namly, o puts N (N +)/4 Z N = S(A)/ da, s N whr S ad da ar as abov. Lt us gralz Thorm 4.4 to ths cas. As bfor, cosdrato of th larg N lmt lads to cosdrato of fat flowrs ad glug of thm. Howvr, th xact atur of glug s ow somwhat dffrt. Namly, th Hrmta cas w had ( j, k l )= δ l δ jk, whch forcd us to glu fat flowrs prsrvg ortato. O th othr had, th ral symmtrc cas =, ad th r product of th fuctoals j o th spac of symmtrc matrcs s gv by ( j, k l )= δ k δ jl + δ l δ jk. Ths mas that bsds th usual (ortato prsrvg) glug of fat flowrs, w ow must allow glug wth a twst of th rbbo by 80 o. Fat graphs thus obtad wll b calld twstd fat graphs. That mas, a twstd fat graph s a surfac wth boudary (possbly ot ortabl), togthr wth a partto to fat flowrs, ad ortatos o ach of thm (whch may or may ot match at th cuts, s Fg. 3). Ths mpls th followg mportat rsult, du to t Hooft. l Z Thorm 4.4. () Thr xsts a lmt W := lm ˆ N N. Ths lmt s gv by th formula N 2 b( Γ) W = ( g ), Γ G c ()[0] b( Γ) a g = ( g ), Γ G c ()[g] Rmark. Gus zro fat graphs ar sad to b plaar bcaus th udrlyg usual graphs ca b put o th 2-sphr (ad hc o th pla) wthout slf-trsctos. Rmark 2. t Hooft s thorm may b trprtd trms of th usual Fyma dagram xpaso. Namly, t mpls that for larg N, th ladg cotrbuto to l(z N ( /N )) coms from th trms th Fyma dagram xpaso corrspodg to plaar graphs (.. thos that admt a mbddg to th 2-sphr). Fgur 3. Twstd-fat graph Now o ca show aalogously to th Hrmta cas that th /N xpaso of l ẐN (whr Z ˆ N = Z N (2 /N )) s gv by th sam formula as bfor, but wth summato ovr th st G tw c () of twstd fat graphs:

4 22 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY Thorm 4.5. l ˆ N 2 2g( Z N = ( g ) Γ) b( Γ). Γ G tw c () Hr th gus g of a (possbly o-ortabl) surfac s dfd by g = χ/2, whr χ s th Eulr charactrstc. Thus th gus of RP 2 s /2, th gus of th Kl bottl s, ad so o. I partcular, w hav th aalog of t Hooft s thorm. Thorm 4.6. () Thr xsts a lmt W := lm N l ˆN. Ths lmt s gv by th formula 4.4. Applcato to a coutg problm. Matrx tgrals ar so rch that v th smplst possbl xampl rducs to a otrval coutg problm. Namly, cosdr th matrx tgral Z N ovr complx Hrmta matrcs th cas S(A) = Tr(A 2 )/2 str(a 2m )/2m, whr s 2 = 0 (.. w work ovr th rg C[s]/(s 2 )). I ths cas w ca st =. Th from Thorm 4.4 w gt Tr(A 2m ) Tr(A2 )/2 da = P m (N), whr P m (N) s a polyomal, gv by th formula P m (N) = g 0 ε g(m)n m+ 2g,ad ε g (m) s th umbr of ways to glu a surfac of gus g from a 2m-go wth labld sds by glug sds prsrvg th ortato. Idd, ths cas w hav oly o fat flowr of valcy 2m, whch has to b glud wth tslf; so a drct applcato of our Fyma ruls lads to coutg ways to glu a surfac of a gv gus from a polygo. Th valu of ths tgral s gv by th followg o-trval thorm. Corollary ( ) 4.8. Th umbr of ways to glu a sphr of a 2m-go s th Catala umbr C m = 2m m+. m Z N 2 b( W = ( g ) Γ G tw ()[0] c Γ), whr G tw c ()[0] dots th st of plaar coctd twstd fat graphs,.. thos whch hav gus zro. (2) Morovr, thr xsts a xpaso l Ẑ N /N 2 = g 0 a gn 2g,whr b( a g = ( g ) Γ G tw c ()[g] Γ), ad G tw c ()[g] dots th st of coctd twstd fat graphs whch hav gus g. Exrcs. Cosdr th matrx tgral ovr th spacq N of quatroc Hrmta matrcs. Show that ths cas th rsults ar th sam as th ral cas, xcpt that ach twstd fat graph couts wth a sg, qual to ( ) m,whr m s th umbr of twstgs (.. msmatchs of ortato at cuts). I othr words, l Ẑ N for quatroc matrcs s qual l Ẑ 2N for ral matrcs wth N rplacd by N. Ht: us that th utary group U(N, H) s a ral form of Sp(2N), ad q N s a ral form of th rprstato of Λ 2 V, whr V s th stadard (vctor) rprstato of Sp(2N). Compar to th cas of ral symmtrc matrcs, whr th rlvat rprstato s S 2 V for O(N), ad th cas of complx Hrmta matrcs, whr t s V V for GL(N). h N Thorm 4.7. (Harr-Zagr, 986) m ( ) P m (x) = (2m)! m x(x )...(x p) 2 p 2 m. m! p (p +)! p=0 Th thorm s provd th xt subsctos. Lookg at th ladg coffct of P m,w gt.

5 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 23 Corollary 4.8 actually has aothr (lmtary combatoral) proof, whch s as follows. For ach parg σ o th st of sds of th 2m-go, lt us coct th mdpots of th sds that ar pard by straght ls (Fg. 4). It s gomtrcally vdt that f ths ls do t trsct th th glug wll gv a sphr. W clam that th covrs s tru as wll. Idd, w ca assum that th statmt s kow for th 2m 2-go. Lt σ b a glug of th 2m-go that gvs a sphr. If thr s a cocto btw two adjact sds, w may glu thm ad go from a 2m-go to a 2m 2-go (Fg. 5). Thus, t s suffct to cosdr th cas wh adjact sds ar vr coctd. Th thr xst adjact sds a ad b whos ls (coctg thm to som c, d) trsct wth ach othr. Lt us ow rplac σ by aothr parg σ, whos oly dffrc from σ s that a s coctd to b ad c to d (Fg. 6). O ss by spcto (chck t!) that ths dos ot dcras th umbr of boudary compots of th rsultg surfac. Thrfor, sc σ gvs a sphr, so dos σ.but σ has adjact sds coctd, th cas cosdrd bfor, hc th clam. Fgur 4. Parg of sds of a 6-go. 6 * Fgur 5 - Fgur 6 Now t rmas to cout th umbr of ways to coct mdpots of sds wth ls wthout trsctos. Suppos w draw o such l, such that th umbr of sds o th lft of t s 2k ad o th rght s 2l (so that k + l = m ). Th w fac th problm of coctg th two sts of 2k ad 2l sds wthout trsctos. Ths shows that th umbr of glugs D m satsfs th rcurso D m = D k D l k+l=m I othr words, th gratg fucto D m x m = + x + satsfs th quato f = xf 2.Ths mpls that f = 4x 2x, whch ylds that D m = C m. W ar do. Corollary 4.8 ca b usd to drv th th followg fudamtal rsult from th thory of radom matrcs, dscovrd by Wgr 955.

6 24 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY Thorm 4.9. (Wgr s smcrcl law) Lt f b a cotuous fucto o R of at most polyomal growth at fty. Th lm Trf (A/ N ) Tr(A2 )/2 2 f (x) 4 x 2 dx. N N h N 2π 2 Ths thorm s calld th smcrcl law bcaus t says that th graph of th dsty of gvalus of a larg radom Hrmta matrx dstrbutd accordg to th Gaussa utary smbl (.. wth dsty Tr(A2 )/2 da) s a smcrcl. Proof. By Wrstrass uform approxmato thorm, w may assum that f s a polyomal. (Exrcs: Justfy ths stp). Thus, t suffcs to chck th rsult f f (x) = x 2m. I ths cas, by Corollary 4.8, th lft had sd 2 s C m. O th othr had, a lmtary computato ylds 2π 2 x 2m 4 x 2 = C m, whch mpls th thorm Hrmt polyomals. Th proof 4 of Thorm 4.7 gv blow uss Hrmt polyomals. So lt us rcall thr proprts. Hrmt s polyomals ar dfd by th formula x 2 d x H 2 (x) = ( ). dx So th ladg trm of H (x) s (2x). W collct th stadard proprts of H (x) th followg thorm. Thorm 4.0. () Th gratg fucto of H (x) s f (x, t) = H (x) t 0! =. 2xt t2 () H (x) satsfy th dffrtal quato f 2xf +2f =0. I othr words, H (x) x 2 /2 ar gfuctos of th oprator L = 2 + x 2 (Hamltoa of th quatum harmoc oscllator) wth 2 2 gvalus +. 2 () H (x) ar orthogoal: (v) O has (f k > m, th aswr s zro). (v) O has x 2 H m (x)h (x)dx = 2!δ m π x 2 x 2m (2m)! H 2k (x)dx = (m k)! π H r r 2 (x) r! = H 2k (x). 2 r r! 2 k k! 2 (r k)! k=0 2 2(k m) Proof. (sktch) () Follows mmdatly from th fact that th oprator d ( ) t! dx maps a fucto g(x) to g(x t). () Follows from () ad th fact that th fucto f (x, t) satsfs th PDE f xx 2xf x +2tf t =0. () Follows from () by drct tgrato (o should comput R f (x, t)f (x, u) x 2 dx usg a shft of coordat). 2m (v) By (), o should calculat R x x 2 dx. Ths tgral quals 2xt t2 ( ) 2m (2m 2p)! x 2m (x t)2 dx = (y + t) 2m y 2 dy == π t 2p 2p 2 m p. (m p)! R R p Th rsult s ow obtad by xtractg dvdual coffcts. 4 I adoptd ths proof from D. Jackso s ots

7 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY 25 (v) By (), t suffcs to show that 2 r+k H 2 r (x)h 2k (x) x 2 r!2 (2k)! dx = k! 2 (r k)! π R To prov ths dtty, lt us tgrat th product of thr gratg fuctos. By (), w hav f (x, t)f (x, u)f (x, v) x 2 dx = 2(tu+tv+uv). π R Extractg th coffct of t r u r v 2k, w gt th rsult Proof of Thorm 4.7. W d to comput th tgral Tr(A 2m ) Tr(A2 )/2 da. h N To do ths, w ot that th tgrad s varat wth rspct to cojugato by utary matrcs. Thrfor, th tgral ca b rducd to a tgral ovr th gvalus λ,...,λ N of A. Mor prcsly, cosdr th spctrum map σ : h N R N /S N. It s wll kow (du to H. Wyl) that th drct mag σ /2 da s gv by th formula σ da = C P λ 2 <j (λ λ j ) 2 dλ, whr C> 0 s a ormalzato costat that wll ot b rlvat to us. Thus, w hav P ( λ 2m ) λ2 /2 P m (N) = RN <j (λ λ j ) 2 dλ P λ 2 R /2 N <j (λ λ j ) 2 dλ To calculat th tgral J m th umrator, w wll us Hrmt s polyomals. Obsrv that sc H (x) ar polyomals of dgr wth hghst coffct 2, w hav <j (λ λ j ) = 2 N (N )/2 dt(h k (λ )), whr k rus through th st 0,,...,. Thus, w fd P J m := ( λ 2m ) λ2 /2 (λ λ j ) 2 dλ = R N <j 2 m+n 2 /2 N λ 2m P λ 2 (λ λ j ) 2 dλ = (2) P 2 m N (N 2)/2 2m λ2 N λ dt(hk (λ j )) 2 dλ = R N P 2 m N (N 2)/2 2m λ2 N λ ( ( ) σ ( ) τ H σ (λ )H τ (λ ))dλ. (Hr ( ) σ dots th sg of σ). Sc Hrmt polyomals ar orthogoal, th oly trms whch ar ozro ar th trms wth σ() = τ () for = 2,...,N. That s, th ozro trms hav σ = τ. Thus, w hav P J λ 2m λ2 m =2 m N (N 2)/2 N ( Hσ (λ ) 2 dλ = R N R N σ,τ S N R N σ S N (3) N x 2 m N (N 2)/2 N!γ 0... γ N x 2m H j (x) 2 2 dx, j=0 whr γ = H (x) 2 x 2 dx ar th squard orms of th Hrmt polyomals. Applyg ths for m = 0 ad dvdg J m by J 0, w fd J m /J 0 =2 m γ j N x x 2m H j (x) 2 2 γ j=0 j Usg Thorm 4.0 () ad (v), w fd: γ =2! π, ad hc N <j dx j 2 m x 2m H 2k (x) J m /J 0 = x 2 dx π R j=0 2 k k! 2 (j k)! k=0

8 26 MATHEMATICAL IDEAS AND NOTIONS OF QUANTUM FIELD THEORY Now, usg (v), w gt (2m)! N j 2 k j! J m /J 0 = = 2 m (m k)!k! 2 (j k)! j=0 k=0 )( ) (2m)! N j ( m j 2 k 2 m m! k k. j=0 k=0 Th sum ovr k ca b rprstd as a costat trm of a polyomal: j ( )( ) 2 k m j = C.T.(( + z) m ( + 2z ) k ). k k k=0 Thrfor, summato ovr j (usg th formula for th sum of th gomtrc progrsso) ylds ( ) (2m)! C.T.(( + z) m ( + 2z ) N (2m)! m )= 2 p m )( N J m /J 0 = 2m m! 2z 2 m m! p p + p=0 W ar do.

3.4 Properties of the Stress Tensor

3.4 Properties of the Stress Tensor cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato