Shift Theorem Involving the Exponential of a Sum of Non-Commuting Operators in Path Integrals. Abstract

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1 YITP-SB-6-4 Sift Teorem Involving te Exponential of a Sum of Non-Commuting Operators in Pat Integrals Fre Cooper 1, an Gouranga C. Nayak 2, 1 Pysics Division, National Science Founation, Arlington VA 2223 arxiv:ep-t/69192v5 19 Jun 29 2 C. N. Yang Institute for Teoretical Pysics, Stony Brook University, SUNY, Stony Brook, NY , USA (Date: Marc 19, 218) Abstract We consier expressions of te form of an exponential of te sum of two non-commuting operators of a single variable insie a pat integration. We sow tat it is possible to sift one of te noncommuting operators from te exponential to oter functions wic are pre-factors an post-factors wen te omain of integration of te argument of tat function is from to +. Tis sift teorem is useful to perform certain integrals an pat integrals involving te exponential of sum of two non-commuting operators. PACS numbers: PACS: q, Me, Cy, Tk Electronic aress: fcooper@nsf.gov Electronic aress: nayak@max2.pysics.sunysb.eu Typeset by REVTEX 1

2 I. INTRODUCTION In backgroun fiel metos in quantum fiel teory one often encounters te exponential of te sum of non-commuting operators insie te pat integration. A simple example of tis type occurs wen looking at pair prouction of carge scalars [1] [2] in te presence of a time epenent backgroun electric fiel E(t) in te longituinal ( z) irection. In te Axial gauge A z = so tat Te action can be written in te form [3] A = E(t)z. (1) S (1) s + = i t < t x < x y < y z < z s [e is[(ˆp +ee(t)z) 2 ˆp 2 z ˆp2 T m2 iǫ] e is(ˆp2 m 2 iǫ) ] z > y > x > t >. (2) Inserting complete set of p T > states 2 p T p T >< p T = 1 we fin (we use te normalization < q p >= 1 2π e iqp ) S (1) = i s s 2 x T e is[( i t +ee(t)z)2 ˆp 2 z ] z > t > 2 p T e is(p2 T +m2 +iǫ) [ t Inserting p z > an p > complete set of states we get S (1) = i s 2 x T 2 p T s e is[( i t +ee(t)z)2 ˆp 2 z ] e izp z e itp p t In te coorinate representation te operators ˆp = 1 i t < t z < z z 1 ]. (3) 4πs p p z p z eis(p2 T +m2 +iǫ) [ t e itp z e izpz z 1 ]. (4) 4πs t an E(t) o not commute wit eac oter. In orer to evaluate tis type of Pat Integral it is quite useful to be able to sift te erivative operator from te exponential to pre-factor an post-factor functions tat occur wen we insert complete sets of states in orer to evaluate te Pat Integral. In particular we woul like to sow tat te following teorem is true: y y x < y < x e [(x+ y )2 +b 2 x 2 +c(y)] x > y > = x < y < x y e [a2 (y)x2 + b 2 x 2 +c(y)] x y > y > (5) 2

3 wic after inserting complete set of states can be written as y p x e ip x(x y ) e iyp y x < y < x e [(x+ y )2 + b 2 x 2 +c(y)] x > y >= 1 y xe iypy e ipx(x p y y ) < p y < p x e a2 (y)x 2 +b 2 x 2+c(y) p x > p y > p y p x (6) were x integration from to + must be performe for eqs. (5) an (6) to be true. Here (wic is equal to i in most of te pysical examples, see eq. (4)) an b are constants an a,c are functions of single variable, suc tat te integration over x is well efine. In wat follows we will assume tat an c(y) are sufficiently ifferentiable, integrable etc. so tat all te formal manipulations are vali. We ave use te normalization < x p x >= 1 2π e ixpx. (7) It can be note tat eq. (6) can not be erive by replacing x x y (8) irectly in eq. (6). Tis is because an x commute wit eac oter in te exponential wereas an y o not commute wit eac oter uner tis replacement. Hence we will use a similarity transformation tecnique to erive te above teorem wic avois tis problem. Te sift teorem leas to te special case W = = 1 y y x < y < x e [(x+ y )2] x > y > < p x e a2 (y)x 2 p x > eip x x p y p y p x p x e ixpx e iypy < p y e ipx y y p y > e ixp x e iyp y. (9) Let us evaluate te left an rigt an sie of te above equation separately. For te left an sie we fin W = = y y x < y < x e [(x+ y )2] x > y > x p y p y < y p y > < x x >< p y e [(x+ y )2] p y >< p y y >. (1) 3

4 Since < p y e [(x+ y )2] p y > is inepenent of we can take < y p y y > to te left. We fin W = y x < p y e [(x+ y )2] p y > = 1 p x p y < p y 2π p y p y Altoug eq. (11) is formally infinite, te x-integral insie W is finite. I(y) = p x < p y y >< y p y >< x p x >< p x x > x e (x+ y )2 p y > (11) x e (x+ y )2 (12) Now evaluating te rigt an sie of eq. (9) we fin W = 1 y x p y p y p x p x e eixpx iypy < p y e ipx < p x e a2 (y)x 2 p x > e ip x = 1 y y p x y > e ixp x e iyp y p y p y p y < p x < p y e a2 (y)x 2 p y > p x > < p It can be sown tat < p y f(y) p y >= y eip x p y p x y p y > e ixp x e iyp y < p y f(y) y >< y p y >= 1 (2π) y p x eiypy e ixpx < p y e ipx y p y > y. (13) y e iy(py p y ) f(y ) (14) is inepenent of y an < p y f(y) y p y >= i y < p y f(y) y >< y p y > p y = ip 1 y y e iy(py p y) f(y ) (2π) (15) is inepenent of y an. Hence we can easily integrate over y in eq. (13). Also since y < p y eip x y p y > is inepenent of W = 1 (2π) < p y eip x = 1 (2π) < p y eip x = 1 < p y e ip x x p y p y y py >< p y e ipx x p y p y y y py >< p y e ipx x p y p y we can bring it to te left. We fin from eq. (13) p y p x p x) x eix(px p y p y >< p x < p (y)x 2 y p e a2 y > p x > p x p x e ix(px p x) x p y y py >< p y e ipx y p y >< p y < p x e a2 (y)x 2 x >< x p x > p + p x p x eix(px p x ) x e ix (p x p x ) p y y > y p y >< p y e a2 (y)x 2 p y >. (16) 4

5 Since te x epenence is only in e ix(px p x ) we can now easily integrate over x to fin W = 1 (2π) < p y e ipx p y p y = 1 (2π) = 1 x 2π = 1 p y 2π p y p y y py >< p y e ipx p y p y p x p x p x p x x y p y >< p y e a2 (y)x 2 p p x < p y e a2 (y)x 2 p y > p x < p y x e a2 (y)x 2 p y > y > x < p y p y >< p y e a2 (y)x 2 p y > (17) Altoug eq. (17) is formally infinite, te x-integral insie W is finite. I(y) = Hence from eqs. (11) an (17) we fin I(y) = x e a2 (y)x 2 (18) x e (x+ y )2 = x e a2 (y)x 2. (19) Tis above teorem eq. (6) can be generalize to involve matrices as follows I ij (y) = [ y x < y < x e [(A(y)x+ y )2 +B 2 x 2 +C(y)] x > y >] ij = 1 [ y x p y p y p x p x e iypy e ixpx < p y e ipx A(y) A(y) < p x e [A2 (y)x 2 + B 2 x 2 +C(y)] p x > eip x y y p y > e ixp x e iyp y ] ij. (2) Here an B are constants an A ij (y), C ij (y), are (i, j) imension matrices wic o not commute wit ; cosen tat te integration over x is well efine. In wat follows we will y assume tat A ij (y) an C ij (y) are sufficiently ifferentiable, integrable etc. so tat all te formal manipulations are vali. Tis sift by erivative tecnique will be very useful wen one stuies particle prouction from arbitrary backgroun fiels via Scwinger-like mecanisms in QED an QCD [1, 2]. Quark an gluon prouction from arbitrary classical cromofiels is expecte to be an important ingreient in te prouction an equilibration of te quark-gluon plasma foun at te RHIC an LHC [4, 5]. 5

6 Tis paper is organize as follows. In section II we provie general erivations of eqs. (6), an (2) by using similarity transformation tecniques. We verify eq. (19) by irectly performing te integration, were we consier te integrals as a function of te variable an assume tat te integrals ave a unique Taylor Series in. We present our conclusions in section IV. II. SIMILARITY TRANSFORMATION APPROACH FOR DERIVING THE SHIFT THEOREM In tis section we provie a general erivation of eqs. (6) an (2) by using similarity transformations. Before giving suc a erivation we consier ere a simple case ( = a=constant) to sow ow te erivative operator acts as a c-number wen x integration is from to +. We fin by using Fourier transformation tecnique = xe (ax+ y )2 f(y) = p xe a2 x 2 f(p)e iyp = xe (ax+ y )2 pf(p)e iyp = p xe (ax+ip)2 f(p)e iyp xe a2 x 2 f(y) (21) In te above we ave assume tat f(y) is well enoug beave so tat it is legal to cange te orer in wic te integrations are taken. In wat follows we will assume tat f(y) is sufficiently ifferentiable, integrable etc. so tat all te formal manipulations are vali. However, tis Fourier transformation tecnique oes not work if is not a constant. Tis is because an y o not commute wit eac oter in te exponential. For tis purpose we use a similarity transformation tecnique to erive te sift teorem. A. Sift Teorem Involving Non-Commuting Operators in te Exponential Consier te following similarity transformations acting on x: Since e y x± y = e± x commutes wit b 2 x 2 we fin (x+ y )2 + b 2 x 2 = e y x [(e y y x x e x e y y x. (22) x x) 2 + b 2 x 2]e y x. (23) 6

7 Hence F(y) = = y e y y x x > y >. x < y < x e [(x+ y )2 + b 2 x 2 +c(y)] x > y > x < y < x e y x e [(e Now inserting complete set of states we fin y x e y x x) 2 +b 2 x 2+e y x c(y)e y x ] F(y) = p y p y p x p x y x < y p y > < x p x > < p y < p x e y x e [(e y x e y x x) 2 +b 2 x 2+e y x c(y)e y x] e y x p x > p y >< p x x >< p y y > = 1 p y p y p x p x y xe iypy e ixpx < p y < p x e y x e [(e y x e y x x) 2 +b 2 x 2+e y x c(y)e y x] e y (24) x p x > p y > e ixp x e iyp y. (25) Unlike te situation in eq. (8) we can now cange te x integration variable to x via x = x y. (26) Tis is because (unlike te left an sie of eq. (24)), an x can not be intercange in te rigt an sie of eq. (24). Hence we can cange te x variable to x via eq. (26) wic involves a erivative. Wit te above cange in integration variable te integration limits for x remain ±. Uner tis cange of integration variable one also as x = x. Wit tese canges we fin from te equation (24) F(y) = 1 p y p y p x p x y xe iypy e ipx(x e [(e y x e y x (x y ))2 +b 2 x 2+e y x c(y)e y x] e y y ) < p y < p x e x p x > p y > y x e ip x (x y ) e iyp y. (27) Using eq. (22) for te similarity transformation of (x ) we fin y y p x e ip x(x x < y < x e [(x+ y )2 + b 2 x 2 +c(y)] x > y >= 1 y xe iypy e ipx(x y ) e iyp y. p y y ) < p y < p x e a2 (y)x 2 +b 2 x 2+c(y) p x > p y > Tis conclues our erivation of eq. (6), an consequently te special case (19). 7 p y p x (28)

8 B. Sift Teorem Involving Matrices an Non-Commuting Operators in te Exponential We next consier te similarity transformation on te matrices xδ ij as follows δ ij x±[ A(y) [ y ]]ij = [e ± were A ij (y) is y epenent matrix. Since [e x A(y) [ y ] ] ij commutes wit Bδ [(A(y)x+[ y ])2 ] ij +δ ij B 2 x = [e 2 x [e x A(y) [ y ] ] lj. x A(y) [ ij 2 y ] xe x 2 we fin A(y) [ y ] ] im [(e Repeating te same logic as use previously, we obtain x A(y) [ x A(y) [ y ] ] ij (29) y ] A(y)e x A(y) [ y ] x) 2 +B 2 I ij (y) = [ y x < y < x e [(A(y)x+ y )2 +B 2 x 2 +C(y)] x > y >] ij = 1 [ y x p y p y p x p x < p eiypy y e ixpx e ipx A(y) A(y) < p x e [A2 (y)x 2 + B 2 x 2 +C(y)] p x > e ixp x e ip x x 2]ml y (3) y p y > e iyp y ]ij. (31) Since tis erivation is rater formal an relies on similarity transformations tat are not very familiar, we will now give examples emonstrating te usefulness an valiity of te special case eq. (19), assuming tat te integrals efine a function wic is Taylor expanable in a series in. III. SOME SPECIAL CASES In tis section we woul like to consier te special case We woul like to sow tat A[,y] = x e (x+ y )2 f(y) (32) A[,y] = x e a2 (y)x 2 f(y) = A[ =,y]. (33) Tootiswewillneetoassumetatf(y)issuctatA[,y]asauniqueTaylorexpansion in te variable. 8

9 To obtain te Taylor series we will use a teorem for two non-commuting operators A, B e (A+B) = e A [1 + n=1 ( 1) n n i=1 xi 1 [ x i e xia B e xia ] ]. (34) Using eq. (34) in (32) we fin e (x+ y )2 f(y) = e A [ x 1 e x 1A B e x 1A x 1 e x 1A B e x 1A x1 x1 1 x 1 e x 1A B e x 1A x 2 e x 2A B e x 2A x 2 e x 2A B e x 2A x2 x 3 e x 3A B e x 3A +... ]f(y) (35) were (x+ y )2 = A+B, A = a 2 (y)x 2 B = 2x y + x y Integrating x from to + in eq. (35) we write y2. (36) A[,y] = were x e A [f(y) I 1 [,y] + I 2 [,y] I 3 [,y] + I 4 [,y] +...] (37) I n [,y] = x e A [ n i=1 [ x i 1 x i e xia B e xia ]]f(y) x (38) e A wit x =1 an n=1,2,3...etc. Te I n consist of a finite number of terms in from n up to 2n Using te expressions for A an B from eq. (36) an performing te x an x i s integrations explicitly in eq (38) we can obtain explicit expressions for all I n. Ten assuming tat after we perform te integrations we can write A[,y] = A n [y] n (39) n= we will fin tat except for n = all te coefficients in te Taylor series in are zero. A. Examples using simple function for an f(y) First, we will consier two examples for an f(y) to emonstrate ow eq. (32) works before giving te result for general an f(y). If we look at eac power of 9

10 in te expression of I n, eac o power of formally vanises because it contains an o integration over x (see eq. (36)). Example I: = y, f(y) = y. Using = y an f(y) = y in eq. (38) we fin I 1 = 2 1 2y, I 2 = (4) 2y 24y 3 wic gives I 1 +I 2 = y 3, inepenent of terms containing two powers of. Similarly we fin I 3 = y3 48y 5, I 4 = y (41) 3 48y5 64y 7 wic gives I 1 + I 2 I 3 + I 4 = y y 7, inepenent of terms containing four powers of. Tis process can be repeate an we fin I 1 +I 2...I n is inepenent of terms containing upto n powers of. Tus if we assume tat our series in B gives us te unique Taylor Series in, ten we fin tat te answer is inepenent of wic is wat we wise to sow. Example II: = 1 y 2, f(y) = e y. Using = 1 y 2 an f(y) = e y in eq. (38) we fin I 1 = e y 2 [ 1+ 2 y + 1 y 2], I 2 = e y { 2 [1 2 y 1 y 2]+4 [ y 5 3y y 4]},(42) wic gives I 1 +I 2 = e y 4 [ y 5 3y y 4 ], inepenent of terms containing two powers of (or ). Similarly we fin y I 3 = e y { 4 [ 1+ 4 y + 2 3y 2 [ 1 2 3y 4] y + 7 6y 7 2 2y 7 4 y y 6]}, I 4 = e y { 4 [ y 5 y + 35 [ 1 2 6y 4] y 49 6y y y y 6] + 8 [ y 1 2y y y y y y 8]} (43) wic gives I 1 +I 2 I 3 +I 4 = e y { 6 [ y 7 y y y y 6] + 8 [ y 1 2y y y y y y 8]} (44) 1

11 inepenent of terms containing four powers of. So continuing tis reasoning to larger n we fin again naively te result is inepenent of. Of course for te above coices of, if one wants to also integrate over y one must exclue te origin in furter integration over y for tis result to make sense. B. General an f(y) an For general an f(y) we fin from eq. (38) ( I 1 = 2 f[y]a [y] 2 + a [y]f [y] 2a[y] 2 a[y] I 2 = 2 ( f[y]a [y] 2 2a[y] 2 4 ( 5f[y]a [y] 4 24a[y] 4 7a [y]f [y] 6a[y] a [y]f [y] a[y] a [y] 3 f [y] a[y] 3 + 7f[y]a [y]a (3) [y] 12a[y] 2 By aing eqs. (45) an (46) we fin I 1 +I 2 = 4 ( 5f[y]a [y] 4 24a[y] 4 4a [y] 2 f [y] 3a[y] 2 f[y]a(4) [y] 6a[y] 7a [y]f [y] 6a[y] ) f(4) [y] f[y]a [y] 2a[y] +f [y] + f[y]a [y] 2a[y] ) + f [y] f[y]a [y] 2 a [y] + 2a [y]f [y]a [y] + 3f[y]a [y] 2 a[y] 3 a[y] 2 8a[y] 2 2f [y]a (3) [y] 3a[y] a [y] 3 f [y] a[y] 3 + 7f[y]a [y]a (3) [y] 12a[y] 2 a [y]f (3) [y] a[y] f[y]a [y] 2 a [y] a[y] 3 2f [y]a (3) [y] 3a[y] wic is inepenent of terms containing two powers of. ) f[y]a(4) [y] 6a[y] (45) + 4a [y] 2 f [y] 3a[y] 2 ) f(4) [y] + 2a [y]f [y]a [y] a[y] 2 + 3f[y]a [y] 2 8a[y] 2 + a [y]f (3) [y] a[y] Similarly evaluating I 3 an I 4 we fin tat I 1 +I 2 I 3 +I 4 oes not contain terms up to four powers of. Tis process can be repeate up to arbitrary powers of so we fin tat te coefficients of n for n=1,2,3... vanis Tus if tere is a unique Taylor series for A[,y], ten we obtain wic is wat we wante to sow. A[,y] = A[ =,y] (48) 11 (47) (46)

12 IV. CONCLUSIONS To conclue, we ave sown tat, remarkably, insie of integrals over te entire real line one can sift te non-commuting erivative operator (not epening on te integration variable) wic occurs in exponentials just as if it were a constant. In particular we ave sown tat y y x < y < x e [(x+ y )2 +b 2 x 2 +c(y)] x > y > = x < y < x y e [a2 (y)x2 + b 2 x 2 +c(y)] x y > y > (49) as well as te extension to Matrix functions were x integration from to + must be performe for te above equation to be true. Tis equation leas to te special case I(y) = x e (x+ y )2 f(y) = x e a2 (y)x 2 f(y) (5) were an b are constants an f, a, b c are functions of single variable cosen tat te integration over x is well efine. Tis sift teorem soul prove useful in te evaluation of Pat Integrals tat occur wen utilizing te backgroun fiel meto. Acknowlegments We tank Warren Siegel, George Sterman an Peter van Nieuwenuizen for iscussions. Tis work was supporte in part by te National Science Founation, grants PHY an PHY One of us (F.C.) woul like to tank bot te Santa Fe Institute an te Pysics Department at Harvar University for teir ospitality wile tis work was one. [1] W. Heisenberg an H. Euler, Z. Pysik 98, (1936) 714. J. Scwinger, Pys. Rev. 82 (1951) 664. V. Weisskopf, Kong. Dans. Vi. Selsk. Mat-fys. Me. XIV No. 6 (1936); Englis translation in: Early Quantum Electroynamics: A Source Book, A. I. Miller, (Cambrige University Press, 1994). [2] G. Nayak an P. van Nieuwenuizen Pys. Rev. D71 (25) 1251; F. Cooper an G. C. Nayak, Pys. Rev. D73 (26) 655; G. C. Nayak, Pys. Rev. D72 (25)

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