8.323 Relativistic Quantum Field Theory I
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1 MI OpenCourseWare Relatvstc Quantum Feld heory I Sprng 2008 For nformaton about ctng these materals or our erms of Use, vst:
2 MASSACHUSES INSIUE OF ECHNOLOGY Physcs Department 8.323: Relatvstc Quantum Feld heory I Prof.Alan Guth Aprl 2, 2008 LECURE NOES 6 PAH INEGRALS, GREEN S FUNCIONS, AND GENERAING FUNCIONALS In these notes we wll extend the path ntegral methods dscussed n Lecture Notes 5 to descrbe Green s functons, whch we defne to be ground state expectaton values of the tme-ordered product of Hesenberg operators.for the case of a nonrelatvstc partcle movng n one dmenson, dscussed n Lecture Notes 5, the Green s functons can be wrtten as G(t N,..., t 1 ) 0 {x(t N )x(t N 1 )... x(t 1 )} 0 = 0 x(t N )x(t N 1 )... x(t 1 ) 0, (6.1) where 0 denotes the ground state, and the second lne assumes that we have labeled the tme arguments so that they are tme-ordered, n the sense that t N t N 1... t 1. (6.2) In the quantum feld theory, the Green s functons wll be defned analogously by G(x N,..., x 1 ) 0 {φ(x N )... φ(x 1 )} 0, = 0 φ(x N )... φ(x 1 ) 0, (6.3) where 0 denotes the vacuum state, and agan the second lne assumes that the tme arguments are tme-ordered.in the nonrelatvstc quantum mechancs example of Eq.(6.1), the Green s functons are not quanttes that are partcularly nterestng, so they are usually never mentoned n a course n quantum mechancs.we wll soon see, however, that the quantum feld theory Green s functons of Eq.(6.3) are very nterestng.in partcular, the entre formalsm for calculatng scatterng cross sectons and decay rates wll be based upon relatng these quanttes to the Green s functons.in addton to showng how to express these Green s functons as path ntegrals, n these notes we wll also see that one can defne a generatng functonal Z[J] n such a way that the Green s functons can be expressed smply n terms of the functonal dervatves of the generatng functonal.
3 8.323 LECURE NOES 6, SPRING 2008 p. 2 Path Integrals, Green s Functons, and Generatng Functonals GREEN S FUNCIONS: o begn, we recall that n Lecture Notes 5 we learned to express the evoluton operator of quantum mechancs as a path ntegral: U f = x(tf )=x f x(0)=x S[x(t)] Dx(t) e h, (6.4) where tf S [x(t)] = dt L(x, ẋ) (L = Lagrangan) 0 t f { } 1 = dt mẋ 2 V (x). 2 0 (6.5) We also know that the Hesenberg feld operators appearng n Eq.(6.1) can be wrtten as x(t) = e Ht x(0) e Ht, (6.6) where x(0) x s the Schrödnger representaton poston operator.eq.(6.1) can S then be rewrtten as G(t N,..., t 1 )= 0 e Ht N x S e H(t N t N 1 ) x S... e H(t 2 t 1 ) x S e Ht 1 0. (6.7) o express ths quantty as a path ntegral, we can nsert at each operator x S a complete set of states n the poston representaton, usng Identty Operator = dx x x, (6.8) whch can be multpled by x S to gve x = dxx x x = dx x x x. (6.9) hen G(t N,..., t 1 )= S S dx 1... dx N 0 e Ht N x N x N x N e H(t N t N 1 ) xn 1 x N 1... x 2 x 2 e H(t 2 t 1 ) x1 x 1 x1 e Ht 1 0. (6.10)
4 8.323 LECURE NOES 6, SPRING 2008 p. 3 Path Integrals, Green s Functons, and Generatng Functonals he matrxelements nthsexpresson canall be wrttenaspathntegrals, except for the ground state matrx elements on the two ends.even these matrx elements can be treated by path ntegral methods, however, by notng that the ground state s defned n terms of the Hamltonan, and path ntegrals can be used to construct matrx elements of exponentals of the Hamltonan.An arbtrary state, such as the state of defnte poston x 0 for some constant x 0, can always be expanded n energy egenstates: x 0 = ψ n ψ n x 0. (6.11) n We can solate the ground state contrbuton to ths equaton by multplyng by sdes by e ξh,where ξ s some real number: e ξh x 0 = e ξe n ψ n ψ n x 0, (6.12) n where on the rght the operator H has been replaced by ts egenvalue E n,where H ψ n = E n ψ n.as ξ becomes large, the ground state contrbuton to the rghthand sde wll be less suppressed than any other state.we can compensate for ths suppresson by multplyng by e ξe 0,where E 0 s the energy of the ground state ψ 0 0.hus, lm e ξe 0 e ξh x 0 = ψ 0 ψ 0 x 0. (6.13) ξ For our path ntegral t wll be more convenent to descrbe the real exponental n the above equaton as a small correcton to a much larger magnary exponental. We ntroduce a varable, wth unts of tme, takng the lmt as approaches nfnty tmes (1 ɛ), where ɛ s a small postve constant.at the end we wll take the lmt ɛ 0, but only after the nfnte lmt s carred out, so that ɛ =. If we assume that ψ 0 x 0 = 0, meanng that we have not chosen x 0 to be a pont where the ground state wave functon vanshes, then we can dvde both sdes of Eq.(6.13) by ths quantty, obtanng e E 0 ψ 0 = lm e H x 0 (1 ɛ) ψ 0 x 0. (6.14) For the bra vector, we can use the analogous relaton E 0 e ψ 0 = lm x 0 e H. (6.15) (1 ɛ) x 0 ψ 0 Note that Eq.(6.15) was not obtaned by smply takng the adjont of Eq.(6.14), because the adjont equaton would nvolve nstead of, whch would not be useful for our current purposes.
5 8.323 LECURE NOES 6, SPRING 2008 p. 4 Path Integrals, Green s Functons, and Generatng Functonals If we use Eqs. (6.14) and (6.15) to replace both ground state matrx elements n Eq.(6.10), we obtan G(t N,..., t 1 ) = lm dx 1... dx N (1 ɛ) x 0 ψ 0 ψ 0 x 0 H( tn ) H(tN t x N 1 ) 0 e x N x N x N e x N 1 x N 1... x 2 x 2 e H(t 2 t 1 ) x1 x 1 x 1 e H( +t 1 ) x0. x(t )=x x(t )=x x(t1 )=x 1 x( )=x 0 x 2 Dx(t) e hī S[x(t)] x 1 Dx(t) e h S[x(t)] (6.17) We see that for any t whch s not equal to one of the set {t 1,...,t N },the quantty x(t) appears as one of the varables of ntegraton n one of the path ntegrals.for each t n the set {t 1,...,t N }, x(t ) s requred by the lmts of ntegraton to be x, whch s then ntegrated from to.hus, x(t) s actually a varable of ntegraton for all values of t, provded that we recognze that x(t ) x.all the path ntegrals can then be combned nto one path ntegral from tme to,so Eq.(6.17) smplfes enormously: e 2E 0 (6.16) hus, we see that the path ntegral s so smart that t can even calculate the ground state wave functon for us.he last matrx element n Eq.(6.16) can also be wrtten as H(t1 ( x )) 1 e x 0, whch can be descrbed as the evoluton operator from tme to tme t 1. If each matrx element n Eq.(6.16) s expressed as a path ntegral by usng Eq.(6.4),we fnd e 2E 0 G(t N,..., t 1 ) = lm dx 1... dx N (1 ɛ) x 0 ψ 0 ψ 0 x 0 x(tn )=x N x( )=x 0 Dx(t) e hī S[x(t)] x N x(tn 1 )=x N 1 x(t N )=x N Dx(t) e h S[x(t)] x N 1... G(t N,..., t 1 )= lm e 2E 0 x( )=x 0 Dx(t) e h S[x(t)] x(t N )... x(t 1 ). (1 ɛ) x 0 ψ 0 ψ 0 x 0 x( )=x 0 (6.18)
6 8.323 LECURE NOES 6, SPRING 2008 p. 5 Path Integrals, Green s Functons, and Generatng Functonals he complcated factor n front of the path ntegral can be cancelled f we dvde the expresson by e 2E 0 x( )=x 0 2H 0 0 = e x 0 ψ 0 ψ 0 x 0 x( )=x 0 Dx(t) S[x(t)] e h, (6.19) whch gves fnally G(t N,..., t 1 ) = lm (1 ɛ) x( )=x0 Dx(t) e h S[x(t)] x(t N )... x(t 1 ) x( )= x 0 x( )=x 0 x( )=x 0 h Dx(t) e S[x(t)]. (6.20) In defnng the Green s functons, we made the explct choce n Eq.(6.3) that we would use tme-ordered products.snce the operators x(t )do not n general commute, we presumably would have found a dfferent answer f we had used a dfferent orderng.nonetheless, n our fnal result (6.20), the orderng s not apparent.he product x(t N )... x(t 1 ) s just a product of c-numbers n the ntegrand, so the product would have the same value f the factors were wrtten n any order.hus, we see that the path ntegral naturally pcks out the tme-ordered product.it wll turn out, however, that the tme-ordered product s exactly what we wll need to calculate cross sectons, so there s a perfect ft between the technque and the needed output. Operator products whch are not tme-ordered are stll well-defned, however, so there ought to be some path ntegral method that would allow one to calculate them f one wanted to.if you read over the prevous dervaton and thnk about what would be dfferent f the operators were not tme-ordered, you would fnd that nothng would change untl the step that turns Eq. (6.16) nto Eq. (6.17). For tme-ordered operators, all the tme arguments appearng n the exponents of Eq.(6.16) are postve semdefnte, snce t n+1 t n.hus, each matrx element s an evoluton operator that evolves forward n tme.if the operators were not tmeordered, then some of the tme arguments would be negatve, correspondng to an evoluton operator backwards n tme.such evoluton operators can be expressed as path ntegrals, too, but the sum s over paths that go backwards n tme.when the rght-hand sde of Eq.(6.17) s combned nto a sngle path ntegral, as n Eq.(6.18), the paths x(t) would have to zgzag n tme, sometmes gong forward and sometmes gong backwards, to reproduce the matrx elements n Eq.(6.16).
7 8.323 LECURE NOES 6, SPRING 2008 p. 6 Path Integrals, Green s Functons, and Generatng Functonals GENERAING FUNCIONALS: Eq.(6.20) can be convenently rewrtten by the use of a generatng functonal, defned by Z[J(t)] = lm Dx(t)e (1 ɛ) h dt[l(x,ẋ)+j(t)x(t)]. (6.21) It wll then be possble to express the Green s functons as dervatves of the generatng functonal. he dervatve of a functonal s called a functonal dervatve, as you mght guess, but the defnton s slghtly ndrect.crudely speakng the functonal Z[J(t)] s a functon of an nfnte number of arguments, J(t) for each valueof t, sothe dervatve should look somethng lke a partal dervatve.partal dervatves are defned n terms of the varaton of the functon when one argument s vared wth the other arguments fxed, but that wll not work for Z[J(t)].If we vary J(t) for one value of t only, Z[J(t)] wll not change at all, snce the one pont would have measure zero n the ntegraton of Eq.(6.21). So, the functonal dervatve s defned by frst thnkng about how a functon of many varables changes when all of ts varables are changed by a small amount.if a functon of N varables s denoted by F (z 1,...,z N ), then ts frst order aylor expanson can be wrtten N F F (z 1 + z 1,...,z N + z N )= F (z 1,...,z N )+ zj + O ( z 2 ). (6.22) z j=1 j Eq.(6.22) could be used as the defnton of F/ z j, whch would be equvalent to the usual defnton.he functonal dervatve δz/δj(t) s defned n analogy to Eq.(6.22): Z[J(t)+ J(t)] Z[J(t)] + dt δz δj(t ) J(t )+ O ( J 2 ). (6.23) o calculate the functonal dervatve of Eq.(6.21), we wrte Z[J(t)+ J(t)] = Dx(t)e h dt[l(x,ẋ)+[j(t)+ J(t)]x(t)] [ ] dt[l(x,ẋ)+j(t)x(t)] = Dx(t)e h 1+ dt J(t ) x(t h ) = Z[J(t)] + dt J(t ) Dx(t)e h h dt[l(x,ẋ)+j(t)x(t)] x(t ). (6.24)
8 8.323 LECURE NOES 6, SPRING 2008 p. 7 Path Integrals, Green s Functons, and Generatng Functonals Comparng Eqs. (6.23) and (6.24), one sees that δz[j(t)] = lm Dx(t)e δj(t ) (1 ɛ) h dt[l(x,ẋ)+j(t)x(t)] x(t ). (6.25) hus, referrng to Eq.(6.20), one sees that 1 δz[j(t)] = G(t). (6.26) Z[J(t)] δj(t) J=0 h Snce Eq.(6.25) mples that the functonal dfferentaton δ/δj(t )brngs down a factor of hī x(t ) n the ntegrand, t s easy to see that successve functonal dfferentatons would brng down successve factors of hī x(t).hus, G(x N,..., x 1 ) 0 {φ(x N )... φ(x 1 )} 0, = ( h) N δ N Z[J(t)] Z[J(t)] δj(t 1 )...δj(t N ) J=0. (6.27)
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