The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space that moves n tme. Ths Hlbert space can be descrbed n any bass we choose: coordnate, x, momentum p, or whatever suts our need. It s also possble to regard the state as fxed and the bass as changng n tme. Ths s the Hesenberg pcture.. Hesenberg operators Consder an operator actng on a state, then projected onto any other state at tme t, χ, t χ,  ψ, t Û t, Â Û t, t0 ψ, χ, Û t, ÂÛ t, ψ, so f we defne a tme-dependent Hesenberg operator,  H  t Û t, ÂSÛ t, then we get the same predcton by lookng at  t actng on the fxed ntal state: χ, t ÂS ψ, t χ, ÂH t ψ, We may replace the Schrodnger equaton wth evoluton equatons for operators. dervatve where Û t, e Ĥt t0, Takng the tme dâh dt tû t, ÂSÛ t, + Û t, ÂS tû t, t, ĤÛ ÂSÛ t, Û t, ÂSÛ t, Ĥ ĤÂH ÂHĤ and we have the Hesenberg equaton of moton, dâh dt Accordng to Sakura, ths was frst wrtten by Drac. [Ĥ,  H ]
.2 Hesenberg bass kets The Hesenberg pcture also requres a change n the bass kets. Snce bass kets are egenkets of partcular operators, and the operators are now tme-dependng, the egenkets also change. We have  S a S a a S where a state n the Schrödnger bass s gven by ψ a, t a ψ, t In the Hesenberg pcture, these become  H t a, t H a a, t H Û Â S Û a, t H a a, t H  S Û a, t H aû a, t H so we must have Û a, t H a S Invertng, a, t H Û a S we see that the Hesenberg bass evolves oppostely to the Schrodnger state to gve the same result..3 Transton ampltudes Gven the tme-dependence of the bass kets, we may ask for the probablty ampltude for a bass ket a, H at tme to be found n another drecton b, H at tme t, b, t a, Ths s called the transton ampltude. For example, the transton ampltude for a system to go from x at tme to x at tme t s x, t x, 2 Propagators We have seen that the tme evoluton of state s gven by ψ, t e Ĥt t0 ψ, when the Hamltonan s ndependent of tme. Insertng an dentty n terms of an energy bass, ψ, t e Ĥt t0 a E a E a ψ, a e Eat t0 E a E a ψ, Now vew the state n a coordnate bass, x ψ, t a e Eat t0 x E a E a ψ, ψ x, t a e Eat t0 x E a E a ψ, 2
Insertng one more dentty n the coordnate bass, we have ψ x, t d 3 x e Eat t0 x E a E a x x ψ, a d 3 x e Eat t0 x E a E a x ψ x, a Now defne the propagator K x, t; x, a x E a E a x e Eat t0 so that we have ψ x, t d 3 x K x, t; x, ψ x, Identfyng the propagator for a gven problem separates the ntal wave functon from the potental, allowng a formal soluton for the wave functon at a later tme and arbtrary poston. Holdng x, fxed, u a x s the statonary state wave functon, and e Eat s ts tme dependence, so K x, t; x, satsfes the tme-dependent Schrödnger equaton. Also, lm K x, t; x, δ 3 x x t Moreover, the propagator s essentally a Green s functon that ncludes the tme evoluton, gvng the probablty ampltude for a partcle ntally at x at to be found at x at the later tme t. In ths way, the propagator s the transton ampltude for the system. We can make ths explct: K x, t; x, a x E a E a x e Eat t0 x e Ht a E a E a e Ht0 x x Û t, 0 Û, 0 x so removng the dentty a E a E a and dentfyng the Hesenberg bass states, Û, 0 x x, H and Û t, 0 x x, t H we have the transton ampltude: K x, t; x, x, t x, Transton ampltudes, or propagators, have a composton property. If we nsert the dentty operator n the form d 3 x x, t x, t where < t < t, nto the transton ampltude, t becomes an ntegral over a product of transton ampltudes: x, t x, d 3 x x, t x, t x, t x, Ths shows that the probablty ampltude for gong from x, to x, t s the product of the probablty ampltudes for gong from x, to an ntermedate state at tme t and the probablty of gong from that state to x, t, summed over all possble ntermedate postons. Ths s just lke the composton of condtonal probabltes: P A gven B P A gven C P C gven B C but t s sgnfcant that t apples to probablty ampltudes nstead of probabltes. Ths fact underles Bell s theorem. 3
3 The Feynman path ntegral We consder a partcle wth Hamltonan of the form Ĥ p2 2m + V x. Applyng the composton property N tmes n gong from x 0, to x N, t N, N x N, t N x 0, d 3 x x N, t N x N, t N x, t x 0, Now look at one of the transton ampltudes, x +, t + x, t x +, t e Ĥt+ t x, t x, t e p x+ x e Ĥt+ t x, t Let N be suffcently large that t + t t becomes nfntesmal. To evaluate the translaton operator and the Hamltonan, we nsert a momentum bass, x +, t + x, t d 3 p x +, t p, t p, t e Ĥt+ t x, t d 3 p x +, t p, t p, t Ĥ t x, t d 3 p x +, t p, t p 2 2m t V x t p, t x, t Now, usng p, t x, t 2π 3/2 e p x the nfntesmal transton ampltude becomes x +, t + x, t d 3 p x +, t p, t p, t e Ĥt+ t x, t d 3 p x +, t p, t p, t Ĥ t x, t 2π 3 d 3 p e p x+ p 2 2m t V x t 2π 3 d 3 p e p x+ x e p 2 2m t V x t [ p x + x p2 2π 3 d 3 p exp 2π 3 d 3 p exp 2π 3 d 3 p exp L p, x dt where we fnd the Hamltonan replaced by the Lagrangan, [ p dx ] dt H dt L p, x dt p ẋ H dt Notce that all operators have been replaced by egenvalues. e p x ] 2m t V x t 4
Now reassemble the full, fnte transton ampltude: N x N, t N x 0, 2π 3N 2π 3N and replacng the sum of nfntesmals by an ntegral, exp N N d 3 x d 3 p N L p, x dt exp d 3 x d 3 p t N exp L p, x dt exp L p, x dt exp S [x t, p t] N L p, x dt where S [x t, p t] s the acton functonal n terms of both poston and momentum. Fnally, we defne the functonal ntegral to be the sum over all ntervenng paths, here n both confguraton and momentum spaces: D [x t] D [p t] N 2π 3N /2 N 2π 3N /2 Wth ths notaton, the transton ampltude, or propagator, s gven by x N, t N x 0, D [x t] D [p t] exp S [x t, p t] Ths s the Feynman path ntegral. Notce agan that the acton here s wrtten as an ndependent functonal of poston and momentum. The nfnte products of ntermedate ntegrals may be nterpreted as meanng that the phase exp S [x t] s to be summed over every value of poston and momentum. As we shall see from examples, the result nvolves some curous normalzatons, but the formulaton s very powerful because t may be mmedately generalzed to feld theory. Any theory of felds Φ havng an acton functonal may be quantzed by averagng exp S [Φ] over all feld confguratons. Φ x, t f Φ x, t D [Φ x, t] exp S [Φ x, t, Π x, t] d 3 x d 3 p S [Φ x, t] t f L Φ x, t, Π x, t d 4 x t Here, the poston and tme are smply parameters, whle the feld and ts conjugate mometum are the dynamcal varables. The most mportant advantage of the path ntegral formulaton s that t allows for a systematc perturbaton theory. If we wrte the partcle Lagrangan as and expand the exponental x N, t N x 0, D [x t] L L 0 + V D [p t] exp t N t N L 0 dt + V dt + 5
t s possble to evaluate the potental terms order by order. The same expanson apples to feld theory, Φ x, t f Φ x, t D [Φ x, t] exp S 0 [Φ x, t] + t N V dt + allowng term by term approxmaton. Ultmately, each term n the expanson nvolves dfferent powers of the potental. We keep track of the large number of requred ntegrals by sets of Feynman dagrams, each dagram correspondng to a partcular set of ntegrals. Typcally, equvalence to other methods holds, but s not demanded. The path ntegral s an ndependent model for quantzaton. 4 The momentum ntegrals For the form of Hamltonan we have chosen, Ĥ p2 2m + V x, t s possble to do all of the momentum ntegrals. Each one s smply a Gaussan: 2π 3 d 3 p exp L p, x dt 2π 3 d 3 p exp p x + x p2 t 2m V x dt 2π 3 d 3 p exp p m dx 2 + m 2m dt 2 x + x 2 V x dt 2π 3 exp m 2 v2 V x dt d 3 p exp 2m p mv 2 Lettng y p m x + x, the ntegral becomes d 3 y exp 2m y2 The magnary unt does not really cause any problem. Addng an nfntesmal part for convergence we have d 3 y exp ε y2 d 3 y exp ε + 2m 2m y2 Each of the three Gaussans gves so lm ε 0 d 3 y exp dy exp αy 2 ε y2 2m The full th ntegral s therefore, 2π 3 d 3 p exp L p, x dt π α 3/2 2mπ lm ε 0 ε + 2πm 3/2 m 3/2 exp 2π 2 mv2 V x dt Combnng these n the full path ntegral, we have x N, t N x 0, N d 3 x 2π 3N d 3 p exp N 3N/2 d 3 x m 2π 3 exp N t N L p, x dt 2 mv2 V x dt 6
and replacng the sum of nfntesmals by an ntegral, and defnng the functonal ntegral measure to be the transton ampltude s N D [x t] d 3 x m 2π 3 3N/2 x N, t N x 0, D [x t] exp t N L x, ẋ dt D [x t] exp S [x t] where S [x t] s now the usual confguraton space acton. Ths s the usual form of the Feynman path ntegral. 7