Path Integrals in Quantum Field Theory. Sanjeev S. Seahra Department of Physics University of Waterloo

Transcription

1 Path Integrals n Quantum Feld Theory Sanjeev S. Seahra Department of Physcs Unversty of Waterloo May, 2000

2 Abstract We dscuss the path ntegral formulaton of quantum mechancs and use t to derve the S matrx n terms of Feynman dagrams. We generalze to quantum feld theory, and derve the generatng functonal Z[J] and n-pont correlaton functons for free scalar feld theory. We develop the generatng functonal for self-nteractng felds and dscuss φ 4 and φ 3 theory.

3 Introducton Thrty-one years ago, Dck Feynman told me about hs sum over hstores verson of quantum mechancs. The electron does anythng t lkes, he sad. It goes n any drecton at any speed, forward and backward n tme, however t lkes, and then you add up the ampltudes and t gves you the wavefuncton. I sad to hm, You re crazy. But he wasn t. F.J. Dyson When we wrte down Feynman dagrams n quantum feld theory, we proceed wth the mnd-set that our system wll take on every confguraton magnable n travelng from the ntal to fnal state. Photons wll splt n to electrons that recombne nto dfferent photons, leptons and ant-leptons wll annhlate one another and the resultng energy wll be used to create leptons of a dfferent flavour; anythng that can happen, wll happen. Each dstnct hstory can be thought of as a path through the confguraton space that descrbes the state of the system at any gven tme. For quantum feld theory, the confguraton space s a Fock space where each vector represents the number of each type of partcle wth momentum k. The key to the whole thng, though, s that each path that the system takes comes wth a probablstc ampltude. The probablty that a system n some ntal state wll end up n some fnal state s gven as a sum over the ampltudes assocated wth each path connectng the ntal and fnal postons n the Fock space. Hence the perturbatve expanson of scatterng ampltudes n terms of Feynman dagrams, whch represent all the possble ways the system can behave. But quantum feld theory s rooted n ordnary quantum mechancs; the essental dfference s just the number of degrees of freedom. So what s the analogue of ths sum over hstores n ordnary quantum mechancs? The answer comes from the path ntegral formulaton of quantum mechancs, where the ampltude that a partcle at a gven pont n ordnary space wll be found at some other pont n the future s a sum over the ampltudes assocated wth all possble trajectores jonng the ntal and fnal postons. The ampltude assocated wth any gven path s just e S, where S s the classcal acton S = L(q, q) dt. We wll derve ths result from the canoncal formulaton of quantum mechancs, usng, for example, the tmedependent Schrödnger equaton. However, f one defnes the ampltude assocated wth a gven trajectory as e S, then t s possble to derve the Schrödnger equaton 2. We can even derve the classcal prncple of least acton from the quantum ampltude e S. In other words, one can vew the ampltude of travelng from one pont to another, usually called the propagator, as the fundamental object n quantum theory, from whch the wavefuncton follows. However, ths formalsm s of lttle Shamelessly lfted from page 54 of Ryder []. 2 Although, the procedure s only vald for velocty-ndependent potentals, see below.

4 use n quantum mechancs because state-vector methods are so straghtforward; the path ntegral formulaton s a lttle lke usng a sledge-hammer to kll a fly. However, the stuaton s a lot dfferent when we consder feld theory. The generalzaton of path ntegrals leads to a powerful formalsm for calculatng varous observables of quantum felds. In partcular, the dea that the propagator Z s the central object n the theory s fleshed out when we dscover that all of the n-pont functons of an nteractng feld theory can be derved by takng dervatves of Z. Ths gves us an easy way of calculatng scatterng ampltudes that has a natural nterpretaton n terms of Feynman dagrams. All of ths comes wthout assumng commutaton relatons, feld decompostons or anythng else assocated wth the canoncal formulaton of feld theory. Our goal n ths paper wll to gve an account of how path ntegrals arse n ordnary quantum mechancs and then generalze these results to quantum feld theory and show how one can derve the Feynman dagram formalsm n a manner ndependent of the canoncal formalsm. 2 Path ntegrals n quantum mechancs To motvate our use of the path ntegral formalsm n quantum feld theory, we demonstrate how path ntegrals arse n ordnary quantum mechancs. Our work s based on secton 5. of Ryder [] and chapter 3 of Baym [2]. We consder a quantum system represented by the Hesenberg state vector ψ wth one coordnate degree of freedom q and ts conjugate momentum p. We adopt the notaton that the Schrödnger representaton of any gven state vector φ s gven by φ, t = e Ht φ, () where H = H(q, p) s the system Hamltonan. Accordng to the probablty nterpretaton of quantum mechancs, the wavefuncton ψ(q, t) s the projecton of ψ, t onto an egenstate of poston q. Hence where we have defned q satsfes the completeness relaton ψ(q, t) = q ψ, t = q, t ψ, (2) q, t = e Ht q. (3) q q = (q q ), (4) whch mples or q ψ = = dq q q q ψ, (5) dq q q. (6) 2

5 q t ( q,t ) f f ( q,t) t = t Fgure : The varous two-legged paths that are consdered n the calculaton of q f, t f q, t Multplyng by e Ht on the left and e Ht on the rght yelds that = dq q, t q, t. (7) Now, usng the completeness of the q, t bass, we may wrte ψ(q f, t f ) = dq q f, t f q, t q, t ψ = dq q f, t f q, t ψ(q, t ). (8) The quantty q f, t f q, t s called the propagator and t represents the probablty ampltudes (expanson coeffcents) assocated wth the decomposton of ψ(q f, t f ) n terms of ψ(q, t ). If ψ(q, t ) has the form of a spatal delta functon (q 0 ), then ψ(q f, t f ) = q f, t f q 0, t. That s, f we know that the partcle s at q 0 at some tme t, then the probablty that t wll be later found at a poston q f at a tme t f s P (q f, t f ; q 0, t ) = q f, t f q, t 0 2. (9) It s for ths reason that we sometmes call the propagator a correlaton functon. Now, usng completeness, t s easly seen that the propagator obeys a composton equaton: q f, t f q, t = dq q f, t f q, t q, t q, t. (0) Ths can be understood by sayng that the probablty ampltude that the poston of the partcle s q at tme t and q f at tme t f s equal to the sum over q of the probablty that the partcle traveled from q to q (at tme t ) and then on to q f. In other words, the probablty ampltude that a partcle ntally at q wll later be seen at q f s the sum of the probablty ampltudes assocated wth all possble 3

6 A 2 B Fgure 2: The famous double-slt experment two-legged paths between q and q f, as seen n fgure. Ths s the meanng of the oft-quoted phrase: moton n quantum mechancs s consdered to be a sum over paths. A partcularly neat applcaton comes from the double slt experment that ntroductory texts use to demonstrate the wave nature of elementary partcles. The stuaton s sketched n fgure 2. We label the ntal pont (q, t ) as and the fnal pont (q f, t f ) as 2. The ampltude that the partcle (say, an electron) wll be found at 2 s the sum of the ampltude of the partcle travelng from to A and then to 2 and the ampltude of the partcle travelng from to B and then to 2. Mathematcally, we say that 2 = 2 A A + 2 B B. () The presence of the double-slt ensures that the ntegral n (0) reduces to the twopart sum n (). When the probablty 2 2 s calculated, nterference between the 2 A A and 2 B B terms wll create the classc ntensty pattern on the screen. There s no reason to stop at two-legged paths. We can just as easly separate the tme between t and t f nto n equal segments of duraton τ = (t f t )/n. It then makes sense to relabel t 0 = t and t n = t f. The propagator can be wrtten as q n, t n q 0, t 0 = dq dq n q n, t n q n, t n q, t q 0, t 0. (2) We take the lmt n to obtan an expresson for the propagator as a sum over nfnte-legged paths, as seen n fgure 3. We can calculate the propagator for small tme ntervals τ = t j+ t j for some j between and n. We have q j+, t j+ q j, t j = q j+ e Ht j+ e +Ht j q j 4

7 t 0 t t 2 t 3 t 4 t 5 q q n f q q f OO Fgure 3: The contnuous lmt of a collecton of paths wth a fnte number of legs = q j+ ( Hτ + O(τ 2 ) q j where we have assumed a Hamltonan of the form = (q j+ q j ) τ q j+ H q j = dp e p(q j+ q j ) τ 2π 2m q j+ p 2 q j τ q j+ V (q) q j, (3) Now, q j+ p 2 q j = H(p, q) = p2 + V (q). (4) 2m dp dp q j+ p p p 2 p p q j, (5) where p s an egenstate of momentum such that p p = p p, q p = 2π e pq, p p = (p p ). (6) Puttng these expressons nto (5) we get q j+ p 2 q j = 2π dp p 2 e p(q j+ q j ), (7) where we should pont out that p 2 s a number, not an operator. Workng on the other matrx element n (3), we get q j+ V (q) q j = q j+ q j V (q j ) = (q j+ q j )V (q j ) = dp e p(q j+ q j ) V (q j ). 2π 5

8 Puttng t all together q j+, t j+ q j, t j = 2π = 2π dp e p(q j+ q j ) [ τh(p, q j ) + O(τ 2 ) ] [ ( dp exp τ p q )] j H(p, q j ), τ where q j q j+ q j. Substtutng ths expresson nto (2) we get n n dq dp ( )) q j q n, t n q 0, t 0 = dp 0 exp τ p j H(p j, q j. (8) 2π τ = In the lmt n, τ 0, we have j=0 n τ j=0 tn t 0 dt, q j τ dq dt = q, dp 0 n = dq dp 2π [dq] [dp], (9) and q n, t n q 0, t 0 = { tn } [dq] [dp] exp dt [p q H(p, q)]. (20) t 0 The notaton [dq] [dp] s used to remnd us that we are ntegratng over all possble paths q(t) and p(t) that connect the ponts (q 0, t 0 ) and (q n, t n ). Hence, we have succeed n wrtng the propagator q n, t n q 0, t 0 as a functonal ntegral over the all the phase space trajectores that the partcle can take to get from the ntal to the fnal ponts. It s at ths pont that we fully expect the reader to scratch ther heads and ask: what exactly s a functonal ntegral? The smple answer s a quantty that arses as a result of the lmtng process we have already descrbed. The more complcated answer s that functonal ntegrals are beasts of a rather vague mathematcal nature, and the arguments as to ther standng as well-behaved enttes are rather nebulous. The phlosophy adopted here s n the sprt of many mathematcally controversal manpulatons found n theoretcal physcs: we assume that everythng works out alrght. The argument of the exponental n (20) ought to look famlar. We can brng ths out by notng that h dp e τ q p { [ τ H(p,q ) = ( ) 2π 2π exp m 2 q τ V (q )]} 2 τ [ dp exp τ ( p m q ) ] 2 2m τ { [ ( m ) ( ) /2 m 2 q = exp τ V (q )]}. 2πτ 2 τ 6

9 Usng ths result n (8) we obtan ( m q n, t n q 0, t 0 = 2πτ N [dq] exp ) n n/2 = tn [ t 0 n dq exp τ j=0 [ m 2 ( dt 2 m q2 V (q) ( ) 2 qj V (q j )] τ )], (2) where the lmt s taken, as usual, for n and τ 0. Here, N s an nfnte constant gven by ( m ) n/2 N = lm. (22) n 2πτ We won t worry too much about the fact that N dverges because we wll later normalze our transton ampltudes to be fnte. Recognzng the Lagrangan L = T V n equaton (2), we have [ tn ] q n, t n q 0, t 0 = N [dq] exp L(q, q) dt = N [dq] e S[q], (23) t 0 where S s the classcal acton, gven as a functonal of the trajectory q = q(t). Hence, we see that the propagator s the sum over paths of the ampltude e S[q], whch s the ampltude that the partcle follows a gven trajectory q(t). Hstorcally, Feynman demonstrated that the Schrödnger equaton could be derved from equaton (23) and tended to regard the relaton as the fundamental quantty n quantum mechancs. However, we have assumed n our dervaton that the potental s a functon of q and not p. If we do ndeed have velocty-dependent potentals, (23) fals to recover the Schrödnger equaton. We wll not go nto the detals of how to fx the expresson here, we wll rather heurstcally adopt the generalzaton of (23) for our later work n wth quantum felds 3. An nterestng consequence of (23) s seen when we restore. Then S[q]/~ q n, t n q 0, t 0 = N [dq] e. (24) The classcal lmt s obtaned by takng 0. Now, consder some trajectory q 0 (t) and neghbourng trajectory q 0 (t)+q(t), as shown n fgure 4. The acton evaluated along q 0 s S 0 whle the acton along q 0 +q s S 0 +S. The two paths wll then make contrbutons exp(s 0 / ) and exp[(s 0 + S)/ ] to the propagator. For 0, the phases of the exponentals wll become completely dsjont and the contrbutons wll n general destructvely nterfere. That s, unless S = 0 n whch case all neghbourng paths wll constructvely nterfere. Therefore, n the classcal lmt the propagator wll be non-zero for ponts that may be connected by a trajectory 3 The generalzaton of velocty-dependent potentals to feld theory nvolves the quantzaton of non-abelan gauge felds 7

10 q( t) q f q( t) + q( t) q Fgure 4: Neghbourng partcle trajectores. If the acton evaluated along q(t) s statonary (.e. S = 0), then the contrbuton of q(t) and t s neghbourng paths q(t) + q(t) to the propagator wll constructvely nterfere and reconstruct the classcal trajectory n the lmt 0. satsfyng S[q] q=q0 ;.e. for paths connected by classcal trajectores determned by Newton s 2 nd law. We have hence seen how the classcal prncple of least acton can be understood n terms of the path ntegral formulaton of quantum mechancs and a correspondng prncple of statonary phase. 3 Perturbaton theory, the scatterng matrx and Feynman rules In practcal calculatons, t s often mpossble to solve the Schrödnger equaton exactly. In a smlar manner, t s often mpossble to wrte down analytc expressons for the propagator q f, t f q, t for general potentals V (q). However, f one assumes that the potental s small and that the partcle s nearly free, one makes good headway by usng perturbaton theory. We follow secton 5.2 n Ryder []. In ths secton, we wll go over from the general confguraton coordnate q to the more famlar x, whch s just the poston of the partcle n a one-dmensonal space. The extenson to hgher dmensons, whle not exactly trval, s not dffcult to do. We assume that the potental that appears n (23) s small, so we may perform an expanson [ tn ] tn exp V (x, t) dt = V (x, t) dt [ tn ] 2 V (x, t) dt +. (25) t 0 t 0 2! t 0 We adopt the notaton that K = K(x n, t n ; x 0, t 0 ) = x n, t n x 0, t 0. Insertng the expanson (25) nto the propagator, we see that K possesses and expanson of the form: K = K 0 + K + K 2 + (26) 8

11 The K 0 term s K 0 = N [ ] [dx] exp 2 mẋ2 dt. (27) If we turned off the potental, the full propagator would reduce to K 0. It s for ths reason that we call K 0 the free partcle propagator, t represents the ampltude that a free partcle known to be at x 0 at tme t 0 wll later be found at x n at tme t n. Gong back to the dscrete expresson: ( m K 0 = lm n 2πτ ) n n/2 = dx exp mτ 2 n (x j+ x j ) 2. (28) Ths s a doable ntegral because the argument of the exponental s a smple quadratc form. We can hence dagonalze t by choosng an approprate rotaton of the x j Cartesan varables of ntegraton. Conversely, we can start calculatng for n = 2 and solve the general n case usng nducton. The result s ( m ) n/2 K 0 = lm n 2πτ n /2 ( 2πτ m Now, (t n t 0 )/n = τ, so we fnally have j=0 ) (n )/2 exp [ m(xn x 0 ) 2 2nτ ]. (29) [ ] m /2 [ m(xn x 0 ) 2 ] K 0 (x n x 0, t n t 0 ) = exp, t n > t 0. (30) 2π(t n t 0 ) 2(t n t 0 ) Here, we ve noted that the substtuton nτ = (t n t 0 ) s only vald for t n > t 0. In fact, f K 0 s non-zero for t n > t 0 t must be zero for t 0 > t n. To see ths, we note that the calculaton of K 0 nvolved ntegratons of the form: e αx2 dx = 2 0 = /2 4 = /2 4 e αx2 dx 0 e αs ds s/2 Θ( s) eαs ds, s/2 where α sgn(τ) = sgn(t n t 0 ). Now, we can ether choose the branch of s /2 to be n ether the left- or rghthand part of the complex s-plane. But, we need to complete the contour n the lefthand plane f α > 0 and the rghthand plane f α < 0. Hence, the ntegral can only be non-zero for one case of the sgn of α. The choce we have mplctly made s the the ntegral s non-zero for α (t n t 0 ) > 0, hence t must vansh for t n < t 0. When we look at equaton (8) we see that K 0 s lttle more than a type of kernel for the ntegral soluton of the free-partcle Schrödnger equaton, whch s really a statement about Huygen s prncple. Our 9

12 choce of K 0 obeys causalty n that the confguraton of the feld at pror tmes determnes the form of the feld n the present. We have hence found a retarded propagator. The other choce for the boundary condtons obeyed by K 0 yelds the advanced propagator and a verson of Huygen s prncple where future feld confguratons determne the present state. The moral of the story s that, f we choose a propagator that obeys casualty, we are justfed n wrtng [ m ] [ ] /2 mx 2 K 0 (x, t) = Θ(t) exp. (3) 2πt 2t Now, we turn to the calculaton of K : [ ] K = N [dx] exp 2 mẋ2 dt dt V (x(t), t). (32) Movng agan to the dscrete case: K = β n/2 dx dx n exp mτ 2 n n (x j+ x j ) 2 τv (x, t ), (33) j=0 = where β = m/2πτ and the lmt n s understood. Let s take the sum over (whch has replaced the ntegral over t) n front of the spatal ntegrals. Also, let s splt up the sum over j n the exponental to a sum runnng from 0 to and a sum runnng from to n. Then n K = τ = β (n )/2 dx β /2 dx dx exp dx + dx n exp mτ 2 mτ 2 (x j+ x j ) 2 V (x, t ) j=0 n (x j+ x j ) 2. (34) We recognze two factors of the free-partcle propagator n ths expresson, whch allows us to wrte n K = τ = j= dx K 0 (x x 0, t t 0 )V (x, t )K 0 (x n x, t n t ). (35) Now, we can replace n = τ by t n t 0 dt and t t n the lmt n. Snce K 0 (x x 0, t t 0 ) = 0 for t < t 0 and K 0 (x n x, t n t) for t > t n, we can extend the lmts on the tme ntegraton to ±. Hence, K = dx dt K 0 (x n x, t n t)v (x, t)k 0 (x x 0, t t 0 ). (36) 0

13 In a smlar fashon, we can derve the expresson for K 2 : K 2 = ( )2 β n/2 dx dx n exp mτ n (x j+ x j ) 2 (37) 2! 2 n = n τv (x, t ) k= j=0 τv (x k, t k ). (38) We would lke to play the same trck that we dd before by splttng the sum over j nto three parts wth the potental terms sandwched n between. We need to construct the mddle j sum to go from an early tme to a late tme n order to replace t wth a free-partcle propagator. But the problem s, we don t know whether t comes before or after t k. To remedy ths, we splt the sum over k nto a sum from to and then a sum from to n. In each of those sums, we can easly determne whch comes frst: t or t k. Gong back to the contnuum lmt: K 2 = ( )2 tn [ t dx dx 2 dt dt 2 K 0 (x n x, t n t ) 2! t 0 t 0 V (x, t )K 0 (x x 2, t t 2 )V (x 2, t 2 )K 0 (x 2 x 0, t 2 t 0 ) tn + dt 2 K 0 (x n x 2, t n t 2 )V (x 2, t 2 )K 0 (x 2 x, t 2 t ) t ] V (x, t )K 0 (x x 0, t t 0 ) But, we can extend the lmts on the t 2 ntegraton to t 0 t n by notng the mddle propagator s zero for t 2 > t. Smlarly, the t 2 lmts on the second ntegral can be extended by observng the mddle propagator vanshes for t > t 2. Hence, both ntegrals are the same, whch cancels the /2! factor. Usng smlar arguments, the lmts of both of the remanng tme ntegrals can be extended to ± yeldng our fnal result: K 2 = ( ) 2 dx dx 2 dt dt 2 K 0 (x n x 2, t n t 2 )V (x 2, t 2 ) (39) K 0 (x 2 x, t 2 t )V (x, t )K 0 (x x 0, t t 0 ). (40) Hgher order contrbutons to the propagator follow n a smlar fashon. The general j th order correcton to the free propagator s K j = ( ) j dx... dx j dt... dt j K 0 (x n x j, t n t j ) V (x j ) V (x )K 0 (x x 0, t t 0 ). (4) We would lke to apply ths formalsm to scatterng problems where we assume that the partcle s ntally n a plane wave state ncdent on some localzed potental.

14 As t ±, we assume the potental goes to zero, whch models the fact that the partcle s far away from the scatterng regon n the dstant past and the dstant future. We go over from one to three dmensons and wrte ψ(x f, t f ) = dx K 0 (x f x, t f t )ψ(x, t ) dx dx dt K 0 (x f x, t f t) V (x, t)k 0 (x x, t t )ψ(x, t ) + (42) We push t nto the dstant past, where the effects of the potental may be gnored, and take the partcle to be n a plane wave state: ψ n (x, t ) = V e p x, (43) where we have used a box normalzaton wth V beng the volume of the box and p x = E t p x. The n label on the wavefuncton s meant to emphasze that t s the form of ψ before the partcle moves nto the scatterng regon. We want to calculate the frst ntegral n (42) usng the 3D generalzaton of (3): where λ = m/2t. Hence, K 0 (x, t) = Θ(t) ( ) λ 3/2 e λx2, (44) π ) 3/2 dx K 0 (x f x, t f t )ψ n (x, t ) = ( λ V π e E t dx e λ(x f x ) 2 +p x. (45) Ths ntegral reduces to ψ n (x f, t f ) as should have been expected, because K 0 s the free partcle propagator and must therefore propagate plane waves nto the future wthout alterng ther form. We also push t f nto the nfnte future where the effects of the potental can be gnored. Then, ψ + (x f, t f ) = ψ n (x f, t f ) dx dx dt K 0 (x f x, t f t) V (x, t)k 0 (x x, t t )ψ n (x, t ) + (46) The + notaton on ψ s there to remnd us that ψ + s the form of the wave functon after t nteracts wth the potental. What we really want to do s Fourer analyze ψ + (x f, t f ) nto momentum egenstates to determne the probablty ampltude for a partcle of momentum p becomng a partcle of momentum p f after nteractng 2

15 wth the potental. Defnng ψ out (x f, t f ) as a state of momentum p f n the dstant future: ψ out (x f, t f ) = e p f x f, (47) V we can wrte the ampltude for a transton from p to p f as S f = ψ out ψ +. (48) Insertng the unt operator = dx f x f, t f x f, t f nto (48) and usng the propagator expanson (46), we obtan S f = (p f p ) dx dx f dx dt ψout(x f, t f )K 0 (x f x, t f t) V (x, t)k 0 (x x, t t )ψ n (x, t ) + (49) The ampltude S f s the f component of what s known as the S or scatterng matrx. Ths object plays a central rôle n scatterng theory because t answers all the questons that one can expermentally ask about a physcal scatterng process. What we have done s expand these matrx elements n terms of powers of the scatterng potental. Our expanson can be gven n terms of Feynman dagrams accordng to the rules:. The vertex of ths theory s attached to two legs and a spacetme pont (x, t). 2. Each vertex comes wth a factor of V (x, t). 3. The arrows on the lnes between vertces pont from the past to the future. 4. Each lne gong from (x, t) to (x, t ) comes wth a propagator K 0 (x x, t t). 5. The past external pont comes wth the wavefuncton ψ n (x, t ), the future one comes wth ψ out(x f, t f ). 6. All spatal coordnates and nternal tmes are ntegrated over. Usng these rules, the S matrx element may be represented pctorally as n fgure 5. We note that these rules are for confguraton space only, but we could take Fourer transforms of all the relevant quanttes to get momentum space rules. Obvously, the Feynman rules for the Schrödnger equaton do not result n a sgnfcant smplfcaton over the raw expresson (49), but t s mportant to notce how they were derved: usng smple and elegant path ntegral methods. 3

16 x f, tf x f, tf x f, tf x 2, t2 S f = x, t + x, t + + x, t x, t x, t Fgure 5: The expanson of S f n terms of Feynman dagrams 4 Sources, vacuum-to-vacuum transtons and tme-ordered products We now consder a alteraton of the system Lagrangan that models the presence of a tme-dependent source. Our dscusson follows secton 5.5 of Ryder [] and chapters and 2 of Brown [3]. In ths context, we call any external agent that may cause a non-relatvstc system to make a transton from one energy egenstate to another a source. For example, a tme-dependent electrc feld may nduce a charged partcle n a one dmensonal harmonc oscllator potental to go from one egenenergy to another. In the context of feld theory, a tme-dependent source may result n spontaneous partcle creaton 4. In ether case, the source can be modeled by alterng the Lagrangan such that L(q, q) L(q, q) + J(t)q(t). (50) The source J(t) wll be assumed to be non-zero n a fnte nterval t [t, t 2 ]. We take T 2 > t 2 and T < t. Gven that the partcle was n t s ground state at T, what s the ampltude that the partcle wll stll be n the ground state at tme T 2? To answer that queston, consder Q 2, T 2 Q, T J = dq dq 2 Q 2, T 2 q 2, t 2 q 2, t 2 q, t J q, t Q, T = dq dq 2 Q 2 e HT 2 e Ht 2 q 2 q 2, t 2 q, t J 4 cf. PHYS 703 March 4, 2000 lecture q e Ht e HT Q = dq dq 2 Q 2 e HT 2 m m e Ht 2 q 2 q 2, t 2 q, t J mn q e HT n n e Ht Q 4

17 J = 0 t 2 J = 0 t J = 0 T Te - Fgure 6: The rotaton of the tme axs needed to solate the ground state contrbuton to the propagator = e (E nt 2 E m T ) φ m (Q 2 )φ n(q ) mn dq dq 2 φ m(q 2, t 2 ) q 2, t 2 q, t J φ n (q, t ), where we have ntroduced a bass of energy egenstates H n = E n n and energy egenfunctons φ n (q, t) = e E nt q m wth φ n (q) = q n. The J subscrpts on the propagators remnd us that the source s to be accounted for. It s mportant to note that φ n (q) s only a true egenfuncton for tmes when the source s not actng;.e. pror to t and later than t 2. The ntegral on the last lne can be thought of as a wavefuncton, φ n (q, t ), that s propagated through the tme when the source s actng by q 2, t 2 q, t J, and s then dotted wth a wavefuncton φ m(q 2, t 2 ). But, φ n (q, t ) and φ m(q 2, t 2 ) are energy egenfunctons for tmes before and after the source, respectvely. Hence, the ntegral s the ampltude that an energy egenstate n wll become an energy egenstate m through the acton of the source. Now, let s perform a rotaton of the tme-axs n the complex plane by some small angle ( > 0), as shown n fgure 6. Under such a transformaton T T + T (5) T 2 T 2 T 2, (52) where we have chosen the axs of rotaton to le between T and T 2. We see that the exponental term e (EnT 2 E mt ) wll acqure a dampng that goes lke e (E n T 2 +E m T ). As we push T and T 2, the dampng wll become nfnte for each term n the sum, except for the ground state whch we can set to have an energy of E 0 0. Therefore, lm T e Q 2, T Q, T J = e E 0(T 2 T ) φ 0 (Q 2 )φ 0(Q ) (53) dq dq 2 φ 0(q 2, t 2 ) q 2, t 2 q, t J φ 0 (q, t ), > 0, where we have set T 2 = T = T for convenence. Now, f we take t 2 and t to ± respectvely, the ntegral reduces to the ampltude that a wavefuncton whch has 5

18 the form of φ 0 (q) n the dstant past wll stll have the form of φ 0 (q) n the dstant future. In other words, t s the ground-to-ground state transton ampltude, whch we denote by 0, 0, J lm T e Q 2, T Q, T J, (54) where the constant of proportonalty depends on Q, Q 2 and T. Now, nstead of rotatng the contour of the the tme-ntegraton, we could have added a small term ɛq 2 /2 to the Hamltonan. Usng frst order perturbaton theory, ths shfts the energy levels by an amount E n = ɛ n q 2 n /2. For most problems (.e. the harmonc oscllator, hydrogen atom), the expectaton value of q 2 ncreases wth ncreasng energy. Assumng that ths s the case for the problem we are dong, we see that the frst order shft n the egenenergy accomplshes the same thng as the rotaton of the tme axs n (54). But, subtractng ɛq 2 /2 from H s the same thng as addng ɛq 2 /2 from L 5. Therefore, { 0, 0, J [dq] exp dt [L(Q, Q) + JQ + 2 ]} ɛq2. (55) Fnally, want to normalze ths result such that f the source s turned off, the ampltude 0, 0, s unty. Defnng [dq] exp { [ dt L(Q, Q) ]} + JQ + 2 ɛq2 Z[J] = [dq] exp { [ dt L(Q, Q) ]}, (56) + 2 ɛq2 we have 0, 0, J = Z[J]. (57) Before movng on the the next secton, we would lke to establsh a result that wll prove very useful later when we consder feld theores. We frst defne the functonal dervatve of Z[J] wth respect to J(t ). Essentally, the functonal dervatve of a functonal f[y], where y = y(x), s the dervatve of the dscrete expresson wth respect to the value of y at a gven x. For example, the dscrete verson of Z[J] s Z (τj(t 0 ), τj(t )... τj(t n )) n exp τ [ L(Q j, Q j ) j=0 + J(t j )Q j + 2 ɛq2 j ] } n = dq, (58) where we have ndcated that the dscrete verson of Z[J] s an ordnary functon of n varables τj(t j ) and omtted the normalzaton factor. We have explctly ncluded 5 An alternatve procedure for snglng out the ground state contrbuton comes from consderng t to be purely magnary,.e. consderaton of Eucldean space. Ths s dscussed n the next secton. 6

19 the weghtng factor τ wth each of the dscrete varables to account for the fact that as n, each J(t k ) covers a smaller and smaller porton of the ntegraton nterval. The functonal dervatve of Z[J] wth respect to J(t k ) s then the partal dervatve of the dscrete expresson wth respect to τj(t k ). Gong back to the contnuum lmt, we wrte the functonal dervatve of Z[J] wth respect to Q(t ) as: Z[J] J(t ) In a smlar fashon, we have [dq]q(t ) exp n Z[J] J(t ) J(t n ) n exp { dt [L(Q, Q) + JQ + 2 ]} ɛq2. (59) [dq]q(t ) Q(t n ) { dt [L(Q, Q) + JQ + 2 ]} ɛq2. (60) We notce a smlarty between ths expresson and the expresson from statstcal mechancs that gves the average value of a mcroscopc varable n the canoncal ensemble. We argue that equaton (60) gves the exact same thng: the expectaton value of Q(t ) Q(t n ). There s one wrnkle, however, whch we now proceed to outlne. Consder, wth t k > t k, q f, t f q(t k )q(t k ) q, t = dq dq n q f, t f q n, t n q k, t k q(t k ) q k, t k q k, t k q(t k ) q k, t k q, t q, t = dq dq n q(t k )q(t k ) q f, t f q n, t n q, t q, t. (6) By pushng t e and t f e, we can repeat our prevous manpulatons to show that dq dq n q(t k )q(t k ) q f, e q n, t n q, t q, e [ ] e = N [dq]q(t k )q(t k ) exp dt L(Q, Q), (62) e whch follows drectly from the arguments of secton 2, and q f, e q(t k )q(t k ) q, e 0 q(t k )q(t k ) 0, (63) whch follows from an argument smlar to the one we used to calculate the matrx element 0, 0, J wth t = t k, t 2 = t k and J = 0. Just as before, the 7

20 rotaton of the tme axs can be acheved by addng ɛq 2 /2 to the Lagrangan. Ths calculaton cannot be repeated for the case t k > t k because the order of q(t k )q(t k ) n q f, t f q(t k )q(t k ) q, t cannot be swtched wthout ntroducng terms nvolvng the commutator of H and q. But, n order to perform the decomposton (6), we need the late q operator appearng to the left of the earler q operator. What ths means s that we must be consderng the tme-ordered product of q(t k ) and q(t k ). Puttng all of ths together along wth the expresson for the functonal dervatves of Z[J], we get 2 Z[J] J(t k )J(t k ) = 2 0 T [q(t k )q(t k )] 0. (64) J=0 Ths s easly generalzed to the tme order product of many q operators: n Z[J] J(t ) J(t n ) = n 0 T [q(t ) q(t n )] 0. (65) J=0 We have demanded strct equalty n these expresson to ensure that the n = 0 case returns 0 0 =. Ths s a very mportant formula for what follows. 5 Free scalar felds We now move on to the quantum feld theory of a scalar feld φ(x). In ths secton we draw on sectons 6. and 6.3 of Ryder [], chapter 2 of Popov [4] and secton 3.2 of Brown [3]. The classcal feld φ s assumed to satsfy the Klen-Gordon equaton ( + m 2 )φ = 0. (66) We defne the vacuum to vacuum transton probablty for ths theory by 0, 0, J = Z[J], (67) where Z[J] = Z 0 { [dφ] exp d 4 x [L(φ) + J(x)φ + 2 ]} ɛφ2, (68) wth Z 0 = { [dφ] exp d 4 x [L(φ) + 2 ]} ɛφ2. (69) In ths expressons, the measure [dφ] s meant to convey an ntegraton over all feld confguratons, whch can be acheved n practce by dvdng spacetme nto N 4 ponts (t m, x, y j, z k ), wth m,, j, k =... N. We can schematcally merge all of these ndces nto a sngle one n =... N 4. The feld s consdered to be a collecton of N 4 ndependent varables, and the functonal ntegraton [dφ] becomes dφ. The functonal Z[J] represents the fundamental object n the theory. Knowledge of the form of Z[J] allows us to derve all the results that we could hope to obtan from 8

21 experments nvolvng the feld φ. Such grandose statements need to be justfed, whch we now proceed to do. We frst substtute the Lagrangan approprate to the Klen-Gordon equaton nto the expresson for Z[J]: Z[J] = Z 0 [dφ] exp ( { d 4 [ x α φ α φ (m 2 ɛ)φ 2] }) + φj. (70) 2 Integratng the α φ α φ term by parts and usng Gauss theorem to dscard the boundary term (assumng φ 0 at nfnty) gves Z[J] = ( { }) [dφ] exp d 4 x Z 0 2 φ( + m2 ɛ)φ φj. (7) Regardng φ as an ntegraton varable, we can change varables accordng to φ(x) φ(x) + φ 0 (x). (72) Notng that d 4 x φ φ 0 = d 4 x φ 0 φ (73) usng Gauss theorem and demandng that φ 0 0 at nfnty, we obtan Z[J] = ( { [dφ] exp d 4 x Z 0 2 φ( + m2 ɛ)φ +φ( + m 2 ɛ)φ 0 + }) 2 φ 0( + m 2 ɛ)φ 0 (φ + φ 0 )J. (74) Let us demand that ( + m 2 ɛ)φ 0 = J(x). (75) We can solve ths equaton by ntroducng the Feynman propagator, whch satsfes ( + m 2 ɛ) F (x) = 4 (x). (76) Hence, φ 0 = F (x y)j(y) d 4 y. (77) We then obtan for Z[J]: Z[J] = Z 0 exp [ 2 [dφ] exp ] J(x) F (x y)j(y) d 4 x d 4 y ] [ 2 φ( + m 2 ɛ)φ d 4 x. (78) 9

22 But we know Z[0] =, so we must have [ Z 0 = [dφ] exp 2 ] φ( + m 2 ɛ)φ d 4 x. (79) Ths leads to our fnal expresson: [ Z[J] = exp 2 ] J(x) F (x y)j(y) d 4 x d 4 y. (80) Before we move forward, we would lke to make a comment on the ncluson of the ɛφ 2 /2 term n our expresson for Z[J]. The reader wll recall that the reason that we added ths term was to smulate the effects of rotatng the tme axs by a small angle n the complex plane. Instead of addng the ɛ-term, we can nstead rotate the t axs by π/2 so that t = τ. The metrc becomes η αβ = dag(,,, ), whch means that we are consderng a Eucldean, not Lorentzan, manfold. In that case, the vacuum-to-vacuum transton ampltude s Z[J] = Z 0 [dφ] exp ( dτdx { [ ( τ φ) 2 + φ 2 + m 2 φ 2] }) φj, (8) 2 where s the del-operator of 3D vector calculus. For J = 0, the argument of the exponental s negatve defnte and the ntegral converges absolutely. So, n some sense, the rotaton of the tme axs ensures that the ntegrals n Z[J] are well behaved. Another aspect the Eucldean manfold comes from the Eucldean verson of Feynman propagator, whch satsfes: ( 2 τ + 2 m 2 ) F (x) = 4 (x). (82) Solvng ths n Eucldean Fourer space, we get F (x) = (2π) 4 d 4 e κ x κ κ 2 + m 2, (83) where κ x = κ 0 τ + k x and κ 2 = (κ 0 ) 2 + k 2 (the s there for convenence). If we change varables accordng to κ 0 = k 0 and τ = t, we get F (x) = C (2π) 4 d 4 e k x k k 2 m 2, (84) where the k 0 ntegraton s to be performed along the contour C shown n fgure 7. But, we can rotate the contour by 90 o clockwse to C, whch s the standard contour used to calculate the Lorentzan Feynman propagator, defned by (76). To some extent, ths explans why the Feynman propagator has the form that t does: t s a drect consequence of the need to rotate the tme axs to solate the vacuum contrbutons to transton ampltudes and make path ntegrals converge. 20

23 Im( k 0 ) Im( k 0 ) C Re( k 0 ) C Re( ) = k 0 Fgure 7: Equvalent contours of ntegraton for the calculaton of Feynman propagator n the complex k 0 plane. Note the poles at ±ω = ± k 2 + m 2. We want to calculate the tme-ordered products gven by n Z[J] n J(x ) J(x n ) = 0 T [φ(x ) φ(x n )] 0. (85) J=0 We recall from the canoncal formulaton of feld theory that 0 T [φ(x)φ(y)] 0 s the ampltude for the creaton of a partcle at y and ts later destructon at x (or vce versa, dependng on the tmes assocated wth x and y). Usng (80) and our prevously mentoned notons of functonal dfferentaton, we have Z[J] = Z[J] J(x ) dy F (x y)j(y). (86) The second order dervatve s Z[J] = Z[J] dy F (x y )J(y ) dy 2 F (x 2 y 2 )J(y 2 ) J(x 2 ) J(x ) + F (x x 2 )Z[J]. Contnung, J(x 3 ) J(x 2 ) J(x ) Z[J] = F (x x 2 )Z[J] dy 3 F (x 3 y 3 )J(y 3 ) F (x x 3 )Z[J] dy 2 F (x 2 y 2 )J(y 2 ) F (x 2 x 3 )Z[J] dy F (x y )J(y ) 2

24 Z[J] dy dy 2 dy 3 F (x y )J(y ) F (x 2 y 2 )J(y 2 ) F (x 3 y 3 )J(y 3 ) Fnally, we wrte down the fourth order dervatve: J(x 4 ) J(x 3 ) J(x 2 ) J(x ) Z[J] = F (x x 2 ) F (x 3 x 4 )Z[J] + F (x x 3 ) F (x 2 x 4 )Z[J] + F (x 2 x 3 ) F (x x 4 )Z[J] + other terms that vansh when J = 0. When J = 0, these expresson gve us the followng tme-ordered products: 0 T [φ(x )] 0 = 0 (87) 0 T [φ(x )φ(x 2 )] 0 = F (x x 2 ) (88) 0 T [φ(x )φ(x 2 )φ(x 3 )] 0 = 0 (89) 0 T [φ(x )φ(x 2 )φ(x 3 )φ(x 4 )] 0 = F (x x 2 ) F (x 3 x 4 ) + F (x x 3 ) F (x 2 x 4 ) + F (x x 4 ) F (x 2 x 3 ) (90) We call 0 T [φ(x ) φ(x n )] 0 an n-pont correlaton functon. Generalzng the pattern above to T products of more feld operators, we fnd that f n s odd, the n-pont functon vanshes. On the other hand, f n s even, the correlaton functon reduces to the sum all possble permutatons of products of 2-pont functons F (x y) wth x, y {x,..., x n }, x y. When ths result s derved from canoncal methods, t s know as Wck s Theorem. It s nterestng to nterpret these results n terms of the Taylor seres expanson of Z[J] about J = 0. To make sense of ths object, recall that the Taylor seres expanson of a functon of a fnte number of varables s F (y,..., y k ) = k n=0 = k n = n! y n F... y k y... y n (9) y =0. Ths s expanson s taken about the zero of all of the ndependent varables. Assumng the varables are weghted approprately, when we go to the contnuum lmt k we obtan that n F F [y] = dx dx n y(x ) y(x n ) n! y(x )... y(x n ). (92) y=0 n=0 Usng ths expanson, we obtan 22

25 ( x x 2 x x 2 x x 2 Z[ J] = + x x x 3 x 4 x 3 x 4 x 3 x 4 ( Fgure 8: Pctoral representaton of the free feld generatng functonal Z[J] Z[J] = + 2 dx dx 2 J(x )J(x 2 ) F (x x 2 ) 2! + 4 dx dx 2 dx 3 dx 4 J(x )J(x 2 )J(x 3 )J(x 4 ) 4! [ F (x x 2 ) F (x 3 x 4 ) + F (x x 3 ) F (x 2 x 4 ) ] + F (x x 4 ) F (x 2 x 3 ) + (93) Ths result s depcted dagrammatcally n fgure 8. followng Feynman rules: In ths fgure, we use the. Each lne s connected to a snk and a source. 2. Each snk or a source s labeled wth a spacetme pont. 3. A lne runnng between x and y comes wth a propagator F (x y). 4. Each source or snk attached to x comes wth a factor J(x). 5. The collecton of all graphs wth n sources and snks s multpled by /n!. 6. All spacetmes coordnates are ntegrated over. Ths frst rule ensures that we only nclude dagrams wth an even number of vertces and that we only consder dsconnected graphs. The Feynman dagrams llustrate the correspondence of Z[J] to the vacuum-to-vacuum transton probablty beautfully. Each term n the graph nvolves the creaton of a partcle at some pont and ts destructon at a later pont. The three terms n the 4-vertex graph account for all possble permutatons of partcle beng created/destroyed at a par of x, x 2, x 3 or x 4. What we would lke to do now s move on to the more nterestng case of nteractng felds. 6 The generatng functonal for self-nteractng felds Whle free feld theory has a certan amount of elegance to t, t s not terrbly nterestng. In ths secton, we consder a self-nteractng feld whose Lagrangan s 23

26 gven by L(φ) = 2 αφ α φ 2 m2 φ 2 + L nt (φ) = L 0 (φ) + L nt (φ). (94) The dscusson follows secton 6.4 of Ryder []. Here, L nt (φ) s the Lagrangan descrbng the self nteracton. The generatng functonal s ( [dφ] exp S + d 4 x Jφ ) Z[J] =, (95) [dφ]e S where S s the classcal acton S = d 4 x [L 0 (φ) + L nt (φ)] = S 0 + S nt. (96) We have dropped the ɛφ 2 /2 term used to sngle out the ground state, whch we can ratonalze by the rotaton of the tme axs. In ths secton, we wll wrte the free feld propagator as ( [dφ] exp S0 + d 4 x Jφ ) Z 0 [J] =. (97) [dφ]e S 0 We would lke to wrte Z[J] n a form partcularly useful for calculatons. We can t really reproduce the manpulatons of the last secton because the L nt n the acton S ntroduces dffcultes when the shft φ φ + φ 0 s performed. We wll nstead derve a dfferental equaton satsfed by Z[J] and then solve t n terms of J(x) and the Feynman propagator. The result wll be somethng that we mght have guessed ntutvely. What s the equaton satsfed by the free feld propagator? Now, we know that J(x) Z 0[J] = Z 0 [J] dy F (x y)j(y). (98) We operate on both sdes wth x +m 2 and use the defnng relaton for the Feynman propagator (76) to get: ( x + m 2 ) J(x) Z 0[J] = J(x)Z 0 [J]. (99) The s the dfferental equaton satsfed by Z 0 [J]. In order to fnd the dfferental equaton satsfed by Z[J], let us defne the functonal Ẑ[φ] = e S [dφ] e S. (00) Then Z[J] = [dφ] ( Ẑ[φ] exp ) d 4 x Jφ, (0) 24

27 whch s the functonal equvalent of the Fourer transform. We can functonally dfferentate Ẑ[φ] wth respect to φ usng [ S = d 4 x 2 αφ α φ ] 2 m2 φ 2 + L nt (φ) [ ] = d 4 x 2 φ( + m2 )φ L nt, (02) where Gauss theorem has been used. The superorty of the latter form s that we can ntegrate twce by parts to change φ φ nto φ φ. So, when we functonally dfferentate, we can make sure that the φ beng acted on by the operator s dfferent from the φ beng acted on by the /φ. We get where Ẑ[φ] = ( + m2 )φ(x)ẑ[φ] L φ(x) nt(φ)ẑ[φ], (03) L nt(φ) = L nt φ (04) and = x. We multply both sdes of (03) by exp[ J(y)φ(y) d 4 y] and ntegrate over φ,.e. we take the Fourer transform. The RHS of (03) becomes RHS = [ ] ( + m 2 ) [dφ]φ(x) exp S + J(y)φ(y) d 4 y Z 0 [ ] [dφ]l Z nt(φ) exp S + J(y)φ(y) d 4 y, (05) 0 where we have wrtten Z 0 = [dφ]e S. (06) Now, t s easy to see Z[J] J(x) = Z 0 [ [dφ] φ(x) exp S + ] J(y)φ(y) d 4 y, (07) whch leads to ( ) n Z[J] = J(x) Z 0 [ [dφ] φ n (x) exp S + ] J(y)φ(y) d 4 y. (08) Now, we assume that L nt (φ) possesses a Taylor seres expanson n φ. We can then reproduce the second term n (05) by addng together a seres of contrbutons of the form of (08). Hence, RHS = ( + m 2 ) ( Z[J] J(x) L nt 25 J(x) ) Z[J]. (09)

28 The Fourer transform of the LHS of (03) becomes LHS = [dφ] Ẑ(φ) [ ] φ(x) exp J(y)φ(y) d 4 y = [ Ẑ(φ) exp J(y)φ(y) d y] 4 φ [ ] [dφ]ẑ(φ) φ(x) exp J(y)φ(y) d 4 y ] = [ [dφ]j(x)ẑ(φ) exp J(y)φ(y) d 4 y = J(x)Z[J]. (0) In the second lne we have performed a functonal ntegraton by parts, whch follows from the fact that the functonal dervatve satsfes the product rule and functonal ntegral obeys the fundamental theorem of calculus. The boundary term must vansh as φ because f t ddn t, the ntegral for Z[J] would dverge. Puttng together our formulae for the LHS and RHS of the Fourer transform of (03): ( + m 2 ) Z[J] J(x) L nt ( J(x) ) Z[J] = J(x)Z[J]. () Ths s the dfferental equaton satsfed by Z[J]. We see that f the nteractng Lagrangan s set to zero, the result reduces to (99). We wll assume a soluton of the dfferental equaton of the form Z[J] = N exp [ d 4 x L nt ( J )] Z 0 [J] (2) partly because ths s what we mght expect from L = L 0 +L nt f we replace /J wth φ, as we usually do, partly because we know t s the rght answer. As usual, N s a normalzng factor. Let s frst establsh and dentty: exp [ d 4 y L nt ( J(y) )] J(x) exp = J(x) L nt ( [ ( )] d 4 y L nt J(y) ). (3) J(x) The frst step s to notce that the commutator of J and / J k when actng on a functon of (J,..., J n ) s gven by the well known relaton [ J, ] = k. (4) J k The contnuum lmt (n ) of ths s [ J(x), ] J(y) 26 = (x y). (5)

29 Now, t s not hard to see that f A and B are operators whose commutator s a number a, then Hence, [ ( ) n ] J(x), J(y) [A, B] = a, [A, B 2 ] = 2aB, [A, B 3 ] = 3aB 2,. [A, B n ] = nab n. ( = (x y)n Assume that L nt (φ) possesses a Taylor seres expanson L nt (φ) = c 0 + c φ + 2! c 2φ 2 + = Snce [A, (B + C)] = [A, B] + [A, C], we can wrte [ ( ) ] J(x), n! c n n = (x y) J(y) But n=0 L nt(φ) = n=0 n n! c nφ n = n= n= Usng ths an ntegratng (8) wth respect to y gves [ ( )] J(x), d 4 y L nt = L nt J(y) J(y) n=0 ( (n )! c n ) n. (6) n! c nφ n. (7) J(y) ) n. (8) (n )! c nφ n. (9) ( J(x) ). (20) where we have scaled L nt by. Fnally, note that the Hausdorff formula gves e B Ae B = A + [A, B] (2) when [A, B] s a number. Puttng (20) nto (2) wth A = J(x) and B = d 4 y L nt ( /J(y)) yelds (3). Usng our assumed form for Z[J] (2) and the dentty (3) we get [ )] J(x)Z[J] = NJ(x) exp Z 0 [J] = N exp [ ( d 4 y L nt J(y) d 4 y L nt ( J(y) 27 )] [ J(x) L nt ( J(x) )] Z 0 [J]

30 [ = N exp ( L nt = ( x + m 2 ) J(x) )] ( x + m 2 ) ( d 4 y L nt J(y) ) [ ( N exp d 4 y L nt ( Z[J] J(x) L nt J(x) ) Z[J]. J(y) J(x) Z 0[J] )] Z 0 [J] In gong from the second to the thrd lne, we have used the dfferental equaton satsfed by Z 0 [J] (99). Ths result s just the dfferental equaton (), whch confrms that we can wrte [ exp ( )] d 4 x L nt J(x) Z 0 [J] Z[J] = [ exp ( )] d 4 x L (22) nt Z 0 [J] J=0 J(x) where we have [ Z 0 [J] = exp 2 ] J(x) F (x y)j(y) d 4 x d 4 y. (23) Wth these two equatons, we have succeedng n wrtng down the generatng functonal entrely n terms of the source J and the Feynman propagator F. 7 φ 4 and φ 3 theory We would lke to demonstrate the calculaton of the generatng functonal Z[J] and some n-pont functons n the case of self-nteractng φ 4 and φ 3 theory. We follow secton 6.5 of Ryder [] and chapter 2 of Popov [4]. We frst consder φ 4 theory, whch has the nteractng Lagrangan: L nt (φ) = λ 4! φ4. (24) It goes wthout sayng that λ s small. The generatng functonal Z 4 [J] for φ 4 s [ exp λ ( ) ] 4 4! d 4 z J(z) Z 0 [J] Z 4 [J] = [ exp λ ( ) ] 4 (25) 4! d 4 z Z 0 [J] J(x) The numerator of Z 4 [J] s, to frst order n λ, s: [ num Z 4 [J] = λ ( ) d 4 4 z + ] Z 0 [J]. (26) 4! J(z) J=0 28

31 Performng the functonal dfferentaton, we obtan { num Z 4 [J] = λ [ ( ) 2 d 4 z 3 2 F (0) + 6 F (0) d 4 x F (z x)j(x) 4! ( ) ] } 4 + d 4 x F (z x)j(x) + Z 0 [J], (27) where we have wrtten F (z z) = F (0) 6. We also get { denom Z 4 [J] = num Z 4 [J] = λ d 4 z [ 3 2 F (0) ] } + Z 0 [J]. (28) J=0 4! Puttng the two results together yelds, to order λ, we have: { Z 4 [J] = λ [ ( ) 2 d 4 z 6 F (0) d 4 x F (z x)j(x) 4! ( ) ] } 4 + d 4 x F (z x)j(x) + Z 0 [J], (29) Now, let s do the same thng for φ 3 theory, where the nteractng Lagrangan s The numerator of the generatng functonal s [ num Z 3 [J] = exp λ d 4 z 3! L nt (φ) = λ 3! φ3. (30) ( J(z) ) 3 ] Z 0 [J]. (3) Expandng to order λ, we get { num Z 3 [J] = λ [ d 4 z 3 F (0) d 4 x F (z x)j(x) 3! ( ) ] } 3 d 4 x F (z x)j(x) + Z 0 [J]. (32) If we set J = 0, the above reduces to unty. So, the denomnator of Z 3 [J] s, to frst order n λ, whch yelds: { Z 3 [J] = λ [ d 4 z 3 F (0) d 4 x F (z x)j(x) 3! ( ) ] } 3 d 4 x F (z x)j(x) + Z 0 [J]. (33) 6 The Feynman propagator evaluated at zero s, of course, nfnte. It s ncluson n the generatng functonal represents the nfnte self-energy of partcles n the theory and must be regulated by renormalzaton, whch we wll not consder here. 29

32 Z J 4 [ ] = 4 x x 2 z 24 x x 2 z x 3 x 4 + x Z J 3[ ] = 2 x z z x x 2 Fgure 9: Pctoral representaton of the generatng functonal for φ 4 and φ 3 theory respectvely We can represent the two generatng functonals Z 4 [J] and Z 3 [J] dagrammatcally wth the followng Feynman rules:. The spacetme pont z s assocated wth nternal ponts, all other varables go wth external ponts (recall that z s the coordnate that occurred n the nteractng Lagrangan). 2. A lne between x and z comes wth a propagator F (x z). 3. Each nternal pont comes wth a factor of λ/4! for φ 4 theory, λ/3! for φ 3 theory. 4. External ponts x come wth a factor J(x). 5. Terms of the form F (0) represent closed loops, or propagators who begn and end at the same pont, joned to nternal ponts. 6. All spacetme ponts are ntegrated over. 7. Each term n the seres s multpled by the free partcle propagator Z 0 [J]. These rules gve the pctures n fgure 9 for Z 4 [J] and Z 3 [J]. These dagrams stress the general structure of the two theores. Because of the power of 4 n the nteractng Lagrangan n φ 4 theory, the vertces n the assocated Feynman dagrams are attached to 4 legs. Smlarly, the vertces n φ 3 theory are attached to 3 legs. A pecularty of φ 3 theory s the dagram wth only one external pont. As we wll see below, ths gves rse to a one pont correlaton functon whch means that partcles n φ 3 can spontaneously self-destruct. 30

33 x x 2 2 x + z x 2 4 x z + Fgure 0: Pctoral representaton of the two-pont functon 0 T [φ(x )φ(x 2 )] 0 4 for φ 4 theory (top) and of the one-pont functon 0 T [φ(x )] 0 3 for φ 3 theory (bottom) Let s calculate the 2-pont functon for φ 4 theory: 0 T [φ(x )φ(x 2 )] 0 4 = 2 Z 4 [J] J(x 2 )J(x ). (34) J=0 The functonal dfferentaton s easy to do from (29) usng the our prevously derved formulae for dervatves of the free feld propagator (87) (90). The result s 0 T [φ(x )φ(x 2 )] 0 4 = F (x x 2 ) λ 2 F (0) d 4 z F (x z) F (z x 2 ) + O(λ 2 ). (35) Ths can be turned nto a Feynman dagram, as n fgure 0, usng rules smlar to the ones for Z[J], except now the only the nternal ponts are ntegrated over. Fgure 0 shows how the ampltude for the creaton of a partcle at pont x and ts later destructon at x 2 s modfed by an (nfnte) loop dagram to frst-order n λ n φ 4 theory. Now, let s get the 2-pont functon n φ 3 theory. It s smply calculated 0 T [φ(x )φ(x 2 )] 0 3 = 2 Z 3 [J] J(x 2 )J(x ) = F (x x 2 ) + O(λ 2 ). (36) J=0 Ths sn t terrbly nterestng, we see that there are no correctons to the free feld result to ths level n perturbaton theory. We shouldn t be surprsed, there s no way to connect to external lnes to a sngle vertex f the vertex must be attached to three lnes. However, there s a way to connect a vertex to one external lne and one nternal lne, as shown n fgure 0. Ths s borne out by the calculaton of the one-pont functon: 0 T [φ(x )] 0 3 = 2 Z 3 [J] J(x ) J=0 = λ 4 F (0) d 4 z F (z x ) + O(λ 2 ). (37) 3

34 x z ( x x z 3 4 x 2 x 3 x 2 z z + x 3 x x x 2 x 3 ( x 2 Fgure : Pctoral representaton of 0 T [φ(x )φ(x 2 )φ(x 3 )] 0 3, the three pont functon, for φ 3 theory Certanly a strange beast, ths represents the ampltude of a partcle beng spontaneously created out of the vacuum, all by tself 7. We can also calculate the three pont functon 2 Z 3 [J] 0 T [φ(x )φ(x 2 )φ(x 3 )] 0 3 = J(x 3 )J(x 2 )J(x ) J=0 = λ d 4 z F (z x ) F (z x 2 ) F (z x 3 ) + λ 4 F (0) d 4 z [ F (z x ) F (x 2 x 3 ) + F (z x 2 ) F (x 3 x ) + F (z x 3 ) F (x x 2 )]. (38) The frst ntegral nvolves all the external legs attached to the same pont,.e. two partcles mergng nto one or one partcle splttng n two. The second ntegral s the product of free 2-pont functons and the -pont functon we have already calculated. The dagram s n fgure. We have hence shown how the generatng functonal and n-pont functons can be found from smple functonal dfferentaton of equatons (22) and (23) and expressed n terms of Feynman dagrams. These dagrams can be converted nto scatterng ampltudes va the LSZ reducton formula 8 : p,..., p n, + q,..., q m, = dsconnected terms + () n+m dy dy n dx dx m e (p y + +p n y n q x q m x m ) ( y + m 2 ) ( yn + m 2 ) ( x + m 2 ) ( xm + m 2 ) 0 T [φ(y ) φ(y n )φ(x )... φ(x m )] 0. 7 However, such a process would be forbdden on energy grounds 8 cf. PHYS 703 March 6, Brown [3] clams that ths formula can be obtaned va path ntegral methods, by he presents a proof usng the canoncal formalsm. 32

35 In practce, these scatterng ampltudes are the only meanngful quanttes n quantum feld theory snce they are the only thngs that can be drectly measured. So, havng arrved at a pont where we can calculate p,..., p n, + q,..., q m, usng the generatng functonal, we have completed our formulaton of self-nteractng feld theores n terms of path ntegrals. More complcated theores, such as QED, can be quantzed n terms of pathntegrals, but there are several ssues that need to be addressed when wrtng down Z[J] for gauge felds. One fnds that Z[J] s nfnte for gauge felds A α, because the [da α ] ntegraton ncludes an nfnte number of contrbutons from felds related by a smple gauge transformaton. The resoluton s the addton of gauge fxng terms to the Lagrangan and the appearance of non-physcal ghost felds. Such thngs are beyond the scope of ths paper. 8 Conclusons We have demonstrated how quantum mechancs can be formulated n terms of the propagator that represents the ampltude of a partcle travels from some ntal poston to some fnal poston. We have determned the form of the propagator to be a sum over paths of the ampltudes assocated wth the partcle travelng along a gven trajectory. The ampltude assocated wth the path q(t) was shown to be e S[q], where S[q] s the classcal Lagrangan. We used ths form of the propagator to derve Feynman rules for the S f matrx n non-relatvstc scatterng problems. Then, we showed that by addng a source J(t) to the Lagrangan and rotatng the tme axs n the complex plane, the propagator Z[J] reduces to the ground stateto-ground state transton ampltude. Functonal dervatves of Z[J] wth respect to the source J(t) were shown to gve the tme-ordered product of confguraton operators q(t). We generalzed the propagator to free scalar felds and expanded Z[J] n terms of Feynman dagrams by expressng t n terms of the Feynman propagator and the source J(x). In the process, we derved some n-pont correlaton functons by dfferentatng Z[J]. We dscussed how the rotaton of the tme axs needed to solate the vacuum-to-vacuum transton probablty s drectly responsble for the promnent role that the Feynman propagator plays n feld theory. We developed a formalsm to deal wth self-nteractng felds and expressed Z[J] entrely n terms of F (x), J(x), and the seres expanson of the nteractng Lagrangan. For both φ 4 and φ 3 theory, we expressed Z[J] and some n-pont functons n terms of Feynman dagrams. Snce the scatterng problem s essentally solved once the procedure for calculatng n-pont functons s specfed (gnorng ssues of renormalzaton), we have hence shown how all of the theory of self-nteractng scalar felds can be derved from path ntegrals. 33