Functional Quantization

Transcription

1 Functional Quantization In quantum mechanics of one or several particles, we may use path integrals to calculate the transition matrix elements as out Ût out t in in D[allx i t] expis[allx i t] Ψ out allx i@t out Ψ in allx in 1 where the paths x i t are independent of each other and are not subject to any boundary conditions, and the action depends on all particles paths as S[allx i t] t out t in Ldt, L i m i ẋ i Vallx i. Let s generalize formula 1 to the quantum field theory. In classical field theory, a finite set of dynamical variables x i becomes a field Φx where x is now a continuous label rather than a dynamical variable of its own. Thus, a set of all the particles paths x i t becomes a field configuration Φx,t, with action functional where the Lagrangian density is t out S[Φx,t] dt d 3 xl 3 t in L 1 µφ µ Φ m Φ λ 4 Φ4 4 for a scalar field, or something more complicated for other kinds of fields. In the quantum field theory, the path integral D[allx i t] i D[x i t] 5 generalizes to the functional integral over the field configurations D[Φx,t] 6 A careful definition of this functional integral involves discretization of both time and space 1

2 to a 4D lattice of N TV/a 4 spacetime points x n and then taking a zero-lattice-spacing limit D[Φx,t] lim a 0 Normalization factor n dφx n. 7 I shall address this issue later this week. For now, let s focus on using the functional integral. The QFT analogue of eq. 1 for the transition matrix element is out Ût out t in in D[Φx,t] expis[φx,t] Ψ out [Φx@t out]ψ in [Φx@t in ]. 8 Note that in the field theory s Hilbert space, the wave-functions Ψ in and Ψ out depend on the entire 3-space field configuration Φx for a fixed time t t in or t t out. In terms of such wave-functions, the out in bracket is given by a 3-space analogue of the function integrals over Φx at a fixed time, out in D[Φx]Ψ out [Φx] Ψ in [Φx]. 9 Now, let s go to the Heisenberg picture and consider the matrix element of the quantum field ˆΦ H x 1,t 1 at some point x µ 1 x 1,t 1. The states and the operators in Heisenberg and Schrödinger pictures are related to each other via the evolution operator Ût t 1, hence out ˆΦ H x 1,t 1 in out Ût out t 1 ˆΦ S x 1 Ût 1 t in in D[Φ 1 x] out Ût out t 1 Φ 1 Φ 1 x 1 Φ 1 Ût 1 t in in D[Φx,t for the whole time] Ψ out [Φx@t out] exp is[φ]fromt 1 tot out Φx 1,t 1 exp is[φ]fromt in tot 1 Ψin [Φx@t in ] D[Φx,t for the whole time] Ψ out[φx@t out ] Ψ in [Φx@t in ] exp is[φ] for the whole time Φx 1,t 1. Likewise, matrix elements of time-ordered products of several fields ˆΦ H x n,t n ˆΦ H x 1,t 1 10

3 for t n > > t 1 are given by functional integrals out ˆΦ H x n,t n ˆΦ H x 1,t 1 in D[Φx,t for the whole time] Ψ out[φx@t out ] Ψ in [Φx@t in ] exp is[φ] for the whole time Φx n,t n Φx 1,t Now, let s take the time interval between the in and out states to infinity in both directions, or rather let s take the limit t out + 1 iǫ, t in 1 iǫ. 1 As we saw back in January cf. class notes, in this limit out e iĥtout Ω c-number factor, e +iĥtin in Ω c-number factor, 13 where Ω is the vacuum state of the quantum field theory. Therefore, we may obtain the correlation functions of n quantum fields G n x 1,x,...,x n Ω TˆΦ H x 1 ˆΦ H x n Ω 14 from the functional integrals G n x 1,x,...,x n const D[Φx] exp is[φ] i d 4 xl Φx 1 Φx n. Note that the action integral S d 4 xl for each field configuration Φx is taken over the whole Minkowski space time, so it s Lorentz invariant, and the correlations functions 15 are also Lorentz invariant. 15 Strictly speaking, the limit 1 implies that all the time coordinates x 0 of all fields in 15 live on the slightly tilted time axis that runs from 1 iǫ to + 1 iǫ. Consequently, the action has a complexified version of the Lorentz symmetry, but after analytic continuation of all the correlation functions back to the real axis, we recover the usual Lorentz symmetry SO3, 1. Alternatively, we may analytically continue to the imaginary times; this gives us SO4 symmetry in the Euclidean spacetime. 3

4 We do not know the constant factor in front of the functional integral 15 and we do not know the overall normalization of the functional integral. But we can get rid of both of these unknown factors by taking ratios of correlation functions. In particular, using G 0 Ω Ω 1 16 we can write all the other correlation functions as normalization-independent ratios G n x 1,x,...,x n Ω TˆΦ H x 1 ˆΦ H x n Ω Ω Ω D[Φx] exp i D[Φx] exp i d 4 xl Φx 1 Φx n. d 4 xl 17 In a few pages, we shall learn how to calculate such ratios in perturbation theory and rederive the Feynman rules. But before we do the perturbation theory, let s see how functional integrals work for the free scalar field. Functional Integrals for the Free Scalar Field The free scalar field has a quadratic action functional S[Φx] d 4 x L 1 µφ µ Φ 1 m Φ 1 d 4 xφx +m Φx. 18 Hence, the functional integral Z D[Φx] exp is[φx] 19 is a kind of a Gaussian integral, generalization of the ordinary Gaussian integral ZA dξ 1 dξ dξ N exp 1 to a continuous family of variables, {ξ j } D[Φx]. N j,k1 A jk ξ j ξ k 0 4

5 Lemma 1: The ordinary Gaussian integral 0 evaluates to ZA πn/ deta. 1 Note: A jk is a symmetric N N matrix, real or complex. To make the integral 0 converge, the real part ReA jk of the A matrix should be positive definite. Proof: Any symmetric matrix can be diagonalized, thus A jk ξ j ξ k k jk B k η k where η 1,...,η N are some independent linear combinations of the ξ 1,...,ξ N. Changing integration variables from the ξ k to η k carries a Jacobian J det ξj ; 3 η k by linearity of the ξ η transform, this Jacobian is constant. Therefore, Z J d N ηexp 1 B k ηk J J N k1 N k1 k dη k exp B k η k π B k πn/ J detb 4 where B is the diagonal matrix B jk jk B k. This matrix is related to A according to B ξ A η ξ, 5 η hence and therefore ξ detb deta det η Z π N/ deta J 6 J detb πn/ deta. 1 5

6 Lemma concerns the rations of Gaussian integrals for the same matrix A: d N ξ exp 1 ij A ijξ i ξ j ξ k ξ l A 1 d N ξ exp 1 ij A kl. 7 ijξ i ξ j Proof: d N ξexp 1 A ij ξ i ξ j ξ k ξ l A kl ij d N ξexp 1 A ij ξ i ξ j, 8 ij hence d N ξ exp 1 d N ξ exp 1 ij A ijξ i ξ j ξ k ξ l log ij A ijξ i ξ A j kl [ d ] ξexp N 1 ij A ijξ i ξ j log πn/ A kl deta + logdeta A kl A 1 kl. 9 Both lemmas apply to the Gaussian functional integrals such as 19. For example, consider the free scalar propagator G x,y. Eq. 17 gives us a formula for this propagator in terms of a ratio of two Gaussian functional integrals G x,y D[Φx] exp i d 4 xφ +m Φ ΦxΦy D[Φx] exp i d 4 xφ +m Φ. 30 The role of the matrix A here is played by the differential operator i +m. This operator is purely anti-hermitian, which corresponds to a purely imaginary matrix A. To make the Gaussian integrals converge, we add an infinitesimal real part, A ij A ij + ǫ ij, or i + m i +m +ǫ. Consequently, lemma tells us that G x,y x 1 d 4 i +m +ǫ y p ie ipx y π 4 p m +iǫ. 31 Note that the rule A A+ǫ we use to make the Gaussina integral converge for an imaginary or anhti-hermitian matrix A immediately leads us to the Feynman propagator 31 rather than some other Green s function. 6

7 To obtain correlation functions G n for n > we need analogues of Lemma for Gaussian integrals involving more than two ξs outside the exponential. For example, for four ξs we have d N ξ exp 1 ij A ijξ i ξ j ξ k ξ l ξ m ξ n 3 d N ξ exp 1 ij A ijξ i ξ j A 1 kl A 1 mn + A 1 km A 1 ln + A 1 kn A 1 lm. Applying this formula to the Gaussian functional integral for the free scalar fields, we obtain for the 4 point function G 4 x,y,z,w G x y G z w + G x z G y w + G x w G y z Likewise, for higher even numbers of ξs, d N ξ exp 1 ij A ijξ i ξ j ξ 1 ξ N d N ξ exp 1 ij A ijξ i ξ j A 1 34 indexpair pairs pairings and similarly G free N x 1,...,x N D[Φx] exp i d 4 xφ +m Φ Φx 1 Φx N D[Φx] exp i d 4 xφ +m Φ pairings pairs. 35 7

8 Feynman Rules for the Interacting Field A self-interacting scalar field has action S[Φx] S free [Φx] λ d 4 xφ 4 x. 36 4! Expanding the exponential e is in powers of the coupling λ, we have exp is[φx] exp is free [Φx] n0 n 1 iλ d 4 xφ x n! 4! This expansion gives us the perturbation theory for the functional integral of the interacting field, D[Φx] exp is[φx] Φx 1 Φx k D[Φx] exp is free [Φx] i d 4 xφ +m Φ n0 iλ n n!4! n d 4 z 1 d 4 z n Φx 1 Φx k n0 n 1 iλ d 4 xφ 4 x n! 4! i D[Φx] exp d 4 xφ +m Φ Φx 1 Φx k Φ 4 z 1 Φ 4 z n using eq. 35 for the free-field functional integrals [ ] i D[Φx] exp d 4 xφ +m Φ iλ n n!4! n d 4 z 1 d 4 z n products of k +4n n0 [ D[Φx] exp n0 pairings i d 4 xφ +m Φ ] free propagators Feynman diagrams with k external and n internal vertices This is how the Feynman rules emerge from the functional integral formalism

9 At this stage, we are summing over all the Feynman diagrams, connected or disconnected, and even the vacuum bubbles are allowed. Also, the functional integral 38 carries an overall factor Z 0 i D[Φx] exp d 4 xφ +m Φ which multiplies the whole perturbation series. Fortunately, this pesky factor as well as all the vacuum bubbles cancel out when we divide eq. 38 by a similar functional integral for k 0 to get the correlation function G k. Indeed, 39 G k x 1,...,x k D[Φx] exp is[φx] Φx1 Φx k D[Φx] exp is[φx] Z 0 all Feynman diagrams with k external vertices Z 0 all Feynman diagrams without external vertices Feynman diagrams with k external vertices and no vacuum bubbles. 40 Generating Functional In the functional integral formalism there is a compact way of summarizing all the connected correlation functions in terms of a single generating functional. Let s add to the scalar field s Lagrangian L a linear source term, LΦ, Φ LΦ, Φ + J Φ 41 where Jx is an arbitrary function of x. Now let s calculate the partition function for Φx as a functional of the source Jx, Z[Jx] D[Φx] exp i d 4 xl+jφ D[Φx] exp is[φx] exp i d 4 xjxφx. 4 To make use of this partition function, we need functional i.e., variational derivatives /Jx. Here is how they work: JyΦyd 4 y Φx, 43 Jx 9

10 µ Jy A µ yd 4 y µ A µ x, 44 Jx Jx exp i JyΦyd 4 y iφx exp i JyΦyd 4 y, 45 etc., etc. Applying these derivatives to the partition function 4, we have and similarly Jx 1 Jx Z[J] D[Φy] exp i d 4 yl+jφ iφx 46 Jx Jx k Z[J] D[Φy] exp i d 4 yl+jφ i k Φx 1 Φx Φx k. Thus, the partition function 4 generates all the functional integrals of the scalar theory: k, D[Φy] exp i d 4 yl Φx 1 Φx k i k Jx 1 47 Jx k Z[J]. Jy 0 Consequently, all the correlation functions 40 follows from the derivatives of the Z[J] according to G k x 1,...,x k ik Z[J] 48 Jx 1 Jx k Z[J]. 49 Jy 0 In terms of the Feynman diagrams, the correlation functions 49 contain both connected and disconnected graphs; only the vacuum bubbles are excluded. To obtain the connected correlation functions G conn k x 1,...,x k connected Feynman diagrams only 50 we should take the /Jx derivatives of log Z[J] rather than Z[J] itself, G conn k x 1,...,x k i k Jx 1 Jx k logz[jy]. 51 Jy 0 In other words, log Z[Jx] is the generating functional for for all the connected correlation functions of the quantum field theory. 10

11 Proof: For the sake of generality, let s allow for G 1 x Φx 0. For k 1, G conn 1 G 1 and eq. 51 is equivalent to eq. 49. For k we use logz[j] JxJy 1 Z Z JxJy 1 Z Z Jx 1 Z Z Jy. 5 Let smultiplythisformulaby i, setj 0aftertaking thederivatives, andapplyeqs. 49 to the right hand side; then i logz[j] Jx Jy G x,y G 1 x G 1 y G conn x,y. 53 J 0 The k 1 and k cases serve as a base of a proof by induction in k. Now we need to prove that is eqs. 51 hold true for all l < k that they also hold true for k itself. A k-point correlation function has a cluster expansion in terms of connected correlation functions, G k x 1,...,x k G conn k x 1,...,x k + cluster decompositions clusters G conn cluster where each cluster is proper subset of x 1,...,x n and we sum over decompositions of the whole set x 1,...,x n into a union of disjoint clusters. For example, for k 3 54 G 3 x,y,z G conn 3 x,y,z + G 1 x G 1 y G 1 z + G conn x,y G 1 z + G conn x,z G 1 y + G conn y,z G 1 x two similar. 55 There is a similar expansion of derivatives of Z in terms of derivatives of logz, 1 k Z Z Jx 1 Jx k k logz Jx 1 Jx k + clusters l logz, Jx 1 Jx l x 1,...,x l cluster decompositions 56 11

12 Now let s set Jy 0 after taking all the derivatives in this expansion. On the left hand side, this gives us i k G k x 1,...,x k by eq. 49. On the right hand side, for each cluster l logz Jx 1 Jx l il G conn l x 1,...,x l 57 by the induction assumption. Altogether, we have G n x 1,...,x n i k k logz Jx 1 Jx k + J 0 cluster decompositions cluster G conn cluster, 58 and comparing this formula to eq. 54 we immediately see that G conn n x 1,...,x n i k k logz Jx 1 Jx k, 51 J 0 quod erat demonstrandum. We may summarize all the equations 51 in a single formula which makes manifest the role of log Z as a generating functional for the connected correlation functions, logz[jx] logz[0] + k1 i k d 4 x 1 d 4 x k G conn k x 1,...,x k Jx 1 Jx k. 59 k! To see how this works, consider the free theory as an example. A free scalar field has only one connected correlation function, namely G conn x y G F x y; all the other G conn k require interactions and vanish for the free field. Hence, the partition function of the free field should have form logz free [Jx] logz free [0] + i d 4 d 4 yg F x y JxJy + nothing else. 60 To check this formula, let s calculate the partition function. For the free field Φ, L free + JΦ 1 Φ +m Φ +JΦ 1 Φ +m Φ + 1 J +m 1 J 61 1

13 up to a total derivative where Φ Φ +m 1 J. In other words, S free [Φx,Jx] def L+JΦd 4 x S free [Φ x,j ] + i d 4 d 4 yjxg F x yjy 6 where Φ x Φx i d 4 yg F x yjy. 63 In the path integral, D[Φx] D[Φ x], hence Z free [Jx] exp exp D[Φx] exp is free [Φx,Jx] D[Φ x] exp d 4 x is free [Φ x,j ] exp 1 d 4 yjxg F x yjy Z free [0]. 1 1 d 4 x d 4 yjxg F x yjy d 4 x d 4 yjxg F x yjy D[Φ x] exp is free [Φ x,j ] Taking the log of both sides of this formula gives us eq. 60, which confirms that logz free [J] is indeed the generating functional for the correlation functions of the free field