Path Integrals in Quantum Field Theory

Sanjeev S. Seahra Department of Physics University of Waterloo

May 11, 2000 Abstract

We discuss the path integral formulation of quantum mechanics and use it to derive the S matrix in terms of Feynman diagrams. We generalize to quantum ﬁeld theory, and derive the generating functional Z[J] and n-point correlation functions for free scalar ﬁeld theory. We develop the generating functional for self-interacting ﬁelds and discuss φ4 and φ3 theory. 1 Introduction

Thirty-one years ago, Dick Feynman told me about his ‘sum over histo- ries’ version of quantum mechanics. ‘The electron does anything it likes’, he said. ‘It goes in any direction at any speed, forward and backward in time, however it likes, and then you add up the amplitudes and it gives you the wavefunction.’ I said to him, ‘You’re crazy’. But he wasn’t.

F.J. Dyson1

When we write down Feynman diagrams in quantum ﬁeld theory, we proceed with the mind-set that our system will take on every conﬁguration imaginable in traveling from the initial to ﬁnal state. Photons will split in to electrons that recombine into diﬀerent photons, leptons and anti-leptons will annihilate one another and the resulting energy will be used to create leptons of a diﬀerent ﬂavour; anything that can happen, will happen. Each distinct history can be thought of as a path through the conﬁguration space that describes the state of the system at any given time. For quantum ﬁeld theory, the conﬁguration space is a Fock space where each vector represents the number of each type of particle with momentum k. The key to the whole thing, though, is that each path that the system takes comes with a probabilistic amplitude. The probability that a system in some initial state will end up in some ﬁnal state is given as a sum over the amplitudes associated with each path connecting the initial and ﬁnal positions in the Fock space. Hence the perturbative expansion of scattering amplitudes in terms of Feynman diagrams, which represent all the possible ways the system can behave. But quantum ﬁeld theory is rooted in ordinary quantum mechanics; the essen- tial diﬀerence is just the number of degrees of freedom. So what is the analogue of this “sum over histories” in ordinary quantum mechanics? The answer comes from the path integral formulation of quantum mechanics, where the amplitude that a particle at a given point in ordinary space will be found at some other point in the future is a sum over the amplitudes associated with all possible trajectories joining the initial and ﬁnal positions. The amplitudeR associated with any given path is just eiS, where S is the classical action S = L(q, q˙) dt. We will derive this result from the canonical formulation of quantum mechanics, using, for example, the time- dependent Schr¨odingerequation. However, if one deﬁnes the amplitude associated with a given trajectory as eiS, then it is possible to derive the Schr¨odingerequation2. We can even “derive” the classical principle of least action from the quantum am- plitude eiS. In other words, one can view the amplitude of traveling from one point to another, usually called the propagator, as the fundamental object in quantum theory, from which the wavefunction follows. However, this formalism is of little

1Shamelessly lifted from page 154 of Ryder [1]. 2Although, the procedure is only valid for velocity-independent potentials, see below.

1 use in quantum mechanics because state-vector methods are so straightforward; the path integral formulation is a little like using a sledge-hammer to kill a ﬂy. However, the situation is a lot diﬀerent when we consider ﬁeld theory. The generalization of path integrals leads to a powerful formalism for calculating various observables of quantum ﬁelds. In particular, the idea that the propagator Z is the central object in the theory is ﬂeshed out when we discover that all of the n-point functions of an interacting ﬁeld theory can be derived by taking derivatives of Z. This gives us an easy way of calculating scattering amplitudes that has a natural interpretation in terms of Feynman diagrams. All of this comes without assuming commutation relations, ﬁeld decompositions or anything else associated with the canonical formulation of ﬁeld theory. Our goal in this paper will to give an account of how path integrals arise in ordinary quantum mechanics and then generalize these results to quantum ﬁeld theory and show how one can derive the Feynman diagram formalism in a manner independent of the canonical formalism.

2 Path integrals in quantum mechanics

To motivate our use of the path integral formalism in quantum ﬁeld theory, we demonstrate how path integrals arise in ordinary quantum mechanics. Our work is based on section 5.1 of Ryder [1] and chapter 3 of Baym [2]. We consider a quantum system represented by the Heisenberg state vector |ψi with one coordinate degree of freedom q and its conjugate momentum p. We adopt the notation that the Schr¨odingerrepresentation of any given state vector |φi is given by

|φ, ti = e−iHt|φi, (1) where H = H(q, p) is the system Hamiltonian. According to the probability inter- pretation of quantum mechanics, the wavefunction ψ(q, t) is the projection of |ψ, ti onto an eigenstate of position |qi. Hence

ψ(q, t) = hq|ψ, ti = hq, t|ψi, (2) where we have deﬁned |q, ti = eiHt|qi. (3) |qi satisﬁes the completeness relation

hq|q0i = δ(q − q0), (4) which implies Z hq|ψi = dq0hq|q0ihq0|ψi, (5) or Z 1 = dq0|q0ihq0|. (6)

2 q

t

(q,tf f )

(q,ti i )

t=t1

Figure 1: The various two-legged paths that are considered in the calculation of hqf , tf |qi, tii

Multiplying by eiHt0 on the left and e−iHt0 on the right yields that Z 1 = dq0|q0, t0ihq0, t0|. (7)

Now, using the completeness of the |q, ti basis, we may write Z

ψ(qf , tf ) = dqihqf , tf |qi, tiihqi, ti|ψi Z

= dqihqf , tf |qi, tiiψ(qi, ti). (8)

The quantity hqf , tf |qi, tii is called the propagator and it represents the probability amplitudes (expansion coeﬃcients) associated with the decomposition of ψ(qf , tf ) in terms of ψ(qi, ti). If ψ(qi, ti) has the form of a spatial delta function δ(q0), then ψ(qf , tf ) = hqf , tf |q0, tii. That is, if we know that the particle is at q0 at some time ti, then the probability that it will be later found at a position qf at a time tf is

2 P (qf , tf ; q0, ti) = |hqf , tf |qi, t0i| . (9)

It is for this reason that we sometimes call the propagator a correlation function. Now, using completeness, it is easily seen that the propagator obeys a composi- tion equation: Z

hqf , tf |qi, tii = dq1hqf , tf |q1, t1ihq1, t1|qi, tii. (10)

This can be understood by saying that the probability amplitude that the position of the particle is qi at time ti and qf at time tf is equal to the sum over q1 of the probability that the particle traveled from qi to q1 (at time t1) and then on to qf . In other words, the probability amplitude that a particle initially at qi will later be seen at qf is the sum of the probability amplitudes associated with all possible

3 A 2

1

B

Figure 2: The famous double-slit experiment

two-legged paths between qi and qf , as seen in ﬁgure 1. This is the meaning of the oft-quoted phrase: “motion in quantum mechanics is considered to be a sum over paths”. A particularly neat application comes from the double slit experiment that introductory texts use to demonstrate the wave nature of elementary particles. The situation is sketched in ﬁgure 2. We label the initial point (qi, ti) as 1 and the ﬁnal point (qf , tf ) as 2. The amplitude that the particle (say, an electron) will be found at 2 is the sum of the amplitude of the particle traveling from 1 to A and then to 2 and the amplitude of the particle traveling from 1 to B and then to 2. Mathematically, we say that

h2|1i = h2|AihA|1i + h2|BihB|1i. (11)

The presence of the double-slit ensures that the integral in (10) reduces to the two- part sum in (11). When the probability |h2|1i|2 is calculated, interference between the h2|AihA|1i and h2|BihB|1i terms will create the classic intensity pattern on the screen. There is no reason to stop at two-legged paths. We can just as easily separate the time between ti and tf into n equal segments of duration τ = (tf − ti)/n. It then makes sense to relabel t0 = ti and tn = tf . The propagator can be written as Z hqn, tn|q0, t0i = dq1 ··· dqn−1 hqn, tn|qn−1, tn−1i · · · hq1, t1|q0, t0i. (12)

We take the limit n → ∞ to obtain an expression for the propagator as a sum over inﬁnite-legged paths, as seen in ﬁgure 3. We can calculate the propagator for small time intervals τ = tj+1 − tj for some j between 1 and n − 1. We have

−iHtj+1 +iHtj hqj+1, tj+1|qj, tji = hqj+1|e e |qji

4 t0 t1 t2 t3 t4 t5

n qf qf qi OO qi

Figure 3: The continuous limit of a collection of paths with a ﬁnite number of legs

2 = hqj+1|(1 − iHτ + O(τ )|qji = δ(q − q ) − iτhq |H|q i jZ+1 j j+1 j 1 iτ = dp eip(qj+1−qj ) − hq |p2|q i 2π 2m j+1 j −iτhqj+1|V (q)|qji, (13) where we have assumed a Hamiltonian of the form p2 H(p, q) = + V (q). (14) 2m

Now, Z 2 0 0 0 2 hqj+1|p |qji = dp dp hqj+1|p ihp |p |pihp|qji, (15) where |pi is an eigenstate of momentum such that 1 p|pi = |pip, hq|pi = √ eipq, hp|p0i = δ(p − p0). (16) 2π Putting these expressions into (15) we get Z 1 hq |p2|q i = dp p2eip(qj+1−qj ), (17) j+1 j 2π where we should point out that p2 is a number, not an operator. Working on the other matrix element in (13), we get

hqj+1|V (q)|qji = hqj+1|qjiV (qj) = δ(q − q )V (q ) jZ+1 j j 1 = dp eip(qj+1−qj )V (q ). 2π j

5 Putting it all together Z 1 £ ¤ hq , t |q , t i = dp eip(qj+1−qj ) 1 − iτH(p, q ) + O(τ 2) j+1 j+1 j j 2π j Z · µ ¶¸ 1 ∆q = dp exp iτ p j − H(p, q ) , 2π τ j where ∆qj ≡ qj+1 − qj. Substituting this expression into (12) we get Z µ ¶ nY−1 dq dp nX−1 ∆q hq , t |q , t i = dp i i exp i τ p j − H(p , q ) . (18) n n 0 0 0 2π j τ j j i=1 j=0

In the limit n → ∞, τ → 0, we have

n−1 Z n X tn ∆q dq Y dq dp τ → dt, j → =q, ˙ dp i i → [dq][dp], (19) τ dt 0 2π j=0 t0 i=1 and Z ½ Z ¾ tn hqn, tn|q0, t0i = [dq][dp] exp i dt [p q˙ − H(p, q)] . (20) t0 The notation [dq][dp] is used to remind us that we are integrating over all possible paths q(t) and p(t) that connect the points (q0, t0) and (qn, tn). Hence, we have succeed in writing the propagator hqn, tn|q0, t0i as a functional integral over the all the phase space trajectories that the particle can take to get from the initial to the ﬁnal points. It is at this point that we fully expect the reader to scratch their heads and ask: what exactly is a functional integral? The simple answer is a quantity that arises as a result of the limiting process we have already described. The more complicated answer is that functional integrals are beasts of a rather vague mathematical nature, and the arguments as to their standing as well-behaved entities are rather nebulous. The philosophy adopted here is in the spirit of many mathematically controversial manipulations found in theoretical physics: we assume that everything works out alright. The argument of the exponential in (20) ought to look familiar. We can bring this out by noting that Z h i ( " µ ¶ #) ∆q 2 1 iτ p i −H(p ,q ) 1 m ∆qi dp e i τ i i = exp iτ − V (q ) 2π i 2π 2 τ i Z " µ ¶ # iτ m∆q 2 × dp exp − p − i i 2m τ ( " µ ¶ #) ³ m ´1/2 m ∆q 2 = exp iτ i − V (q ) . 2πiτ 2 τ i

6 Using this result in (18) we obtain " # Z n−1 n−1 µ ¶ ³ m ´n/2 Y X m ∆q 2 hq , t |q , t i = dq exp i τ j − V (q ) n n 0 0 2πiτ i 2 τ j i=1 j=0 Z · Z µ ¶¸ tn 1 → N [dq] exp i dt mq˙2 − V (q) , (21) t0 2 where the limit is taken, as usual, for n → ∞ and τ → 0. Here, N is an inﬁnite constant given by ³ m ´n/2 N = lim . (22) n→∞ 2πiτ We won’t worry too much about the fact that N diverges because we will later normalize our transition amplitudes to be ﬁnite. Recognizing the Lagrangian L = T − V in equation (21), we have Z · Z ¸ Z tn iS[q] hqn, tn|q0, t0i = N [dq] exp i L(q, q˙) dt = N [dq] e , (23) t0 where S is the classical action, given as a functional of the trajectory q = q(t). Hence, we see that the propagator is the sum over paths of the amplitude eiS[q], which is the amplitude that the particle follows a given trajectory q(t). Historically, Feynman demonstrated that the Schr¨odingerequation could be derived from equa- tion (23) and tended to regard the relation as the fundamental quantity in quantum mechanics. However, we have assumed in our derivation that the potential is a function of q and not p. If we do indeed have velocity-dependent potentials, (23) fails to recover the Schr¨odingerequation. We will not go into the details of how to ﬁx the expression here, we will rather heuristically adopt the generalization of (23) for our later work in with quantum ﬁelds3. An interesting consequence of (23) is seen when we restore ~. Then Z iS[q]/~ hqn, tn|q0, t0i = N [dq] e . (24)

The classical limit is obtained by taking ~ → 0. Now, consider some trajectory q0(t) and neighbouring trajectory q0(t)+δq(t), as shown in ﬁgure 4. The action evaluated along q0 is S0 while the action along q0 +δq is S0 +δS. The two paths will then make contributions exp(iS0/~) and exp[i(S0 + δS)/~] to the propagator. For ~ → 0, the phases of the exponentials will become completely disjoint and the contributions will in general destructively interfere. That is, unless δS = 0 in which case all neighbouring paths will constructively interfere. Therefore, in the classical limit the propagator will be non-zero for points that may be connected by a trajectory

3The generalization of velocity-dependent potentials to ﬁeld theory involves the quantization of non-Abelian gauge ﬁelds

7 q q(t) f

q(t)+ dq(t)

qi

Figure 4: Neighbouring particle trajectories. If the action evaluated along q(t) is stationary (i.e. δS = 0), then the contribution of q(t) and it’s neighbouring paths q(t) + δq(t) to the propagator will constructively interfere and reconstruct the clas- sical trajectory in the limit ~ → 0 .

satisfying δS[q]|q=q0 ; i.e. for paths connected by classical trajectories determined by Newton’s 2nd law. We have hence seen how the classical principle of least action can be understood in terms of the path integral formulation of quantum mechanics and a corresponding principle of stationary phase.

3 Perturbation theory, the scattering matrix and Feynman rules

In practical calculations, it is often impossible to solve the Schr¨odingerequation exactly. In a similar manner, it is often impossible to write down analytic expressions for the propagator hqf , tf |qi, tii for general potentials V (q). However, if one assumes that the potential is small and that the particle is nearly free, one makes good headway by using perturbation theory. We follow section 5.2 in Ryder [1]. In this section, we will go over from the general conﬁguration coordinate q to the more familiar x, which is just the position of the particle in a one-dimensional space. The extension to higher dimensions, while not exactly trivial, is not diﬃcult to do. We assume that the potential that appears in (23) is “small”, so we may perform an expansion

· Z ¸ Z ·Z ¸2 tn tn 1 tn exp −i V (x, t) dt = 1 − i V (x, t) dt − V (x, t) dt + ··· . (25) t0 t0 2! t0

We adopt the notation that K = K(xn, tn; x0, t0) = hxn, tn|x0, t0i. Inserting the expansion (25) into the propagator, we see that K possesses and expansion of the form: K = K0 + K1 + K2 + ··· (26)

8 The K0 term is Z · Z ¸ 1 K = N [dx] exp i mx˙ 2 dt . (27) 0 2

If we turned oﬀ the potential, the full propagator would reduce to K0. It is for this reason that we call K0 the free particle propagator, it represents the amplitude that a free particle known to be at x0 at time t0 will later be found at xn at time tn. Going back to the discrete expression: Z n−1 n−1 ³ ´n/2 Y X m imτ 2 K0 = lim dxi exp (xj+1 − xj) . (28) n→∞ 2πiτ 2 i=1 j=0

This is a doable integral because the argument of the exponential is a simple quadratic form. We can hence diagonalize it by choosing an appropriate rotation of the xj Cartesian variables of integration. Conversely, we can start calculating for n = 2 and solve the general n case using induction. The result is

³ ´ µ ¶(n−1)/2 · 2 ¸ m n/2 1 2πiτ im(xn − x0) K0 = lim exp . (29) n→∞ 2πiτ n1/2 m 2nτ

Now, (tn − t0)/n = τ, so we ﬁnally have

· ¸1/2 · 2 ¸ m im(xn − x0) K0(xn − x0, tn − t0) = exp , tn > t0. (30) 2πi(tn − t0) 2(tn − t0)

Here, we’ve noted that the substitution nτ = (tn − t0) is only valid for tn > t0. In fact, if K0 is non-zero for tn > t0 it must be zero for t0 > tn. To see this, we note that the calculation of K0 involved integrations of the form: Z ∞ Z ∞ 2 1 2 eiαx dx = eiαx dx −∞ 2 0 Z i−1/2 i∞ eαs = ds 1/2 4 0 s Z i−1/2 i∞ eαs = Θ(−is) ds, 1/2 4 −i∞ s

−1/2 where α ∝ sign(τ) = sign(tn − t0). Now, we can either choose the branch of s to be in either the left- or righthand part of the complex s-plane. But, we need to complete the contour in the lefthand plane if α > 0 and the righthand plane if α < 0. Hence, the integral can only be non-zero for one case of the sign of α. The choice we have implicitly made is the the integral is non-zero for α ∝ (tn − t0) > 0, hence it must vanish for tn < t0. When we look at equation (8) we see that K0 is little more than a type of kernel for the integral solution of the free-particle Schr¨odingerequation, which is really a statement about Huygen’s principle. Our

9 choice of K0 obeys causality in that the conﬁguration of the ﬁeld at prior times determines the form of the ﬁeld in the present. We have hence found a retarded propagator. The other choice for the boundary conditions obeyed by K0 yields the advanced propagator and a version of Huygen’s principle where future ﬁeld conﬁgurations determine the present state. The moral of the story is that, if we choose a propagator that obeys casuality, we are justiﬁed in writing · ¸ h m i1/2 imx2 K (x, t) = Θ(t) exp . (31) 0 2πit 2t

Now, we turn to the calculation of K1: Z · Z ¸ Z 1 K = −iN [dx] exp i mx˙ 2 dt dt V (x(t), t). (32) 1 2

Moving again to the discrete case: Z imτ nX−1 nX−1 K = −iβn/2 dx ··· dx exp (x − x )2 τV (x , t ), (33) 1 1 n−1 2 j+1 j i i j=0 i=1 where β = m/2πiτ and the limit n → ∞ is understood. Let’s take the sum over i (which has replaced the integral over t) in front of the spatial integrals. Also, let’s split up the sum over j in the exponential to a sum running from 0 to i − 1 and a sum running from i to n − 1. Then Z Z nX−1 imτ Xi−1 K = −i τ dx βi/2 dx ··· dx exp (x − x )2 V (x , t ) 1 i 1 i−1 2 j+1 j i i i=1 j=0 Z imτ nX−1 ×β(n−i)/2 dx ··· dx exp (x − x )2 . (34) i+1 n−1 2 j+1 j j=i

We recognize two factors of the free-particle propagator in this expression, which allows us to write nX−1 Z K1 = −i τ dx K0(x − x0, ti − t0)V (x, ti)K0(xn − x, tn − ti). (35) i=1 P R Now, we can replace n−1 τ by tn dt and t → t in the limit n → ∞. Since i=1 t0 i K0(x − x0, t − t0) = 0 for t < t0 and K0(xn − x, tn − t) for t > tn, we can extend the limits on the time integration to ±∞. Hence, Z K1 = −i dx dt K0(xn − x, tn − t)V (x, t)K0(x − x0, t − t0). (36)

10 In a similar fashion, we can derive the expression for K2: Z (−i)2 imτ nX−1 K = βn/2 dx ··· dx exp (x − x )2 (37) 2 2! 1 n−1 2 j+1 j j=0 nX−1 nX−1 × τV (xi, ti) τV (xk, tk). (38) i=1 k=1 We would like to play the same trick that we did before by splitting the sum over j into three parts with the potential terms sandwiched in between. We need to construct the middle j sum to go from an early time to a late time in order to replace it with a free-particle propagator. But the problem is, we don’t know whether ti comes before or after tk. To remedy this, we split the sum over k into a sum from 1 to i − 1 and then a sum from i to n − 1. In each of those sums, we can easily determine which comes ﬁrst: ti or tk. Going back to the continuum limit: Z Z ·Z (−i)2 tn t1 K2 = dx1 dx2 dt1 dt2K0(xn − x1, tn − t1) 2! t0 t0 ×V (x1, t1)K0(x1 − x2, t1 − t2)V (x2, t2)K0(x2 − x0, t2 − t0) Z tn + dt2K0(xn − x2, tn − t2)V (x2, t2)K0(x2 − x1, t2 − t1) t1 ¸

×V (x1, t1)K0(x1 − x0, t1 − t0) (39)

But, we can extend the limits on the t2 integration to t0 → tn by noting the middle propagator is zero for t2 > t1. Similarly, the t2 limits on the second integral can be extended by observing the middle propagator vanishes for t1 > t2. Hence, both integrals are the same, which cancels the 1/2! factor. Using similar arguments, the limits of both of the remaining time integrals can be extended to ±∞ yielding our ﬁnal result: Z 2 K2 = (−i) dx1 dx2 dt1 dt2 K0(xn − x2, tn − t2)V (x2, t2)

×K0(x2 − x1, t2 − t1)V (x1, t1)K0(x1 − x0, t1 − t0). (40)

Higher order contributions to the propagator follow in a similar fashion. The general jth order correction to the free propagator is Z j Kj = (−i) dx1 . . . dxj dt1 . . . dtj K0(xn − xj, tn − tj)

×V (xj) ··· V (x1)K0(x1 − x0, t1 − t0). (41)

We would like to apply this formalism to scattering problems where we assume that the particle is initially in a plane wave state incident on some localized potential.

11 As t → ±∞, we assume the potential goes to zero, which models the fact that the particle is far away from the scattering region in the distant past and the distant future. We go over from one to three dimensions and write Z

ψ(xf , tf ) = dxiK0(xf − xi, tf − ti)ψ(xi, ti) Z

−i dxi dx dt K0(xf − x, tf − t)

×V (x, t)K0(x − xi, t − ti)ψ(xi, ti) + ··· (42)

We push ti into the distant past, where the eﬀects of the potential may be ignored, and take the particle to be in a plane wave state: 1 −ipi·xi ψin(xi, ti) = √ e , (43) V where we have used a box normalization with V being the volume of the box and pi ·x = Eiti −pi ·xi. The “in” label on the wavefunction is meant to emphasize that it is the form of ψ before the particle moves into the scattering region. We want to calculate the ﬁrst integral in (42) using the 3D generalization of (31):

µ ¶3/2 λ 2 K (x, t) = −iΘ(t) eλx , (44) 0 π where λ = im/2t. Hence, Z µ ¶ i λ 3/2 dx K (x − x , t − t )ψ (x , t ) = −√ i 0 f i f i in i i π Z V 2 −iEiti λ(xf −xi) +ipi·xi ×e dxie . (45)

This integral reduces to ψin(xf , tf ) as should have been expected, because K0 is the free particle propagator and must therefore propagate plane waves into the future without altering their form. We also push tf into the inﬁnite future where the eﬀects of the potential can be ignored. Then, Z + ψ (xf , tf ) = ψin(xf , tf ) − i dxi dx dt K0(xf − x, tf − t)

×V (x, t)K0(x − xi, t − ti)ψin(xi, ti) + ··· (46)

The “+” notation on ψ is there to remind us that ψ+ is the form of the wave function after it interacts with the potential. What we really want to do is Fourier analyze + ψ (xf , tf ) into momentum eigenstates to determine the probability amplitude for a particle of momentum pi becoming a particle of momentum pf after interacting

12 with the potential. Deﬁning ψout(xf , tf ) as a state of momentum pf in the distant future: 1 −ipf ·xf ψout(xf , tf ) = √ e , (47) V we can write the amplitude for a transition from pi to pf as

+ Sfi = hψout|ψ i. (48) R Inserting the unit operator 1 = dxf |xf , tf ihxf , tf | into (48) and using the propa- gator expansion (46), we obtain Z ∗ Sfi = δ(pf − pi) − i dxi dxf dx dt ψout(xf , tf )K0(xf − x, tf − t)

×V (x, t)K0(x − xi, t − ti)ψin(xi, ti) + ··· (49)

The amplitude Sfi is the fi component of what is known as the S or scattering matrix. This object plays a central rˆolein scattering theory because it answers all the questions that one can experimentally ask about a physical scattering process. What we have done is expand these matrix elements in terms of powers of the scattering potential. Our expansion can be given in terms of Feynman diagrams according to the rules:

1. The vertex of this theory is attached to two legs and a spacetime point (x, t).

2. Each vertex comes with a factor of −iV (x, t).

3. The arrows on the lines between vertices point from the past to the future.

0 0 0 0 4. Each line going from (x, t) to (x , t ) comes with a propagator K0(x −x, t −t).

5. The past external point comes with the wavefunction ψin(xi, ti), the future ∗ one comes with ψout(xf , tf ). 6. All spatial coordinates and internal times are integrated over.

Using these rules, the S matrix element may be represented pictorially as in ﬁgure 5. We note that these rules are for conﬁguration space only, but we could take Fourier transforms of all the relevant quantities to get momentum space rules. Obviously, the Feynman rules for the Schr¨odingerequation do not result in a signiﬁcant sim- pliﬁcation over the raw expression (49), but it is important to notice how they were derived: using simple and elegant path integral methods.

13 x,tf f x,tf f x,tf f

x,t2 2 … Sfi = + x,t1 1 + + x,t1 1

x,ti i x,ti i x,ti i

Figure 5: The expansion of Sfi in terms of Feynman diagrams

4 Sources, vacuum-to-vacuum transitions and time-ordered products

We now consider a alteration of the system Lagrangian that models the presence of a time-dependent “source”. Our discussion follows section 5.5 of Ryder [1] and chapters 1 and 2 of Brown [3]. In this context, we call any external agent that may cause a non-relativistic system to make a transition from one energy eigenstate to another a “source”. For example, a time-dependent electric ﬁeld may induce a charged particle in a one dimensional harmonic oscillator potential to go from one eigenenergy to another. In the context of ﬁeld theory, a time-dependent source may result in spontaneous particle creation4. In either case, the source can be modeled by altering the Lagrangian such that

L(q, q˙) → L(q, q˙) + J(t)q(t). (50)

The source J(t) will be assumed to be non-zero in a ﬁnite interval t ∈ [t1, t2]. We take T2 > t2 and T1 < t1. Given that the particle was in it’s ground state at T1 → −∞, what is the amplitude that the particle will still be in the ground state at time T2 → ∞? To answer that question, consider Z

hQ2,T2|Q1,T1iJ = dq1 dq2hQ2,T2|q2, t2ihq2, t2|q1, t1iJ hq1, t1|Q1,T1i Z −iHT2 iHt2 = dq1 dq2hQ2|e e |q2ihq2, t2|q1, t1iJ

−iHt1 iHT1 ×hq1|e e |Q1i X Z −iHT2 iHt2 = dq1 dq2hQ2|e |mihm|e |q2ihq2, t2|q1, t1iJ mn −iHT1 iHt1 ×hq1|e |nihn|e |Q1i

4cf. PHYS 703 March 14, 2000 lecture

14 J= 0 t -d 1 J= 0 T J= 0 t2 -d Te-id

Figure 6: The rotation of the time axis needed to isolate the ground state contribu- tion to the propagator

X −i(EnT2−EmT1) ∗ = e φm(Q2)φn(Q1) mnZ ∗ × dq1 dq2 φm(q2, t2)hq2, t2|q1, t1iJ φn(q1, t1), where we have introduced a basis of energy eigenstates H|ni = En|ni and energy −iEnt eigenfunctions φn(q, t) = e hq|mi with φn(q) = hq|ni. The J subscripts on the propagators remind us that the source is to be accounted for. It is important to note that φn(q) is only a true eigenfunction for times when the source is not acting; i.e. prior to t1 and later than t2. The integral on the last line can be thought of as a wavefunction, φn(q1, t1), that is propagated through the time when the source ∗ is acting by hq2, t2|q1, t1iJ , and is then dotted with a wavefunction φm(q2, t2). But, ∗ φn(q1, t1) and φm(q2, t2) are energy eigenfunctions for times before and after the source, respectively. Hence, the integral is the amplitude that an energy eigenstate |ni will become an energy eigenstate |mi through the action of the source. Now, let’s perform a rotation of the time-axis in the complex plane by some small angle −δ (δ > 0), as shown in ﬁgure 6. Under such a transformation

T1 → T1 + i|T1|δ (51)

T2 → T2 − i|T2|δ, (52) where we have chosen the axis of rotation to lie between T1 and T2. We see that the exponential term e−i(EnT2−EmT1) will acquire a damping that goes like −δ(En|T |+Em|T |) e 2 1 . As we push T1 → −∞ and T2 → ∞, the damping will become inﬁnite for each term in the sum, except for the ground state which we can set to have an energy of E0 ≥ 0. Therefore,

−iE0(T2−T1) ∗ lim hQ2,T |Q1, −T iJ = e φ0(Q2)φ (Q1) (53) −iδ 0 T →∞ e Z ∗ × dq1 dq2 φ0(q2, t2)hq2, t2|q1, t1iJ φ0(q1, t1), δ > 0, where we have set T2 = −T1 = T for convenience. Now, if we take t2 and t1 to ±∞ respectively, the integral reduces to the amplitude that a wavefunction which has

15 the form of φ0(q) in the distant past will still have the form of φ0(q) in the distant future. In other words, it is the ground-to-ground state transition amplitude, which we denote by h0, ∞|0, −∞iJ ∝ lim hQ2,T |Q1, −T iJ , (54) T →∞ e−iδ where the constant of proportionality depends on Q1, Q2 and T . Now, instead of rotating the contour of the the time-integration, we could have added a small term −i²q2/2 to the Hamiltonian. Using ﬁrst order perturbation theory, this shifts 2 the energy levels by an amount δEn = −i²hn|q |ni/2. For most problems (i.e. the harmonic oscillator, hydrogen atom), the expectation value of q2 increases with increasing energy. Assuming that this is the case for the problem we are doing, we see that the ﬁrst order shift in the eigenenergy accomplishes the same thing as the rotation of the time axis in (54). But, subtracting i²q2/2 from H is the same thing as adding i²q2/2 from L5. Therefore,

Z ½ Z ∞ · ¸¾ 1 2 h0, ∞|0, −∞iJ ∝ [dQ] exp i dt L(Q, Q˙ ) + JQ + i²q . (55) −∞ 2 Finally, want to normalize this result such that if the source is turned oﬀ, the amplitude h0, ∞|0, −∞i is unity. Deﬁning R n R h io ∞ ˙ 1 2 [dQ] exp i −∞ dt L(Q, Q) + JQ + 2 i²Q Z[J] = R n R h io , (56) ∞ ˙ 1 2 [dQ] exp i −∞ dt L(Q, Q) + 2 i²Q we have h0, ∞|0, −∞iJ = Z[J]. (57) Before moving on the the next section, we would like to establish a result that will prove very useful later when we consider ﬁeld theories. We ﬁrst deﬁne the functional derivative of Z[J] with respect to J(t0). Essentially, the functional derivative of a functional f[y], where y = y(x), is the derivative of the discrete expression with respect to the value of y at a given x. For example, the discrete version of Z[J] is Z nX−1 h Z (τJ(t ), τJ(t ) . . . τJ(t )) ∝ exp iτ L(Q , Q˙ ) 0 1 n−1 j j j=0 ¸) 1 nY−1 + J(t )Q + i²Q2 dQ , (58) j j 2 j i i=1 where we have indicated that the discrete version of Z[J] is an ordinary function of n variables τJ(tj) and omitted the normalization factor. We have explicitly included

5An alternative procedure for singling out the ground state contribution comes from considering t to be purely imaginary, i.e. consideration of Euclidean space. This is discussed in the next section.

16 the weighting factor τ with each of the discrete variables to account for the fact that as n → ∞, each J(tk) covers a smaller and smaller portion of the integration interval. The functional derivative of Z[J] with respect to J(tk) is then the partial derivative of the discrete expression with respect to τJ(tk). Going back to the continuum limit, we write the functional derivative of Z[J] with respect to Q(t1) as:

Z ½ Z ∞ · ¸¾ δZ[J] 1 2 ∝ i [dQ]Q(t1) exp i dt L(Q, Q˙ ) + JQ + i²Q . (59) δJ(t1) −∞ 2 In a similar fashion, we have Z δnZ[J] ∝ in [dQ]Q(t ) ··· Q(t ) δJ(t ) ··· δJ(t ) 1 n 1 n ½ Z · ¸¾ ∞ 1 × exp i dt L(Q, Q˙ ) + JQ + i²Q2 . (60) −∞ 2 We notice a similarity between this expression and the expression from statistical mechanics that gives the average value of a microscopic variable in the canonical ensemble. We argue that equation (60) gives the exact same thing: the expectation value of Q(t1) ··· Q(tn). There is one wrinkle, however, which we now proceed to outline. Consider, with tk > tk0 , Z

hqf , tf |q(tk)q(tk0 )|qi, tii = dq1 ··· dqn−1hqf , tf |qn−1, tn−1i · · ·

hqk, tk|q(tk)|qk−1, tk−1i · · · hqk0 , tk0 |q(tk0 )|qk0−1, tk0−1i · · · hq , t |q , t i Z 1 1 i i

= dq1 ··· dqn−1 q(tk)q(tk0 )hqf , tf |qn−1, tn−1i · · ·

hq1, t1|qi, tii. (61)

−iδ −iδ By pushing ti → −∞ e and tf → ∞ e , we can repeat our previous manipula- tions to show that Z −iδ −iδ dq1 ··· dqn−1 q(tk)q(tk0 )hqf , ∞ e |qn−1, tn−1i · · · hq1, t1|qi, −∞ e i " # Z Z ∞ e−iδ ˙ = N [dq]q(tk)q(tk0 ) exp i dt L(Q, Q) , (62) −∞ e−iδ which follows directly from the arguments of section 2, and

−iδ −iδ hqf , ∞ e |q(tk)q(tk0 )|qi, −∞ e i ∝ h0|q(tk)q(tk0 )|0i, (63) which follows from an argument similar to the one we used to calculate the matrix element h0, ∞|0, −∞iJ with t1 = tk0 , t2 = tk and J = 0. Just as before, the

17 rotation of the time axis can be achieved by adding i²q2/2 to the Lagrangian. This calculation cannot be repeated for the case tk > tk0 because the order of q(tk)q(tk0 ) in hqf , tf |q(tk)q(tk0 )|qi, tii cannot be switched without introducing terms involving the commutator of H and q. But, in order to perform the decomposition (61), we need the late q operator appearing to the left of the earlier q operator. What this means is that we must be considering the time-ordered product of q(tk) and q(tk0 ). Putting all of this together along with the expression for the functional derivatives of Z[J], we get ¯ 2 ¯ δ Z[J] ¯ 2 ¯ = i h0|T [q(tk)q(tk0 )]|0i. (64) δJ(tk)δJ(tk0 ) J=0 This is easily generalized to the time order product of many q operators: ¯ n ¯ δ Z[J] ¯ n ¯ = i h0|T [q(t1) ··· q(tn)]|0i. (65) δJ(t1) ··· δJ(tn) J=0 We have demanded strict equality in these expression to ensure that the n = 0 case returns h0|0i = 1. This is a very important formula for what follows.

5 Free scalar ﬁelds

We now move on to the quantum ﬁeld theory of a scalar ﬁeld φ(x). In this section we draw on sections 6.1 and 6.3 of Ryder [1], chapter 2 of Popov [4] and section 3.2 of Brown [3]. The classical ﬁeld φ is assumed to satisfy the Klein-Gordon equation

(¤ + m2)φ = 0. (66)

We deﬁne the vacuum to vacuum transition probability for this theory by

h0, ∞|0, −∞iJ = Z[J], (67) where Z ½ Z · ¸¾ 1 1 Z[J] = [dφ] exp i d4x L(φ) + J(x)φ + i²φ2 , (68) Z0 2 with Z ½ Z · ¸¾ 1 Z = [dφ] exp i d4x L(φ) + i²φ2 . (69) 0 2 In this expressions, the measure [dφ] is meant to convey an integration over all ﬁeld conﬁgurations, which can be achieved in practice by dividing spacetime into N 4 points (tm, xi, yj, zk), with m, i, j, k = 1 ...N. We can schematically merge all of 4 these indices into a single one n = 1 ...N . The ﬁeld is considered to be a collectionR 4 Q of N independent variables, and the functional integration [dφ] becomes i dφi. The functional Z[J] represents the fundamental object in the theory. Knowledge of the form of Z[J] allows us to derive all the results that we could hope to obtain from

18 experiments involving the ﬁeld φ. Such grandiose statements need to be justiﬁed, which we now proceed to do. We ﬁrst substitute the Lagrangian appropriate to the Klein-Gordon equation into the expression for Z[J]: Z µ Z ½ ¾¶ 1 4 1 £ α 2 2¤ Z[J] = [dφ] exp i d x ∂αφ∂ φ − (m − i²)φ + φJ . (70) Z0 2

α Integrating the ∂αφ∂ φ term by parts and using Gauss’ theorem to discard the boundary term (assuming φ → 0 at inﬁnity) gives Z µ Z ½ ¾¶ 1 1 Z[J] = [dφ] exp −i d4x φ(¤ + m2 − i²)φ − φJ . (71) Z0 2 Regarding φ as an integration variable, we can change variables according to

φ(x) → φ(x) + φ0(x). (72)

Noting that Z Z 4 4 d x φ¤φ0 = d x φ0¤φ (73) using Gauss’ theorem and demanding that φ0 → 0 at inﬁnity, we obtain Z µ Z ½ 1 1 Z[J] = [dφ] exp −i d4x φ(¤ + m2 − i²)φ Z 2 0 ¾¶ 1 +φ(¤ + m2 − i²)φ + φ (¤ + m2 − i²)φ − (φ + φ )J . (74) 0 2 0 0 0

Let us demand that 2 (¤ + m − i²)φ0 = J(x). (75) We can solve this equation by introducing the Feynman propagator, which satisﬁes

2 4 (¤ + m − i²)∆F (x) = −δ (x). (76)

Hence, Z 4 φ0 = − ∆F (x − y)J(y) d y. (77)

We then obtain for Z[J]: · Z ¸ 1 i Z[J] = exp − J(x)∆ (x − y)J(y) d4x d4y Z 2 F 0Z · Z ¸ i × [dφ] exp − φ(¤ + m2 − i²)φ d4x . (78) 2

19 But we know Z[0] = 1, so we must have Z · Z ¸ i Z = [dφ] exp − φ(¤ + m2 − i²)φ d4x . (79) 0 2

This leads to our ﬁnal expression: · Z ¸ i Z[J] = exp − J(x)∆ (x − y)J(y) d4x d4y . (80) 2 F

Before we move forward, we would like to make a comment on the inclusion of the i²φ2/2 term in our expression for Z[J]. The reader will recall that the reason that we added this term was to simulate the eﬀects of rotating the time axis by a small angle −δ in the complex plane. Instead of adding the ²-term, we can instead rotate the t axis by −π/2 so that t = −iτ. The metric becomes ηαβ = diag(−1, −1, −1, −1), which means that we are considering a Euclidean, not Lorentzian, manifold. In that case, the vacuum-to-vacuum transition amplitude is Z µ Z ½ ¾¶ 1 1 £ 2 2 2 2¤ Z[J] = [dφ] exp − dτdx (∂τ φ) + ∇φ + m φ − φJ , (81) Z0 2 where ∇ is the del-operator of 3D vector calculus. For J = 0, the argument of the exponential is negative deﬁnite and the integral converges absolutely. So, in some sense, the rotation of the time axis ensures that the integrals in Z[J] are well behaved. Another aspect the Euclidean manifold comes from the Euclidean version of Feynman propagator, which satisﬁes:

2 2 2 4 (∂τ + ∇ − m )∆F (x) = δ (x). (82)

Solving this in Euclidean Fourier space, we get Z i e−iκ·x ∆ (x) = − d4κ , (83) F (2π)4 κ2 + m2 where κ · x = κ0τ + k · x and κ2 = (κ0)2 + k2 (the i is there for convenience). If we change variables according to κ0 = −ik0 and τ = it, we get

Z −ik·x 1 4 e ∆F (x) = − 4 d k 2 2 , (84) (2π) C0 k − m

0 where the k0 integration is to be performed along the contour C shown in ﬁgure 7. But, we can rotate the contour by 90o clockwise to C, which is the standard contour used to calculate the Lorentzian Feynman propagator, deﬁned by (76). To some extent, this explains why the Feynman propagator has the form that it does: it is a direct consequence of the need to rotate the time axis to isolate the vacuum contributions to transition amplitudes and make path integrals converge.

20 k k Im( 0) Im( 0)

C

C Re(k ) -w k 0 = Re( 0) -w +w +w

Figure 7: Equivalent contours of integration for the calculationp of Feynman propa- 2 2 gator in the complex k0 plane. Note the poles at ±ω = ± |k| + m .

We want to calculate the time-ordered products given by ¯ n ¯ 1 δ Z[J] ¯ n ¯ = h0|T [φ(x1) ··· φ(xn)]|0i. (85) i δJ(x1) ··· δJ(xn) J=0 We recall from the canonical formulation of ﬁeld theory that h0|T [φ(x)φ(y)]|0i is the amplitude for the creation of a particle at y and its later destruction at x (or vice versa, depending on the times associated with x and y). Using (80) and our previously mentioned notions of functional diﬀerentiation, we have Z 1 δ Z[J] = −Z[J] dy ∆F (x1 − y)J(y). (86) i δJ(x1) The second order derivative is Z Z 1 δ 1 δ Z[J] = Z[J] dy1 ∆F (x1 − y1)J(y1) dy2 ∆F (x2 − y2)J(y2) i δJ(x2) i δJ(x1) + i∆F (x1 − x2)Z[J].

Continuing, 1 δ 1 δ 1 δ Z[J] = i δJ(x ) i δJ(x ) i δJ(x ) 3 2 Z 1

− i∆F (x1 − x2)Z[J] dy3 ∆F (x3 − y3)J(y3) Z

− i∆F (x1 − x3)Z[J] dy2 ∆F (x2 − y2)J(y2) Z

− i∆F (x2 − x3)Z[J] dy1 ∆F (x1 − y1)J(y1)

21 Z

− Z[J] dy1 dy2 dy3 ∆F (x1 − y1)J(y1)

× ∆F (x2 − y2)J(y2)∆F (x3 − y3)J(y3) Finally, we write down the fourth order derivative:

1 δ 1 δ 1 δ 1 δ Z[J] = i∆F (x1 − x2)i∆F (x3 − x4)Z[J] i δJ(x4) i δJ(x3) i δJ(x2) i δJ(x1) + i∆F (x1 − x3)i∆F (x2 − x4)Z[J]

+ i∆F (x2 − x3)i∆F (x1 − x4)Z[J] + other terms that vanish when J = 0.

When J = 0, these expression give us the following time-ordered products:

h0|T [φ(x1)]|0i = 0 (87)

h0|T [φ(x1)φ(x2)]|0i = i∆F (x1 − x2) (88)

h0|T [φ(x1)φ(x2)φ(x3)]|0i = 0 (89)

h0|T [φ(x1)φ(x2)φ(x3)φ(x4)]|0i = i∆F (x1 − x2)i∆F (x3 − x4)

+ i∆F (x1 − x3)i∆F (x2 − x4)

+ i∆F (x1 − x4)i∆F (x2 − x3) (90)

We call h0|T [φ(x1) ··· φ(xn)]|0i an n-point correlation function. Generalizing the pattern above to T products of more ﬁeld operators, we ﬁnd that if n is odd, the n-point function vanishes. On the other hand, if n is even, the correlation func- tion reduces to the sum all possible permutations of products of 2-point functions i∆F (x − y) with x, y ∈ {x1, . . . , xn}, x 6= y. When this result is derived from canonical methods, it is know as Wick’s Theorem. It is interesting to interpret these results in terms of the Taylor series expansion of Z[J] about J = 0. To make sense of this object, recall that the Taylor series expansion of a function of a ﬁnite number of variables is ¯ X∞ Xk Xk n ¯ 1 ∂ F ¯ F (y1, . . . , yk) = ··· yi1 . . . yik ¯ . (91) n! ∂yi1 . . . ∂yin n=0 i1=1 in=1 yi=0 This is expansion is taken about the zero of all of the independent variables. As- suming the variables are weighted appropriately, when we go to the continuum limit k → ∞ we obtain that Z ¯ X∞ 1 δnF ¯ F [y] = dx ··· dx y(x ) ··· y(x ) ¯ . (92) n! 1 n 1 n δy(x ) . . . δy(x )¯ n=0 1 n y=0 Using this expansion, we obtain

22 x x x x x 1 2 x1 2 x x 1 2 Z[J]=1+ 1 2 + + + + … x x3 4 x x x x ( 3 4 3 4 (

Figure 8: Pictorial representation of the free ﬁeld generating functional Z[J]

Z i2 Z[J] = 1 + dx dx J(x )J(x )i∆ (x − x ) 2! 1 2 1 2 F 1 2 Z i4 + dx dx dx dx J(x )J(x )J(x )J(x ) 4! 1 2 3 4 1 2 3 4 ·

× i∆F (x1 − x2)i∆F (x3 − x4) + i∆F (x1 − x3)i∆F (x2 − x4) ¸

+ i∆F (x1 − x4)i∆F (x2 − x3) + ··· (93)

This result is depicted diagrammatically in ﬁgure 8. In this ﬁgure, we use the following Feynman rules: 1. Each line is connected to a sink and a source.

2. Each sink or a source is labeled with a spacetime point.

3. A line running between x and y comes with a propagator i∆F (x − y). 4. Each source or sink attached to x comes with a factor iJ(x).

5. The collection of all graphs with n sources and sinks is multiplied by 1/n!.

6. All spacetimes coordinates are integrated over. This ﬁrst rule ensures that we only include diagrams with an even number of vertices and that we only consider disconnected graphs. The Feynman diagrams illustrate the correspondence of Z[J] to the vacuum-to-vacuum transition probability beauti- fully. Each term in the graph involves the creation of a particle at some point and its destruction at a later point. The three terms in the 4-vertex graph account for all possible permutations of particle being created/destroyed at a pair of x1, x2, x3 or x4. What we would like to do now is move on to the more interesting case of interacting ﬁelds.

6 The generating functional for self-interacting ﬁelds

While free ﬁeld theory has a certain amount of elegance to it, it is not terribly interesting. In this section, we consider a self-interacting ﬁeld whose Lagrangian is

23 given by 1 1 L(φ) = ∂ φ ∂αφ − m2φ2 + L (φ) = L (φ) + L (φ). (94) 2 α 2 int 0 int

The discussion follows section 6.4 of Ryder [1]. Here, Lint(φ) is the Lagrangian describing the self interaction. The generating functional is R ¡ R ¢ [dφ] exp iS + i d4x Jφ Z[J] = R , (95) [dφ]eiS where S is the classical action Z 4 S = d x [L0(φ) + Lint(φ)] = S0 + Sint. (96)

We have dropped the i²φ2/2 term used to single out the ground state, which we can rationalize by the rotation of the time axis. In this section, we will write the free ﬁeld propagator as R ¡ R ¢ 4 [dφ] exp iS0 + i d x Jφ Z0[J] = R . (97) [dφ]eiS0

We would like to write Z[J] in a form particularly useful for calculations. We can’t really reproduce the manipulations of the last section because the Lint in the action S introduces diﬃculties when the shift φ → φ + φ0 is performed. We will instead derive a diﬀerential equation satisﬁed by Z[J] and then solve it in terms of J(x) and the Feynman propagator. The result will be something that we might have guessed intuitively. What is the equation satisﬁed by the free ﬁeld propagator? Now, we know that Z 1 δ Z [J] = −Z [J] dy ∆ (x − y)J(y). (98) i δJ(x) 0 0 F

2 We operate on both sides with ¤x+m and use the deﬁning relation for the Feynman propagator (76) to get: 1 δ (¤ + m2) Z [J] = J(x)Z [J]. (99) x i δJ(x) 0 0

The is the diﬀerential equation satisﬁed by Z0[J]. In order to ﬁnd the diﬀerential equation satisﬁed by Z[J], let us deﬁne the functional

eiS Zˆ[φ] = R . (100) [dφ] eiS

Then Z µ Z ¶ Z[J] = [dφ] Zˆ[φ] exp i d4x Jφ , (101)

24 which is the functional equivalent of the Fourier transform. We can functionally diﬀerentiate Zˆ[φ] with respect to φ using Z · ¸ 1 1 S = d4x ∂ φ ∂αφ − m2φ2 + L (φ) 2 α 2 int Z · ¸ 1 = − d4x φ(¤ + m2)φ − L , (102) 2 int where Gauss’ theorem has been used. The superiority of the latter form is that we −→ ←− can integrate twice by parts to change φ¤φ into φ¤φ. So, when we functionally diﬀerentiate, we can make sure that the φ being acted on by the ¤ operator is diﬀerent from the φ being acted on by the δ/δφ. We get

δZˆ[φ] i = (¤ + m2)φ(x)Zˆ[φ] − L0 (φ)Zˆ[φ], (103) δφ(x) int where ∂L L0 (φ) = int (104) int ∂φ R 4 and ¤ = ¤x. We multiply both sides of (103) by exp[i J(y)φ(y) d y] and integrate over φ, i.e. we take the Fourier transform. The RHS of (103) becomes Z · Z ¸ 1 RHS = (¤ + m2) [dφ]φ(x) exp iS + i J(y)φ(y) d4y Z 0 Z · Z ¸ 1 0 4 − [dφ]Lint(φ) exp iS + i J(y)φ(y) d y , (105) Z0 where we have written Z iS Z0 = [dφ]e . (106)

Now, it’s easy to see Z · Z ¸ 1 δZ[J] 1 = [dφ] φ(x) exp iS + i J(y)φ(y) d4y , (107) i δJ(x) Z0 which leads to µ ¶ Z · Z ¸ 1 δ n 1 Z[J] = [dφ] φn(x) exp iS + i J(y)φ(y) d4y . (108) i δJ(x) Z0

0 Now, we assume that Lint(φ) possesses a Taylor series expansion in φ. We can then reproduce the second term in (105) by adding together a series of contributions of the form of (108). Hence, µ ¶ 1 δZ[J] 1 δ RHS = (¤ + m2) − L0 Z[J]. (109) i δJ(x) int i δJ(x)

25 The Fourier transform of the LHS of (103) becomes Z · Z ¸ δZˆ(φ) LHS = i [dφ] exp i J(y)φ(y) d4y δφ(x) · Z ¸ ¯ ¯ ˆ 4 ¯ = iZ(φ) exp i J(y)φ(y) d y ¯ φ→∞ Z · Z ¸ δ −i [dφ]Zˆ(φ) exp i J(y)φ(y) d4y δφ(x) Z · Z ¸ = [dφ]J(x)Zˆ(φ) exp i J(y)φ(y) d4y

= J(x)Z[J]. (110)

In the second line we have performed a functional integration by parts, which follows from the fact that the functional derivative satisﬁes the product rule and functional integral obeys the fundamental theorem of calculus. The boundary term must vanish as φ → ∞ because if it didn’t, the integral for Z[J] would diverge. Putting together our formulae for the LHS and RHS of the Fourier transform of (103): µ ¶ 1 δZ[J] 1 δ (¤ + m2) − L0 Z[J] = J(x)Z[J]. (111) i δJ(x) int i δJ(x) This is the diﬀerential equation satisﬁed by Z[J]. We see that if the interacting Lagrangian is set to zero, the result reduces to (99). We will assume a solution of the diﬀerential equation of the form · Z µ ¶¸ 1 δ Z[J] = N exp i d4x L Z [J] (112) int i δJ 0 partly because this is what we might expect from L = L0 +Lint if we replace −iδ/δJ with φ, as we usually do, partly because we know it’s the right answer. As usual, N is a normalizing factor. Let’s ﬁrst establish and identity: h R ³ ´i h R ³ ´i 4 1 δ 4 1 δ exp −i d y Lint i δJ(y) J(x) exp i d y Lint i δJ(y) ³ ´ 0 1 δ = J(x) − Lint i δJ(x) . (113)

The ﬁrst step is to notice that the commutator of Ji and ∂/∂Jk when acting on a function of (J1,...,Jn) is given by the well known relation · ¸ 1 ∂ Ji, = iδik. (114) i ∂Jk The continuum limit (n → ∞) of this is · ¸ 1 δ J(x), = iδ(x − y). (115) i δJ(y)

26 Now, it’s not hard to see that if A and B are operators whose commutator is a number a, then

[A, B] = a, [A, B2] = 2aB, [A, B3] = 3aB2, . . [A, Bn] = naBn−1.

Hence, · µ ¶ ¸ µ ¶ 1 δ n 1 δ n−1 J(x), = iδ(x − y)n . (116) i δJ(y) i δJ(y)

Assume that Lint(φ) possesses a Taylor series expansion

1 X∞ 1 L (φ) = c + c φ + c φ2 + ··· = c φn. (117) int 0 1 2! 2 n! n n=0 Since [A, (B + C)] = [A, B] + [A, C], we can write " µ ¶ # µ ¶ X∞ 1 1 δ n X∞ 1 1 δ n−1 J(x), c = iδ(x − y) c . (118) n! n i δJ(y) (n − 1)! n i δJ(y) n=0 n=1 But X∞ n X∞ 1 L0 (φ) = c φn−1 = c φn−1. (119) int n! n (n − 1)! n n=0 n=1 Using this an integrating (118) with respect to y gives · Z µ ¶¸ µ ¶ 1 δ 1 δ J(x), i d4y L = −L0 . (120) int i δJ(y) int i δJ(x) where we have scaled Lint by i. Finally, note that the Hausdorﬀ formula gives

e−BAeB = A + [A, B] (121) whenR [A, B] is a number. Putting (120) into (121) with A = J(x) and B = 4 i d y Lint (−iδ/δJ(y)) yields (113). Using our assumed form for Z[J] (112) and the identity (113) we get · Z µ ¶¸ 1 δ J(x)Z[J] = NJ(x) exp i d4y L Z [J] int i δJ(y) 0 · Z µ ¶¸ · µ ¶¸ 1 δ 1 δ = N exp i d4y L J(x) − L0 Z [J] int i δJ(y) int i δJ(x) 0

27 · Z µ ¶¸ 1 δ 1 δ = N exp i d4y L (¤ + m2) Z [J] int i δJ(y) x i δJ(x) 0 µ ¶ · Z µ ¶¸ 1 δ 1 δ −L0 N exp i d4y L Z [J] int i δJ(x) int i δJ(y) 0 µ ¶ 1 δZ[J] 1 δ = (¤ + m2) − L0 Z[J]. x i δJ(x) int i δJ(x)

In going from the second to the third line, we have used the diﬀerential equation satisﬁed by Z0[J] (99). This result is just the diﬀerential equation (111), which conﬁrms that we can write h R ³ ´i 4 1 δ exp i d x Lint i δJ(x) Z0[J] Z[J] = h R ³ ´i ¯ (122) 4 1 δ ¯ exp i d x Lint Z0[J]¯ i δJ(x) J=0 where we have · Z ¸ i Z [J] = exp − J(x)∆ (x − y)J(y) d4x d4y . (123) 0 2 F

With these two equations, we have succeeding in writing down the generating func- tional entirely in terms of the source J and the Feynman propagator ∆F .

7 φ4 and φ3 theory

We would like to demonstrate the calculation of the generating functional Z[J] and some n-point functions in the case of self-interacting φ4 and φ3 theory. We follow section 6.5 of Ryder [1] and chapter 2 of Popov [4]. We ﬁrst consider φ4 theory, which has the interacting Lagrangian: λ L (φ) = − φ4. (124) int 4!

4 It goes without saying that λ is small. The generating functional Z4[J] for φ is · ¸ R ³ ´4 iλ 4 1 δ exp − 4! d z i δJ(z) Z0[J] Z [J] = · ¸ (125) 4 R ³ ´4 ¯ iλ 4 1 δ ¯ exp − 4! d z i δJ(x) Z0[J]¯ J=0

The numerator of Z4[J] is, to ﬁrst order in λ, is: " Z µ ¶ # iλ 1 δ 4 num Z [J] = 1 − d4z + ··· Z [J]. (126) 4 4! i δJ(z) 0

28 Performing the functional diﬀerentiation, we obtain ( Z " µZ ¶ iλ 2 num Z [J] = 1 − d4z −3∆2 (0) + 6i∆ (0) d4x ∆ (z − x)J(x) 4 4! F F F # ) µZ ¶4 4 + d x ∆F (z − x)J(x) + ··· Z0[J], (127)

6 where we have written ∆F (z − z) = ∆F (0) . We also get ¯ ½ Z ¾ ¯ £ ¤ ¯ iλ 4 2 denom Z4[J] = num Z4[J]¯ = 1 − d z −3∆F (0) + ··· Z0[J]. (128) J=0 4! Putting the two results together yields, to order λ, we have: ( Z " µZ ¶ iλ 2 Z [J] = 1 − d4z 6i∆ (0) d4x ∆ (z − x)J(x) 4 4! F F # ) µZ ¶4 4 + d x ∆F (z − x)J(x) + ··· Z0[J], (129)

Now, let’s do the same thing for φ3 theory, where the interacting Lagrangian is λ L (φ) = − φ3. (130) int 3! The numerator of the generating functional is " Z µ ¶ # iλ 1 δ 3 num Z [J] = exp − d4z Z [J]. (131) 3 3! i δJ(z) 0 Expanding to order λ, we get ½ Z · Z iλ num Z [J] = 1 − d4z −3i∆ (0) d4x ∆ (z − x)J(x) 3 3! F F # ) µZ ¶3 4 − d x ∆F (z − x)J(x) + ··· Z0[J]. (132)

If we set J = 0, the above reduces to unity. So, the denominator of Z3[J] is 1, to ﬁrst order in λ, which yields: ½ Z · Z iλ Z [J] = 1 − d4z −3i∆ (0) d4x ∆ (z − x)J(x) 3 3! F F # ) µZ ¶3 4 − d x ∆F (z − x)J(x) + ··· Z0[J]. (133)

6The Feynman propagator evaluated at zero is, of course, inﬁnite. It’s inclusion in the generating functional represents the inﬁnite self-energy of particles in the theory and must be regulated by renormalization, which we will not consider here.

29 x 1 x2 l il Z [J]=1 + - z … 4 x x + 4 1 z 2 24 x3 x4

x1 z Z J - l x + il x … 3[ ]=1 1 z 3 + 2 6

x2

Figure 9: Pictorial representation of the generating functional for φ4 and φ3 theory respectively

We can represent the two generating functionals Z4[J] and Z3[J] diagrammati- cally with the following Feynman rules:

1. The spacetime point zi is associated with internal points, all other variables go with external points (recall that zi is the coordinate that occurred in the interacting Lagrangian).

2. A line between x and z comes with a propagator ∆F (x − z). 3. Each internal point comes with a factor of iλ/4! for φ4 theory, iλ/3! for φ3 theory.

4. External points x come with a factor J(x).

5. Terms of the form ∆F (0) represent closed loops, or propagators who begin and end at the same point, joined to internal points.

6. All spacetime points are integrated over.

7. Each term in the series is multiplied by the free particle propagator Z0[J].

These rules give the pictures in ﬁgure 9 for Z4[J] and Z3[J]. These diagrams stress the general structure of the two theories. Because of the power of 4 in the interact- ing Lagrangian in φ4 theory, the vertices in the associated Feynman diagrams are attached to 4 legs. Similarly, the vertices in φ3 theory are attached to 3 legs. A peculiarity of φ3 theory is the diagram with only one external point. As we will see below, this gives rise to a one point correlation function which means that particles in φ3 can spontaneously self-destruct.

30 l i x1 x - + … 2 x x 2 1 z 2

il x1 + … 4 z

Figure 10: Pictorial representation of the two-point function h0|T [φ(x1)φ(x2)]|0i4 for 4 3 φ theory (top) and of the one-point function h0|T [φ(x1)]|0i3 for φ theory (bottom)

Let’s calculate the 2-point function for φ4 theory: ¯ 2 ¯ δ Z4[J] ¯ h0|T [φ(x1)φ(x2)]|0i4 = − ¯ . (134) δJ(x2)δJ(x1) J=0 The functional diﬀerentiation is easy to do from (129) using the our previously derived formulae for derivatives of the free ﬁeld propagator (87) – (90). The result is Z λ h0|T [φ(x )φ(x )]|0i = i∆ (x − x ) − ∆ (0) d4z ∆ (x − z)∆ (z − x ) 1 2 4 F 1 2 2 F F 1 F 2 + O(λ2). (135) This can be turned into a Feynman diagram, as in ﬁgure 10, using rules similar to the ones for Z[J], except now the only the internal points are integrated over. Figure 10 shows how the amplitude for the creation of a particle at point x1 and its later destruction at x2 is modiﬁed by an (inﬁnite) loop diagram to ﬁrst-order in λ in φ4 theory. Now, let’s get the 2-point function in φ3 theory. It is simply calculated ¯ 2 ¯ δ Z3[J] ¯ 2 h0|T [φ(x1)φ(x2)]|0i3 = − ¯ = i∆F (x1 − x2) + O(λ ). (136) δJ(x2)δJ(x1) J=0 This isn’t terribly interesting, we see that there are no corrections to the free ﬁeld result to this level in perturbation theory. We shouldn’t be surprised, there is no way to connect to external lines to a single vertex if the vertex must be attached to three lines. However, there is a way to connect a vertex to one external line and one internal line, as shown in ﬁgure 10. This is borne out by the calculation of the one-point function: ¯ δ2Z [J]¯ h0|T [φ(x )]|0i = −i 3 ¯ 1 3 δJ(x ) ¯ 1 Z J=0 iλ = ∆ (0) d4z ∆ (z − x ) + O(λ2). (137) 4 F F 1

31 x1 x x x -l z l 1 z 2 z 3 z x3 + + + + … 4 x x x x x x ( 2 3 3 1 1 2 ( x2

Figure 11: Pictorial representation of h0|T [φ(x1)φ(x2)φ(x3)]|0i3, the three point function, for φ3 theory

Certainly a strange beast, this represents the amplitude of a particle being sponta- neously created out of the vacuum, all by itself7. We can also calculate the three point function ¯ δ2Z [J] ¯ h0|T [φ(x )φ(x )φ(x )]|0i = i 3 ¯ 1 2 3 3 δJ(x )δJ(x )δJ(x )¯ Z 3 2 1 J=0 4 = −λ d z ∆F (z − x1)∆F (z − x2)∆F (z − x3) Z λ + ∆ (0) d4z [∆ (z − x )∆ (x − x ) 4 F F 1 F 2 3 + ∆F (z − x2)∆F (x3 − x1)

+ ∆F (z − x3)∆F (x1 − x2)]. (138)

The ﬁrst integral involves all the external legs attached to the same point, i.e. two particles merging into one or one particle splitting in two. The second integral is the product of free 2-point functions and the 1-point function we have already calculated. The diagram is in ﬁgure 11. We have hence shown how the generating functional and n-point functions can be found from simple functional diﬀerentiation of equations (122) and (123) and expressed in terms of Feynman diagrams. These diagrams can be converted into scattering amplitudes via the LSZ reduction formula8:

hp ,..., p , +∞|q ,..., q , −∞i = disconnected terms 1 n 1 m Z n+m + (i) dy1 ··· dyn dx1 ··· dxm

× ei(p1·y1+···+pn·yn−q1·x1−···−qm·xm) 2 2 × (¤y1 + m ) ··· (¤yn + m ) 2 2 × (¤x1 + m ) ··· (¤xm + m ) × h0|T [φ(y1) ··· φ(yn)φ(x1) . . . φ(xm)]|0i.

7However, such a process would be forbidden on energy grounds 8cf. PHYS 703 March 16, 2000. Brown [3] claims that this formula can be obtained via path integral methods, by he presents a proof using the canonical formalism.

32 In practice, these scattering amplitudes are the only meaningful quantities in quan- tum ﬁeld theory since they are the only things that can be directly measured. So, having arrived at a point where we can calculate hp1,..., pn, +∞|q1,..., qm, −∞i using the generating functional, we have completed our formulation of self-interacting ﬁeld theories in terms of path integrals. More complicated theories, such as QED, can be quantized in terms of path- integrals, but there are several issues that need to be addressed when writing down α RZ[J] for gauge ﬁelds. One ﬁnds that Z[J] is inﬁnite for gauge ﬁelds A , because the [dAα] integration includes an inﬁnite number of contributions from ﬁelds related by a simple gauge transformation. The resolution is the addition of gauge ﬁxing terms to the Lagrangian and the appearance of non-physical “ghost ﬁelds”. Such things are beyond the scope of this paper.

8 Conclusions

We have demonstrated how quantum mechanics can be formulated in terms of the propagator that represents the amplitude of a particle travels from some initial position to some ﬁnal position. We have determined the form of the propagator to be a sum over paths of the amplitudes associated with the particle traveling along a given trajectory. The amplitude associated with the path q(t) was shown to be eiS[q], where S[q] is the classical Lagrangian. We used this form of the propagator to derive Feynman rules for the Sfi matrix in non-relativistic scattering problems. Then, we showed that by adding a source J(t) to the Lagrangian and rotating the time axis in the complex plane, the propagator Z[J] reduces to the ground state- to-ground state transition amplitude. Functional derivatives of Z[J] with respect to the source J(t) were shown to give the time-ordered product of conﬁguration operators q(t). We generalized the propagator to free scalar ﬁelds and expanded Z[J] in terms of Feynman diagrams by expressing it in terms of the Feynman propagator and the source J(x). In the process, we derived some n-point correlation functions by diﬀerentiating Z[J]. We discussed how the rotation of the time axis needed to isolate the vacuum-to-vacuum transition probability is directly responsible for the prominent role that the Feynman propagator plays in ﬁeld theory. We developed a formalism to deal with self-interacting ﬁelds and expressed Z[J] entirely in terms of 4 ∆F (x), J(x), and the series expansion of the interacting Lagrangian. For both φ and φ3 theory, we expressed Z[J] and some n-point functions in terms of Feynman diagrams. Since the scattering problem is essentially solved once the procedure for calculating n-point functions is speciﬁed (ignoring issues of renormalization), we have hence shown how all of the theory of self-interacting scalar ﬁelds can be derived from path integrals.

33 References

[1] Lewis H. Ryder. Quantum Field Theory, 2nd ed.. Cambridge: 1996.

[2] Gordon Baym. Lecture Notes on Quantum Mechanics. Benjamin/Cummings: 1969.

[3] Lowell S. Brown. Quantum Field Theory. Cambridge: 1992.

[4] Viktor N. Poppv. Functional Integrals in Quantum Field Theory and Statistical Physics. Reidel: 1983.

34