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The Path Integral Formulation of QFT

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The Path Integral Formulation of QFT April 3, 2016 version 1.0 Quantum Field Theory II Lectures Notes Part I: The Path Integral formulation of QFT Prof. Dr. Gino Isidori ETH & UZH, Spring Semester 2016 i ii Contents 1 The Path Integral formulation of Quantum Field Theory 1 1.1 The Action Functional in Classical Mechanics . 1 1.2 Path Integral formulation of QM . 2 1.3 Path-integral formulation of QM for a genetic Hamiltonian . 4 1.4 Path-Integral formulation of Scalar Fields . 6 1.4.1 PI formulation of the two-point correlation function . 7 1.5 The Generating Functional . 8 1.5.1 Generating functional for a free scalar field . 9 1.5.2 Interacting scalar field theory . 12 1.6 Path integral formulation of an Abelian Field Theory . 13 1.7 Quantization of spinor field . 16 1.7.1 Grassmann numbers . 16 1.7.2 Path integral formulation for the Dirac field . 18 1.7.3 Explicit calculation on functional determinants . 19 1.8 Symmetries in Path Integral . 20 1.8.1 Ward-Takahashi identity in QED . 22 iii iv Chapter 1 The Path Integral formulation of Quantum Field Theory 1.1 The Action Functional in Classical Mechanics Within Classical Mechanics, the equation of motions can be derived from a simple principle: the principle of Least Action, or the requirement that the trajectory of the system minimize a functional called the Action. The Action Functional can be constructed also for quantum systems and, as we shall see in this chapter, it allows us to derive a simple \bridge" between Classical Mechanics, Quantum Mechanics (QM), and Quantum Field Theory (QFT). To start, we briefly recall the principle of Least Action in Classical Mechanics. Consider a classical, non-relativistic 3-dimensional system with a point-like particle whose trajectory is described by the vector ~x(t) = fx1(t); x2(t); x3(t)g. The equations of motion reads: @V mx¨i(t) = − ; (1.1) @xi where V (~x) is the potential energy of the particle. Solving the equations of motion, and making use of the initial conditions (position and velocity) at time t1 we can find the position ~x(t) of the particle at any time t. The principle of Least Action states that the trajectory of the system can be obtained by the requirement δS[x(t)] = 0 ; (1.2) where S is the following functional Z t2 Z t2 1 _ 2 S[~x(t)] = dt [Ekin − Epot] = dt m~x − V (~x) ; (1.3) t1 t1 2 namely the difference between kinetic and potential energy, integrated along the trajectory. In order to solve (1.2), some boundary conditions on the trajectory need to be given. In this case it is more convenient to choose them to be the position of the trajectory at two different times (e.g. the initial and the final position): ~x(t1) = ~x(1) ; ~x(t2) = ~x(2) : (1.4) In order to prove that (1.2) leads to the classical equations of motion, consider the variation of S under a small perturbation δ~x of the trajectory, which does not affect the endpoints 1 coordinates (boundary conditions): Z t2 1 S[~x + δ~x] = dt m (~x_ + δ~x_)2 −V (~x + δ~x) : (1.5) 2 t1 | {z } ~x_ 2 + 2δ~x_ · ~x_ Integrating by parts we obtain: Z t2 h i Z t2 d S[~x + δ~x] = S[~x] + dtδ~x −m~x¨ − r~ V (~x) + m dt δ~x · ~x_ : (1.6) t1 t1 dt The last term is zero because of the fixed endpoints. We thus recover the equations of motion from the principle δS = 0. Note, however, that the boundary conditions on the trajectory cannot be derived by the principle and must be given. The Action Functional is also particularly useful to identify the conserved quantities follow- ing from the invariance of the system under specific transformations (corresponding to symme- tries of the system). For instance, let's assume that the system is invariant under a rotation of the coordinates. The infinitesimal transformation of coordinates (for rotations along the axis k) is (k) (k) (k) xi ! xi + (δα x)i ; (δα x)i = α kijxj ; (1.7) (k) where α is an infinitesimal parameter and kij is the completely antisymmetric tensor (123 = (k) 1). Because of the symmetry hypothesis, we expect δα S = 0 under the transformation (1.7). Proceeding this way we derive the same result as in (1.6); however, in this case the boundary (k) term no longer vanishes. In order to obtain δα S = 0 we must impose (k) _ (k) _ δα ~x(t2)~x(t2) = δα ~x(t1)~x(t1) : (1.8) We thus deduce the existence of a conserved quantity, which in this case is the angular momen- (k) tum L = mkijxix_ j. In summary, the Action Functional in Classical Mechanics allow us to i) determine the trajectories of the system once fixed boundary conditions are given; ii) determine the conserved quantities of the system under specific symmetry transformations. As we shall see in the rest of this chapter, these two properties hold {with some peculiar modifications, especially in the case i){ also in QM and QFT. 1.2 Path Integral formulation of QM Consider a one-dimensional quantum system with the following Hamiltonian p2 H^ = + V (x) : (1.9) 2m In the Schr¨odinger picture, the probability amplitude for the state jx1i at time t = 0 to evolve to the state jx2i at time t = T is − i HT^ U(xf ; xi; T ) = hxf je ~ jxii ; (1.10) which solves the Schr¨odingerequation @ i U = HU:^ (1.11) ~@T 2 Feynman proposed a way to connect U(xf ; xi; T ) to the Action Functional. In particular, he proposed the following expression Z i S[x(t)] U(xf ; xi; T ) = Dx(t)e ~ ; (1.12) Z where the symbol of integration over all paths Dx(t) will be defined soon. If S[x(t)] ~ the phase of the integrand in (1.12) oscillates very rapidly over the various trajectories. This implies a net vanishing contribution to the integral but for the stationary trajectory characterized by δ (S[x(t)]) = 0 ; (1.13) δx xcl that is nothing but the classical trajectory. The expression (1.12) thus allow us to recover in a simple and intuitive way the classical limit. We will now present a proof of the consistency of (1.10) with the principles of QM {in particular with the Schr¨odingerequation{ via the discretization of the path. We divide the time interval between 0 and T in n infinitesimal time slices of duration ∆t = . The action becomes: Z T n−1 2 1 X 1 (xk+1 − xk) xk+1 + xk S = dt mx_ 2 − V (x) −! m − V ; (1.14) 2 2 2 2 0 k=0 where x0 = xi and xn = xf . We then define the (discretized) path integral by Z Z Z Z n Z 1 def 1 dx1 dx2 dxn−1 1 Y dxk Dx(t) = ··· = ; (1.15) N() N() N() N() N() N() k=1 −∞ where N() is a normalisation factor to be determined later. In other words, we integrate on all possible values of the intermediate coordinates x1 : : : xn−1. We want now to demonstrate the consistency of (1.10) by induction method on n, the number of time slices. To this purpose, we assume (1.10) to be true over the time interval (0;T − ), and then compute explicitly the effect of the last time slice. This implies Z 1 i S[x] U(x ; x ; T ) = dx e ~ U(x ; x ; T − ) f i N() n−1 n−1 i Z 0 2 0 1 0 i 1 (xf − x ) x + xf 0 = dx exp m − V U(x ; xi; T − ) : N() ~ 2 2 2 0 0 Since the kinetic term is rapidly oscillating unless xf ≈ x , we can (Taylor) expand for x close to xf , 0 ! i x +xf − V 2 e ~ z }| { Z (x −x0)2 1 0 i m f i U(xf ; xi; T ) = dx e 2~ 1 − V (xf ) + ··· N() ~ " 2 # 0 @ 1 0 2 @ × 1 + (x − xf ) + (x − xf ) 2 + ··· U(xf ; xi; T − ) : @xf 2 @xf | 0 {z } U(x ;xi;T −) 3 We can perform the integration over x0 using the following Gaussian integration formulae: r r Z 2 π Z 2 Z 2 1 π dx e−ax = ; dx xe−ax = 0 ; dx x2e−ax = : (1.16) a 2a a In principle, these identities hold only for Re[a] > 0, that is not our case. However, as we shall discuss later on, we can overcome this problem adding small \damping terms" (small real coefficients in the exponents) that later on are set to zero. Leaving aside this technical point, we obtain " r #" 2 # 1 2π~ i i~ @ 2 U(xf ; xi; T ) = 1 − V (xf ) + 2 + O( ) U(xf ; xi; T − ) : (1.17) N() −im ~ 2m @xf In order to give a meaning to the limit ! 0 we deduce that we need to impose the condition " # 1 r2π ~ = 1 : (1.18) N() −im We are now able to take the limit ! 0, that yields " 2 2 # @ ~ @ ^ i~ U(xf ; xi; T ) = − 2 + V (xf ) U(xf ; xi; T ) = HU(xf ; xi; T ) : (1.19) @T 2m @xf As can be seen, we recover the Schr¨odingerequation, demonstrating the consistency of the Feynman Path-Integral formulation of QM in this simple system. 1.3 Path-integral formulation of QM for a genetic Hamil- tonian Consider now a generic system with N coordinates ~q = fqig and a generic Hamiltonian H (~q; ~p).
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