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April 3, 2016 version 1.0

Quantum 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 Field Theory 1 1.1 The Functional in Classical ...... 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 Fields ...... 6 1.4.1 PI formulation of the two-point correlation ...... 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 of 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

1.1 The Action Functional in

Within Classical Mechanics, the equation of 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, (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) = {x1(t), x2(t), x3(t)}. The equations of reads: ∂V mx¨i(t) = − , (1.1) ∂xi where V (~x) is the potential energy of the particle. Solving the , 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¨ − ∇~ 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 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 = mkijxix˙ 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 Hˆ = + V (x) . (1.9) 2m

In the Schr¨odinger picture, the for the state |x1i at time t = 0 to evolve to the state |x2i at time t = T is

− i HTˆ U(xf , xi; T ) = hxf |e ~ |xii , (1.10) which solves the Schr¨odingerequation ∂ i U = HU.ˆ (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 ∞ 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 # ∂ ~ ∂ ˆ 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 = {qi} and a generic Hamiltonian H (~q, ~p).

The transition amplitude from the initial state ~qI to the ﬁnal state ~qF is given by

ˆ −i HT h~qF | e ~ |~qI i . (1.20)

Q R i i i Inserting the identity I = i dqk |qki hqk| in each time slice, the transition amplitude can be expressed by the product of terms of the type (~qn ≡ ~qF , ~q1 ≡ ~qI ) ! ˆ ˆ −i H H h~qk+1| e ~ |~qki = h~qk+1| I − i + ··· |~qki . (1.21) ~

Let’s ﬁrst assume that H has the following “separable” form1

Hˆ = fˆ(~q) +g ˆ(~p) . (1.22)

a) Evaluation of the contribution from fˆ(~q)

    Z i ~p (~q −~q ) ~qk+1 + ~qk Y i i ~qk+1 + ~qk Y dpk i k k+1 k h~q | fˆ(~q) |~q i = f δ(q − q ) = f e ~ , k+1 k 2 k+1 k 2 2π i i ~ (1.23)

1 The “hat” on the various terms is there to remind us that these are operators.

4 where we have used fˆ(~q) |~qi = f(~q) |~qi and Z dp eipq = δ(q) . 2π

b) Evaluation of the contribution fromg ˆ(~p)

Y 1 Z h~q | gˆ(~p) |~q i = h~q | dpi g(~p ) |~p i h~p |~q i , (1.24) k+1 k k+1 2π k k k k k i ~ R dp where we have introduced eigenstates via the insertion of I = 2π |pi hp|. i i ~ − ~qk·~pk + ~pk·~qk Using also h~pk|~qki = e ~ and h~qk|~pki = e ~ we get

1 Z i Y i − ~qk·~pk h~q |g(~p )|~q i = dp h~q |~p ig(~p )e ~ k+1 k k 2π k k+1 k k i ~ 1 Z i Y i ~pk·(~qk+1−~qk) = dp g(~p )e ~ . (1.25) 2π k k i ~ Taking the full Hamiltonian, we ﬁnally get

i   Z dp ~q + ~q i Y k k k+1 ~pk·(~qk+1−~qk) h~q |Hˆ |~q i = H , ~p e ~ . (1.26) k+1 k 2π 2 k i ~

Note that on the right hand side H is just a function of the variables, whereas of the left hand side it is an operator. For generic Hˆ we have a potential problem since p and q don’t commute: we need to specify their order. To this purpose, it is convenient to deﬁne the so-called Weyl ordering of Hˆ . This is deﬁned by (1.26), namely Hˆ is Weyl ordered if   Z ~q + ~q i k k+1 ~pk·(~qk+1−~qk) h~q |Hˆ |~q i ∝ d~p H , ~p e ~ . (1.27) k+1 k k 2 k

It is then easy to check that, for instance, Hˆ = p2q2 is not Weyl ordered, while 1 Hˆ = p2q2 + q2p2 + 2qp2q (1.28) 4 it is Weyl ordered. If Hˆ is not Weyl ordered, it can always be put in a Weyl-ordered form using the commutation relations between q and p. This way we can continue our discussion in full generality, relaxing the assumption of a separable structure for Hˆ , as in Eq. (1.22), but simply requiring Hˆ is Weyl ordered. We can now go back to the single time slice term in (1.21):

i  ~q +~q  iHˆ Z dp i k k+1 i − Y k − H 2 ,~pk ~pk·(~qk+1−~qk) h~q |e ~ |~q i = e ~ e ~ (1.29) k+1 k 2π i ~ Z i i h ~qk+1−~qk  ~qk+~qk+1 i Y dp + ~pk· −H ,~pk = k e ~  2 . (1.30) 2π i ~ The underlined term has the following limit in the continuum ( → 0):  ~q − ~q ~q + ~q  h i ~p · k+1 k − H k k+1 , ~p → ~p · ~q˙ − H (~q , ~p ) . (1.31) k  2 k k k k k

5 Then taking all time slices into account, we ﬁnally obtain

i i iHTˆ Z dq dp  P ˙ − Y Y k k i [~pk· ~qk−H(qk,~pk)] h~q |e ~ |~q i = e ~ k (1.32) F I 2π k i ~ Z  i Z T h i = DqDp exp dt ~p · ~q˙ − H (~q, ~p) (1.33) ~ 0 where the integral over the trajectories ~q(t) is constrained at t = 0 and t = T , while the one on ~p(t) is free. The expression in (1.32) can be interpreted as the formal deﬁnition of the path integral for generic QM systems. As can be seen, the formulation is quite simple and elegant. It is also worth to notice that in this case the of the integral is the standard quantum phase-space measure.

1.4 Path-Integral formulation of Scalar Fields

The Path Integral (PI) formulation discussed in the previous section, valid for a system with N coordinates, is well suited for a generalization to the case of a scalar ﬁeld theory. To this purpose, we need to generalize the result to a system with an inﬁnite number of coordinates: one quantum coordinate for each point of the 3-dimensional space: ∂L ~qk(t) → φ(~x,t) , ~pk(t) → Π(~x,t) ≡ , (1.34) ∂φ˙(~x,t) with the following Hamiltonian: Z 1 1  Hˆ = d3~x Π2 + (∇φ)2 + V (φ) . (1.35) 2 2

From now on, except when explicitly stated, we set ~ = 1. Generalizing the result in (1.33), the transition amplitude from the initial state conﬁguration φI (~x) at time t = −T to the ﬁnal state conﬁguration φF (~x) at time t = +T is

ˆ Z R +T 4 ˙ 1 2 1 2 −iH(tF −tI ) +i d x[Πφ− Π − (∇φ) −V (φ)] hφF (x)|e |φI (x)i = DφDΠe −T 2 2 (1.36)

˙ Z +T Z +T Π=Πe −φ +i R d4x 1 (∂ φ)−V (φ) i R d4xL(φ,∂ φ) −→ ∝ Dφe −T [ 2 µ ] = Dφe −T µ (1.37)

Some comments are in order: • In going from (1.36) to (1.37) we have performed a Gaussian integral over the shift “momentum” variable Π.e This integral leads to an overall constant (φ-independent) factor that we can “re-absorb” in the deﬁnition of R Dφ. • The integral extends over all the the ﬁeld conﬁgurations connecting the two boundary

conditions at tI = −T and tF = +T , namely φF (~x, +T ) and φI (~x, −T ). The result in manifestly Lorentz invariant except for these boundary conditions. • From now on we adopt (1.37) as the deﬁnition of the PI formulation of a Theory. This implies we will characterize the diﬀerent ﬁeld in terms of the Lagrangian of the system rather than the Hamiltonian.

6 1.4.1 PI formulation of the two-point

A further step toward a more convenient formulation of QFT is obtained by getting rid of the boundary conditions. What we are really interested in, is the set of time-ordered correlation functions among ﬁelds at diﬀerent space-time points, evaluated on the of the system (also know as Green’s functions). Indeed, as we know from QFT-1, starting from these correlation functions the LSZ reduction formulae allow us to extract the probability amplitudes for physical processes, i.e. the elements of the S . The simplest of such correlations is the two-point function ˆ ˆ hΩ|T {φH (x1)φH (x2)}|Ωi, (1.38) ˆ where |Ωi denotes the ground state (of the interacting theory) and φH the operators associated to the ﬁelds in the Heisenberg’s picture. In order to ﬁnd a PI formulation of (1.38), consider the following functional integral, with generic boundary conditions at t = ±T Z i R +T d4xL(φ) I12 = Dφ(x)φ(x1)φ(x2)e −T . (1.39)

The main functional integral R Dφ(x) can be broken up as Z Z Z Dφ(x) ≡ Dφ1(~x)Dφ2(~x) Dφ(x) , (1.40)

0 φ(x1, ~x) = φ1(~x) 0 φ(x2, ~x) = φ2(~x)

0 0 namely introducing two speciﬁc boundary conditions at times x1 and x2 (in addition to those at the endpoints −T and T ), but integrating separately over the corresponding ﬁxed-time ﬁeld 0 0 conﬁgurations φ1(~x) and φ2(~x). Consider ﬁrst the case T > x2 > x1 > −T . Under this time ordering, the exponential term in (1.39) can be decomposed as

0 0 x T x i R 2 d4xL(φ) R T 4 i R d4xL(φ) i R 1 d4xL(φ) 0 i 0 d xL(φ) e −T = e −T × e x1 × e x2 . (1.41)

After the decomposition of the main integral, the extra factors φ(x1) and φ(x2) in (1.39) become φ1(~x1) and φ2(~x2), and can be taken outside the inner integral. We thus obtain Z I12(x2 > x1) = Dφ1(~x)Dφ2(~x)φ1(~x1)φ2(~x2) (1.42)

−iH(T −x0) −iH(x0−x0) −iH(x0+T ) × hφF |e 2 |φ2i hφ2|e 2 1 |φ1i hφ1|e 1 |φI i . | {z } | {z } | {z } i R T d4xL(φ) x0 x0 R x0 i R 2 d4xL(φ) R i R 1 d4xL(φ) Dφe 2 x0 Dφe −T R Dφe 1

The factors φi(~xi) can be transformed into insertions of Schr¨odingeroperators, by using Z ˆ 0 φS(xi , ~xi) ≡ Dφi(~x)φi(~xi) |φii hφi| , (1.43) leading to

0 0 0 0 −iH(T −x2) ˆ 0 −iH(x2−x1) ˆ 0 −iH(x1+T ) I12(x2 > x1) = hφF | e φS(x2, ~x2)e φS(x1, ~x1)e |φI i (1.44) −iHT ˆ ˆ −iHT = hφF | e φH (x2)φH (x1)e |φI i . (1.45)

7 Putting together the case x2 > x1 and x1 > x2 we arrive to

Z R +T 4 i −T d xL −iHT ˆ ˆ −iHT I12 = Dφ(x)φ(x1)φ(x2)e = hφF | e T {φH (x2)φH (x1)}e |φI i . (1.46)

We thus obtained the desired time-ordered product, but evaluated on the generic endpoint ﬁeld conﬁgurations, rather than on the ground state of the system.2

Consider now the ratio I12/I0, where Z i R +T d4xL −iH(2T ) I0 = Dφ(x)e −T = hφF | e |φI i , (1.47) and consider further the limit T → +∞(1 − i), where  is an arbitrarily small real parameter.

In order to evaluate this limit, we can project the initial state |φI i into the complete sum of the eigenstates |ni of the Hamiltonian. For T → +∞(1 − i), the action of e−iHT selects only the ground state component which is the least suppressed(the additional terms being exponentially suppressed):

−iHT X −iHT −iE0T e |φI i = e |ni hn |φI i −→ e |Ωi hΩ |φI i . (1.48) T →+∞(1−i) n

A similar procedure can be applied to the the ﬁnal state |φF i. As a result, we obtain

−2iE0T I12 = e hφF | ΩihΩ |φI i hΩ| T {φ(x1)φ(x2)} |Ωi ∝ hΩ| T {φ(x1)φ(x2)} |Ωi . (1.49)

The proportionality constant on the r.h.s. of (1.49) drops out in the ratio I12/I0, leading to

R Dφ(x)φ(x )φ(x )eiS hΩ| T {φ(x )φ(x )} |Ωi = 1 2 (1.50) 1 2 R Dφ(x)eiS where Z S = d4L(x) .

N.B.: The ratio in (1.50) is the reason why we can often neglect the overall normalization of the functional integral.

1.5 The Generating Functional

We now need a method able to generate in a systematic way all the relevant correlation func- tions, namely all the correlation functions involving an arbitrary number of ﬁelds at diﬀerent space-time points. As we shall show, all these correlation functions are encoded in one main transition amplitude: the groundstate-to-groundstate transition amplitude in presence of ar- bitrary driving J(x), or the Generating Functional. The basic physical picture behind the Generating Functional is the following: by means of an arbitrary source J(x) we create an arbitrary excitation of intermediate states that, in turn, give us access to all possible correlation functions.

Technical tool: functional derivatives. Similarly to the Action, S[φ], that is a functional of φ, the Generating Functional, W[J], is a functional of J. In order to extract the correlation

2 ˆ From now on we omit the speciﬁc subscripts to denote the : φH (x) → φ(x).

8 functions from W [J] we need to introduce the technical tool of functional derivatives. The main rules of functional derivatives are listed below δ δ Z J(y) = δ4(x − y) , d4yJ(y)f(y) = f(x) , (1.51) δJ(x) δJ(x) δ F [J(y)] = F [J → δ4(x − y), separately for each J] . (1.52) δJ(x) Similarly to ordinary derivatives, it follows that

δ R 4 R 4 ei d yJ(y)φ(y) = iφ(x)ei d yJ(y)φ(y) , (1.53) δJ(x) δ Z d4y∂ J(y)φ(y) P.I.= −∂ φ(x) . (1.54) δJ(x) µ µ

We are now ready to go back to W [J]. The Generating Functional for a scalar ﬁeld theory is deﬁned by Z i R d4x[L(φ)+J(x)φ(x)] W [J] = hΩ|ΩiJ = Dφe , (1.55) where J(x) is an arbitrary function. Starting from this deﬁnition, it is easy to verify that     Z δ δ i R d4x[L(φ)+J(x)φ(x)] −i −i W [J] = Dφ(x)φ(x1)φ(x2)e . (1.56) δJ(x1) δJ(x2) As a result, the two-point correlation function can be written as     1 δ δ hΩ|T {φ(x1)φ(x2)}|Ωi = −i −i W [J] . (1.57) W [0] δJ(x1) δJ(x2) J=0 This result can be easily generalised to arbitrary correlation functions. We can then express W [J] be means of a (functional) Taylor expansion as

∞ X in Z W [J] = W [0] × dx ··· dx J(x ) ··· J(x )G(n)(x , ··· , x ) , (1.58) n! 1 n 1 n 1 n n=0

(n) where the coeﬃcient of the expansion, namely the functions G (x1, ··· , xn), are the n-point Green’s functions n   (n) 1 Y δ G (x1, ··· , xn) = −i W [J] W [0] δJ(x ) i=1 i J=0

= hΩ|T {φ(x1)φ(x2) . . . φ(xn)}|Ωi . (1.59)

1.5.1 Generating functional for a free scalar ﬁeld

For a free scalar ﬁeld, the argument at the exponent of the generating functional is Z 1 1  iS = i d4x (∂ φ)(∂µφ) − m2φ2 + Jφ J 2 µ 2 Z 1 1  −→i i d4x (∂ φ)(∂µφ) − (m2 − iφ2) + Jφ 2 µ 2 Z 1  = i d4x φ −∂2 − m2 + i φ + Jφ (1.60) 2

9 where the i term has been introduced in order to make the integral convergent, and the last identity has been obtained through integration by parts. In (1.60) we have obtained a quadratic form in φ that can be integrated explicitly (↔ Gaussian integral). To this purpose, we need to ﬁnd the “inverse” of the operator (−∂2−m2+i), or the function D(x − y) deﬁned by

(−∂2 − m2 + i)D(x − y) = iδ4(x − y) . (1.61)

As known from QFT-1, this function is

Z ddk i D(x − y) = e−ik·(x−y) . (1.62) (2π)4 k2 − m2 + i

In a formal sense, that acquire a well-deﬁned meaning inside functional integrals by means of (1.61), we can write −iD(x − y) “ = ” (−∂2 − m2 + i)−1 . (1.63) This allows us to complete the square in (1.60) via the shift Z φ(x) → φ0(x) = φ − i d4yD(x − y)J(y) “ = ” φ(x) + (−∂2 − m2 + i)−1 × J(x) , (1.64) obtaining

Z 1  1 Z iS = i d4x φ0(−∂2 − m2 + i)φ0 − d4xd4yJ(x)D(x − y)J(y) . (1.65) J 2 2

The change of variable doesn’t aﬀect the measure, since the Jacobian of the transformation is one. The integral on Dφ0 is not well deﬁned, but this only leads to an irrelevant (J-independent) normalization factor. We thus obtain the following simple expression for W [J] in the free ﬁeld case3 Z 4 4 − 1 2 d xd yJ(x)D(x − y)J(y) 1 − JxDxyJy W0[J] = W0[0]e = W0[0]e 2 , (1.66) where the second identity is nothing but a convenient compact notation.

As far as the normalization W0[0] is concerned, its formal expression is given by Z Z i d4xφ0[−∂2 − m2 + i]φ0 0 2 W0[0] = Dφ e . (1.67)

By analogy with the ﬁnite-dimensional Gaussian integrals,

Z 1 1 Y − xiBij xj − dxk e 2 ∝ (det B) 2 , (1.68) k we can express W0[0] as 2 2 − 1 W0[0] ∝ [det(−∂ − m + i)] 2 . (1.69) This result would be well deﬁned if we were able to prove that the determinant is a positive quantity and it converges (the number of eigenvalues could be inﬁnite). This can be achieved

3 The subscript “0” in W0[J] denotes the free-ﬁeld case.

10 assuming appropriate boundary conditions on the system. However, as anticipated, for most of the applications we are interested in, the explicit expression of W [0] is irrelevant.

Using the explicit expression of W0[J] we can now compute some correlation functions (in the free-ﬁeld case). For the two-point function we ﬁnd

1 δ δ − JxDxyJy hΩ|T φ(x1)φ(x2)|Ωi = − e 2 δJ1 δJ2 J=0

  1 δ 1 1 − JxDxyJy = − − D2yJy − D2xJx e 2 δJ1 2 2 J=0 1 1 = D + D = D = D(x − x ) , (1.70) 2 12 2 21 12 1 2 i.e. we recover the expression of the free scalar . For the four-point function we obtain

1 δ δ δ − JxDxyJy hΩ|T φ1φ2φ3φ4|Ωi = − [−JxDx4]e 2 δJ1 δJ2 δJ3 J=0

1 δ δ − JxDxyJy = − [D34 + JxDx4JyDy3]e 2 δJ1 δJ2 J=0

1 δ − JxDxyJy = − [D34JxDx2 + D24JyDy3 + JxDx4D23]e 2 δJ1 J=0 = D34D12 + D24D13 + D14D23, (1.71) namely the sum of three terms, each containing two disconnected contributions.

Besides W [J], we can introduce also the functional Z[J], deﬁned by

W [J] = eiZ[J] , (1.72) that in the free-ﬁeld case assumes the form 1 iZ [J] = − J D J + const. (1.73) 0 2 x xy y

The Taylor decomposition of Z[J] can be written in terms of new set of Green functions:

∞ n Z X i (n) iZ[J] = dx ··· dx J(x ) ··· J(x )G (x , ··· , x ) . (1.74) n! 1 n 1 n C 1 n n=0

In the free-ﬁeld case we ﬁnd

(2) (n>2) Gc = D(x − y) ,Gc = 0 . (1.75)

As we will show in the next paragraph, by means of the explicit calculation in the case of the 4 interacting φ theory, the GC functions are nothing but the the connected Green functions of the theory.

11 1.5.2 Interacting scalar ﬁeld theory

We will consider now an interacting Lagrangian of the form

L = L0(φ) − V (φ) . (1.76) We have just seen that

δ R 4 R 4 −i ei d xJ(x)φ(x) = φ(y)ei d xJ(x)φ(x) , (1.77) δJ(y) then for a generic (analytic) functional F (φ) we have   R 4 δ R 4 F (φ)ei d xJφ = F −i ei d xJφ (1.78) δJ and thus

Z R 4 δ R 4 R 4 δ 1 −i d xV (−i ) i d x[L0+Jφ] −i d xV (−i ) − JxDxyJy W [J] = Dφe δJx e = e δJx e 2 W0[0] . (1.79) | {z } W0[J] We now want to analyse the case in which V can be trated as a small perturbation. In this case we can perform a perturbative expansion of e−iV around 1. We can thus write W [J] as

n −1 h −i R d4xV (−i δ ) i o W [J] = W0[J] 1 + W0[J] e δJx − 1 W0[J] (1.80) and recalling that W [J] = eiZ[J] we ﬁnd

n h R 4 δ i o −iZ0[J] −i d V (−i ) iZ0[J] iZ[J] = iZ0[J] + ln 1 + e e δJx − 1 e + const., (1.81) where we have deﬁned

1 1 iZ0[J] − JxDxyJy W0[J] = e = e 2 . (1.82) W0[0] Let’s now apply these generic formulae to the speciﬁc example, namely the case of the potential λ V = φ4 . (1.83) 4! For small enough λ we get Z 4 −i R d4xV (−i δ ) λ 4 4 δ 2 e δJx − 1 ≈ −i d z(−i) + O(λ ) (1.84) 4! δJ(z)4 and the logarithm in (1.81) becomes λ Z δ4 ln {1 + ···} = −i e−iZ0[J] d4z eiZ0[J] + O(λ2) (1.85) 4! δJ(z)4 where Z 4 Z 3 4 δ − 1 J D J 4 δ  − 1 J D J  d z e 2 x xy y = d z −D J e 2 x xy y δJ(z)4 δJ(z)3 xz x Z 2 4 δ h − 1 J D J i = d z (−D + D J D J )e 2 x xy y δJ(z)2 zz xz x yz y Z 4 δ h − 1 J D J i = d z (D D J + 2D D J − D J D J D J )e 2 x xy y δJ(z) zz yz y zz yz y xz x yz y αz α

Z 1 4  2  − 2 JxDxyJy = d z 3Dzz − 6DzzDyzDxzJyJx + DxzDyzDαzDbzJxJyJαJb e (1.86)

12 As anticipated, we have only connected Green’s functions, that in the last line we have con- veniently expressed by means of Feynman diagrams. The ﬁnal result for the functional F [J] is then 1 λ iZ[J] = C − J D J − i [3D2 − 6D D D J J 2 x xy y 4! zz zz yz xz y x 2 + DxzDyzDαzDbzJxJyJαJb] + O(λ ) . (1.87) Using this result in Eq. (1.82) we can get back to W [J] obtaining

Since bubbles always cancel in the correlation functions, for simplicity, it is absorbed into W0[0], which becomes W [0]. Computing G(x1, x2, x3, x4) we get the same results as in the free case plus the insertion of the interacting terms at O(λ).

1.6 Path integral formulation of an Abelian Field The- ory

We now proceed extending the path-integral formulation of QFT to theories with diﬀerent type of ﬁelds (and correspondingly diﬀerent free actions). In this section we consider the case of Abelian gauge ﬁelds, namely the massless vector ﬁelds (Aµ) appearing in the Maxwell Lagrangian: 1 L = − F F µν ,F = ∂ A − ∂ A . (1.88) 4 µν µν µ ν ν µ By construction, the generating functional for this ﬁeld is

Z R µ i{S[A]+ JµA } W [Jµ] = DA e (1.89) with Z  1  S[A] = d4x − (∂ A − ∂ A )(∂µAν − ∂νAµ) 4 µ ν ν µ 1 Z = − d4x [∂ A ∂µAν − ∂ A ∂µAν] 2 µ ν ν µ 1 Z = d4xA [∂2gµν − ∂µ∂ν]A , (1.90) 2 µ ν

13 where the last relation follows from integration by parts. To proceed as in the case of the scalar theory we need to identify the free-ﬁeld propagator, or the function Dµν(x − y) solving the following equation

2 νρ ρ 4 (∂ gµν − ∂µ∂ν)D (x − y) = igµδ (x − y) (1.91) 2 νρ ρ (−k gµν + kµkν + i)De (k) = igµ (in momentum space) . (1.92)

2 However, it turns out that we cannot proceed since (−k gµν + kµkν) is a singular (or not invertible) operator, given that

2 ν (−k gµν + kµkν)k = 0 . (1.93)

This fact is a consequence of the gauge invariance of the action:

1 S[A] = S[A0] for A0 = A + ∂ α(x) , (1.94) µ µ e µ that, in turn, implies that in the path-integral (1.89) we are redundantly integrating over a con- tinuous inﬁnity of physically equivalent ﬁeld conﬁgurations. To ﬁx the problem, we would need to isolate the interesting part of the functional integral, counting each physical conﬁguration only once. In order to select a speciﬁc gauge we introduce a gauge-ﬁxing condition, such as

µ ∂µA = ω(x) . (1.95)

µ Deﬁning the function G(A) = ∂µA − ω(x), the above gauge-ﬁxing condition is equivalent to impose G(A) = 0. Let’s now consider the gauge-transformed ﬁeld

1 Aα(x) = A (x) + ∂ α(x) , (1.96) µ µ e µ 1 G(Aα) = ∂ Aµ + ∂2α − ω(x) . (1.97) µ e

Generalizing the following identity

Z !   Y ∂gi 1 = da δ(n)(g(a , ··· , a )) det (1.98) i 1 n ∂a i j to functional integrals we can write

select gauge conﬁguration Z Z  α  α z }| α { δG(A ) α I = DG(A ) δ[ G(A ) ] = Dα(x) det δ[G(A )] . (1.99) | {z } δα gauge ﬁxing condition | {z } 1 2 det( e ∂ )

The last expression is the key relation that allows us to split in the functional integral (1.89) the redundant part from the non-trivial integration over physical inequivalent ﬁeld conﬁgurations.

14 Indeed inserting the above identity in the generating functional we can write  α  Z Z δG(A ) R 4 W [J ] = DA Dα δ[G(Aα)] det eiS[A]ei d x J·A µ δα | {z } C

Z Z α R 4 α = C DAα Dα δ[G(Aα)]eiS[A ]ei d x J·A

α Z Z R 4 A =→A C Dα DA δ[G(A)]eiS[A]ei d x J·A | {z } C0

Z R 4 = C0 DA δ[G(A)]eiS[A]ei d x J·A (1.100)

The overall factor C0 is badly divergent, but this is not a problem since we can reabsorb this divergence in the deﬁnition of the “measure” in the DA space and, as we have already seen, the overall factor in functional integral drops out in physical correlation functions. Summarizing, we can obtain Eq. (1.100) starting from Eq. (1.89) thanks to the following key observations: • S[A] = S[Aα] R 4 µ R 4 µ • d xJµA = d xJµAα if Jµ is a conserved current •DA = DAα [the measure is unchanged under gauge transormations]  δG(Aα)  1 2 • det δα = det e ∂ is independent of A. As we will discuss later on in this course, the last point does not hold for non-Abelian gauge theories. Let’s go back to the result in Eq. (1.100). We would like to further transform it in order to obtain a simple Gaussian integral (as in the case of the scalar ﬁeld). To this purpose, we note that the choice of ω(x) is arbitrary. We can thus consider a properly weighted combination of diﬀerent ω(x) function. In particular, we can consider the following combination

Z 2 Z 00 −i R d4x ω µ iS[A] i R d4x J·A W [Jµ] = C (ξ) Dω e 2ξ DA δ(∂µA − ω)e e | {z } Gaussian integral centered in ω = 0, with variance ξ

Z (∂ Aµ)2 00 iS[A] −i R d4x µ i R d4x J(x)A(x) = C (ξ) DA e e 2ξ e    Z Z 1 1 R 4 = C00(ξ) DA exp i d4x − F F µν − (∂A)2 ei d x J(x)A(x) 4 µν 2ξ | {z } ”eﬀective” term in the Lagrangian     Z Z 1 R 4 = C00(ξ) DA exp i d4x Aµ ∂2g − ∂ ∂ + ∂ ∂ Aν ei d x J(x)A(x) . (1.101) µν µ ν ξ µ ν | {z } this operator can now be inverted This way we have ﬁnally reached an invertible quadratic form in the exponent. This allows us to obtain a well-behaved propagator, Dνρ(x − y), deﬁned by4,   1 (∂2 − i)g − ∂ ∂ 1 − Dνρ(x − y) = ig ρδ4(x − y) (1.102) µν µ ν ξ µ

4The i term is added to make the integral convergent.

15 or, in the momentum space,   1 (−k2 − i)g + k k 1 − Dνρ(k) = ig ρ . (1.103) µν µ ν ξ µ Using the general ansatz Dνρ(k) = Agνρ + Bkνkρ , (1.104) it is easy to ﬁnd that −i  kνkρ  Dνρ(k) = gνρ − (1 − ξ) . (1.105) k2 + i k2

For obvious reasons the parameter ξ is denoted “gauge-ﬁxing parameter”. Two notable choice for it, that allows to get particularly simple , are

ξ = 0 [Landau gauge] and ξ = 1 [Feynman gauge] . (1.106)

The ξ-dependence should drop in any correlation function involving gauge-invariant operators, O(x), namely

R iSξ[A] DAO(x1) ··· O(xn)e h0| T {O(x1) ··· O(xn)} |0i = , (1.107) R DA eiSξ[A]

Z  1 1  where S [A] = d4x − F F µν − . ξ 4 µν 2ξ

1.7 Quantization of spinor ﬁeld

1.7.1 Grassmann numbers

In order to generalize the path-integral method to spinor ﬁelds we need to introduce a new type of variables: the Grassmann numbers. The deﬁning property of two Grassmann numbers θ and η is

{θ, η} = 0, θη = −ηθ =⇒ θ2 = η2 = 0 (1.108)

In this section we will use latin letters for ordinary numbers and greek letters for Grassmann numbers. Any function f(θ) can be expanded to the ﬁrst order as ( a + βθ = a − θβ f(θ) = (1.109) a + bθ = a + θb with β a Grassmann variable and b an ordinary number. We also deﬁne the derivative with respect to a Grassmann variable by the relation  d  , θ = 1 , (1.110) dθ such that ( d d (βθ) = − d (θβ) = −β f(θ) = dθ dθ . (1.111) dθ d dθ (bθ) = b

16 The integral over a Grassmann variable is deﬁned by Z Z dθ1 = 0 , dθθ = 1 , (1.112) such that Z Z Z  dθ(βθ) = − dθ(θβ) = −β dθf(θ) = Z (1.113)  dθ(bθ) = b This implies Z Z dθf(θ) = dθf(θ + η) , (1.114) namely the invariance of the integral under a shift of variables. It is also useful to deﬁne complex Grassmann variables, θ + iθ θ − iθ θ = 1 √ 2 , θ∗ = 1 √ 2 , (1.115) 2 2 with θ1 and θ2 ordinary (real) Grassmann variables. It is easy to check that 1 1 θθ∗ = (θ + iθ )(θ − iθ ) = (−iθ θ + iθ θ ) = −iθ θ (1.116) 2 1 2 1 2 2 1 2 2 1 1 2 1 1 θ∗θ = (θ − iθ )(θ + iθ ) = (iθ θ − iθ θ ) = +iθ θ (1.117) 2 1 2 1 2 2 1 2 2 1 1 2 that implies that θ and θ∗ can be treated as independent (Grassmann) variables: {θ, θ∗} = 0. Finally we deﬁne (θη)∗ = η∗θ∗ = −θ∗η∗ (1.118)

We are now ready to evaluate Gaussian-type integrals over complex Grassmann variables. The simplest example is Z Z Z Z Z dθ∗dθe−bθ∗θ = dθ∗dθ[1 − bθ∗θ] = −b dθ∗dθθ∗θ = b dθ∗θ∗ dθdθ = b . (1.119)

This has to be compared with the analog integeral over ordinary complex variables,

Z ∗ 2π dz∗dze−bz z = . (1.120) b Similarly

Z ∗ Z Z 1 dθ∗dθ(θθ∗)e−bθ θ = dθ∗dθ(θθ∗)[1 − bθ∗θ] = dθ∗dθ(θθ∗) = 1 = b , (1.121) b can be compared with

Z ∗ 2π 2π 1 dz∗dz(z∗z)e−bz z = = . (1.122) b2 b b

Considering a generic Hermitian matrix Bij with non-vanishing eigenvalues bi, and generalizing the above results, one obtains Z Z Y ∗ Y ∗ ∗ −θi Bij θj ∗ −θi biθi dθi dθie = dθi dθie = det(B) . (1.123) i i Z Y ∗ ∗ ∗ −θi Bij θj −1 dθi dθiθkθl e = det(B)(B )kl . (1.124) i

17 1.7.2 Path integral formulation for the Dirac ﬁeld

We deﬁne the Grassmann ﬁeld ψ(x) as X ψ(x) = ψiφi(x) , (1.125) where ψi are Grassmann numbers and φi(x) is an ordinary function of x. To describe the Dirac ﬁeld, we identify the φi with the components of a (4-component) . We can then to introduce the generating functional Z 4   Z i d x ψ¯(i∂/ − m)ψ +ηψ ¯ + ψη¯ W [¯η, η] = Dψ¯Dψe , (1.126) whereη ¯ and η are external Grassmannian sources. Proceeding as in the case of scalar ﬁelds, we extract the free-ﬁeld propagator by solving

4 (i∂/ − m + i)DF (x − y) = iδ (x − y) , (1.127) and obtaining Z i D (x − y) = d4k e−ik(x−y) . (1.128) F k/ − m + i Then by a shift of variables in the functional integral we can re-write the generating functional as Z 4 4 − d xd yη¯(x)D(x − y)η(y) W [¯η, η] = W0e . (1.129) It is straightforward to check that, as expected,     ¯ 1 −iδ +iδ h0|T {ψ(x1)ψ(x2)}|0i = W [η, η¯] W0 δη(x) δη¯(x) η=¯η=0 Z ¯ ¯ iS DψDψψ(x1)ψ(x2)e = Z . (1.130) Dψ¯DψeiS

Having introduced a path-integral formulation for Dirac and Abeilan gauge ﬁelds, it is easy to derive the expression of the Generating functional of QED. By construction the latter is Z   4 1 µν 1 µ 2 ¯ Z i d x − FµνF − (∂µA ) + ψ(iD/ − m)ψ ¯ 4 2ξ W [¯η, η, Jµ] = N DψDψDAµ e Z 4  ¯ µ i d x ηψ¯ + ψη + AµJ × e (1.131) where Dµ = ∂µ + ieAµ. Treating the term in the action, namely

¯ µ ∆L = −eψ(x)γ ψ(x)Aµ(x) (1.132) as a perturbation, and proceeding as in Sect. 1.5.2, one recovers the Feynman rules of QED derived using the formulation of QFT discussed in QFT-1 ().

18 1.7.3 Explicit calculation on functional determinants

As outlined above and shown explicitly in Sect. 1.5.2, the path-integral formulation of QFT allows us to recover known results obtained by means of canonical quantization. However, the path-integral formulation is more general and, in some cases, allow to derive in a simple and transparent way results that are not easy to obtain by means of the canonical-quantization formalism. Two examples of this statement are show in this and in the following section. As a ﬁrst example, we present here the explicit calculation of the functional integral for the Dirac ﬁeld in presence of an external vector ﬁeld (that could possibly by identiﬁed with the electromagnetic ﬁeld in the limit where we treat the latter as a non-dynamical background ﬁeld). The generating functional we are interested in is Z 4  µ µ  Z i d x ψ¯(i∂/ − m)ψ − eJ ψγ¯ ψ ¯ W [Jµ] = DψDψ e   i  ∝ det(iD/ − m) = det (i∂/ − m) 1 − (−ieJ/) J i∂/ − m  i  = W × det 1 − (−ieJ/) . (1.133) 0 i∂/ − m To evaluate explicitly this we ﬁrst note that, for a generic matrix A with eigenvalues ai, we can write P Y Y log(ai) log(ai) Tr[log A] det(A) = ai = e = e i = e . (1.134) i i

Z[Jµ] Then deﬁning W [Jµ] = e , and applying the above decomposition to the functional deter- minant in Eq .(1.133) we obtain the following formal expression for Z[Jµ]   i  Z[J ] = Z × Tr log 1 − (−ieJ/) µ 0 i∂/ − m ∞ n X  1   i   = Z × − Tr (−ieJ/) (1.135) 0 n i∂/ − m n=1 | {z } Cn In order to interpret the meaning of this expression it is convenient to consider the limit of discretized and ﬁnite space time. In this case the various (functional) operators we are considering can be identiﬁed as matrices leaving in a vector spaces labeled by the (ﬁnite) number of space-time points [e.g.: D(x − y) → Dxi yj = D(xi − xj)]. With this identiﬁcation is easy to realize that X X Z Tr = → dx (1.136)

xi,r r where r denotes the spinor indices. Recalling also that the inverse of the operator (i∂/x − m) is nothing but the propagator of the Dirac ﬁeld, we obtain the following expressions: Z 4 C1 = (−1)(−ie) d x tr[D(x − x)J/(x)] (1.137) 1 Z C = − (−ie)2 d4xd4y tr[D(x − y)J/(y)D(y − x)J/(x)] (1.138) 2 2 1 Z C = − (−ie)n d4x ··· d4x tr[D(x − x )J/(x ) ··· D(x − x )J/(x )] (1.139) n n 1 n 1 2 2 n 1 1

19 where now the trace refers only to spinor indices. Diagrammatically, the above result corre- sponds to

As expected, the functional Z[J] contains only connected Green functions. Exponentiating Z[J] we obtain the complete functional integral with connected and disconnected diagram. The calculation of this functional determinant was particularly simple due to non-interacting nature of the external ﬁeld. Still, it served to the purpose of illustrating the simplicity and the powerfulness of this method.

1.8 Symmetries in Path Integral

Another aspect where the path integral formulation of QFT is particularly useful is in the exploitation of the symmetries of the theory. Let’s start considering an example, namely the global Abelian symmetry of a complex scalar ﬁeld theory. The Lagrangian of the system is

∗ µ 2 ∗ L = ∂µφ ∂ φ − m φ φ (1.140) and is invariant under the global U(1) transformation

φ(x) → eiαφ(x) ≈ φ(x) + iαφ(x) . (1.141)

At the classical level, thanks to Noether’s theorem, we know that this invariance implies the µ existence of a conserved current (∂µJ = 0). In this speciﬁc example, the conserved current is

µ ∂L ∂L ∗ µ ∗ µ ∗ J = iφ + ∗ (−iφ ) = i(∂ φ φ − ∂ φφ ) . (1.142) ∂(∂µφ) ∂(∂µφ )

What happens at the quantum level? The derivation of Noether’s theorem relies on the as- sumption that the system obeys the classical equations of motions. This fact no longer holds at the quantum level: in the functional integral we consider also trajectories that do not minimize µ the action. It is therefore not obvious what happens to the relation ∂µJ = 0 when we consider a quantized system. In order to investigate what happens at the quantum level, let’s consider the following two-point correlation function, Z ∗ ∗ iS[φ] h0|T {φ(x1)φ (x2)}|0i = N Dφφ(x1)φ (x2)e . (1.143)

20 Let’s then apply a change of variables (in the ﬁelds) corresponding to a local U(1) transforma- tion, namely

φ(x) → φ0(x) = eiα(x)φ(x) ≈ φ(x) + iα(x)φ(x) (1.144)

The measure Dφ is unchanged, since the change of variables is a unitary transformation, we therefore ﬁnd Z Z Z ∗ iS[φ] 0 0 ∗0 iS[φ0] 0 ∗0 iS[φ0] Dφφ(x1)φ (x2)e = Dφ φ (x1)φ (x2)e = Dφφ (x1)φ (x2)e . (1.145)

Considering the ﬁrst and the last term in this identity, and expanding the latter to ﬁrst order in α we obtain Z  Z iδS  0 = Dφ iα(x )φ(x )φ∗(x ) − iα(x )φ(x )φ∗(x ) + φ(x )φ(x ) d4x α(x) eiS[φ] (1.146) 1 1 2 2 1 2 1 2 δα where δS δ Z = d4yL(φ + iαφ) δα δα(x) Z   δ 4 ∂L ∂L ∗ ∂L ∂L ∗ = d y L(φ) + δφ + ∗ δφ + i(∂µα)φ − ∗ i(∂µα)φ δα(x) ∂φ ∂φ ∂(∂µφ) ∂(∂µφ ) | {z } =iα(y)φφ∗−iα(y)φφ∗=0 Z δ 4 µ ∗ µ = d y∂µα(y) i(∂ φ φ − ∂ φφ∗) . (1.147) δα(x) | {z } Jµ Given the action is invariant under global transformations, the variation of the action under the local transformation is necessarily proportional to ∂µα, and we found that the proportionality factor is nothing but the current J µ (i.e. the current that is conserved at the classical level). Note that, contrary to Noether’s theorem, to obtain this result we have not used the fact that the ﬁeld satisfy the equations of motion. Then integrating by parts we ﬁnally obtain δS = −∂ J µ(x) . (1.148) δα µ Using this result in Eq. (1.146) we get Z  Z  ∗ ∗ 4 µ iS[φ] 0 = Dφ iα(x1)φ(x1)φ (x2) − iα(x2)φ(x1)φ (x2) − iφ(x1)φ(x2) d xα(x)∂µJ (x) e Z Z 4 ∗ ∗ = d xα(x) Dφ[iδ(x − x1)φ(x1)φ (x2) − iδ(x − x2)φ(x1)φ (x2)

µ iS[φ] −iφ(x1)φ(x2)∂µJ (x)]e . (1.149)

Since the above result must be valid for any α(x), we ﬁnally obtain the following identity

µ ∗ h0|T {∂µJ (x)φ(x1)φ(x2)}|0i = h0|T {φ(x1)δ(x − x1)φ (x2)}|0i ∗ − h0|T {φ(x1)δ(x − x2)φ (x2)}|0i . (1.150)

This result is a representative example of the modiﬁcation of the concept of conserved current in QFT: the current is conserved up to “contact terms”. In the canonical-quantization for- malism the later appear due to the non-trivial commutation relations among the ﬁelds. In the

21 path-integral formulation these contact terms simply arise by the deﬁnition of the correlation functions, as illustrated above. It should be clear that the above result can be generalized to generic n-point functions, and also to more complex sets of symmetry transformations. The corresponding relations are known as Ward-Takahashi identities (for continuos global transformations) or, more generally, Schwinger-Dyson equations. The general expression of the latter, obtained proceeding as in the example above but for: i) starting from a generic n-point correlation function; ii) considering a generic (unitary) change of variables in the ﬁelds is n X h0| T {∆(x)φ(x1) ··· φ(xn)} |0i = h0| T {∆(x)φ(x1) ··· (−iδ(x − xi)) ··· φ(xn)} |0i (1.151) i=1 where δS[φ] ∂L  ∂L  ∆(x) = = − ∂µ . (1.152) δφ(x) ∂φ ∂(∂µφ) This result can be considered the quantum version of Euler-Lagrangian equations.

1.8.1 Ward-Takahashi identity in QED

A notable application of the Ward-Takahashi discussed above is the connection between 3- and

2-point functions in QED, that leads to the relation Z1 = Z2 derived in at O(e2) in QFT-1. The relevant (global) symmetry is ψ(x) → ψ0(x) = eiαψ(x) , (1.153) µ ¯ µ leading to the conserved current Je.m. = ψγ ψ. Proceeding as in the previous section (taking into account the various signs due to the Dirac algebra), we get ∂ i h0|T {J µ(x)ψ¯(x )ψ(x )}|0i = −i h0|T {ψ(x )ψ¯(x )}|0i δ(x − x ) ∂xµ 1 2 1 2 1 ¯ + i h0|T {ψ(x1)ψ(x2)}|0i δ(x − x2) . (1.154) If we then perform a Fourier transformation, via Z Z Z 4 −iqx 4 ip1x1 4 −ip2x2 d xe d x1e d x2e , (1.155) we obtain the following identity among correlation function in momentum space

µ −iqµM (q; p1, p2) = −iM(p1 − q, p2) + iM(p1, q + p2) . (1.156) or, using QFT-1 notation (see Peskin and Schroeder),

µ −iqµΓ (p1, p2) = iΣ(p2) − iΣ(q + p2) , (1.157) Here Γµ(p0, p) is the vertex function describing the interaction of the with the ﬁeld and Σ(p) is the one-particle irreducible correction to the electron propagator. Expanding 2 2 these correlation functions for p1,2 → m and q → 0 −1 Σ(p) = Σ(p)|p2=m2 + Z2 × (p/ − m) + ... µ 0 −1 µ Γ (p , p) = Z1 γ + ... (1.158) we ﬁnally get the relation Z1 = Z2. Note that, while this relation was derived by means of an explicit calculation at O(e2) in QFT-1, we have now derived it in a general way that is valid to all orders in perturbation theory.

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