aRudolf Peierls Centre for Theoretical Physics University of Oxford OX1 3PN Oxford, United Kingdom
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1 Path Integrals in Quantum Field Theory
In the solid-state physics part of the lecture you have seen how to formulate quantum me- chanics (QM) in terms of path integrals. This has led to an intuitive picture of the transition between classical and quantum physics. In this lecture notes I will show how to apply path integrals to the quantization of ﬁeld theories. We start the discussion by recalling the most important feature of path integrals in QM.
1.1 QM Flashback It has already been explained how QM can be formulated in terms of path integrals. One important ﬁnding was that the matrix element between two position eigenstates is given by
Z Z t x, t x0, t0 = x e−iH(t−t0) x0 x exp i dt00 (x, x˙) . (1.1) h | i h | | i ∝ D t0 L
iHt 0 0 iHt0 0 Notice that here x, t = e x S and x , t = e x S are Heisenberg picture states. We also learned| i in the lectures| i on| interactingi | quantumi ﬁelds that the central objects to compute in quantum ﬁeld theory (QFT) are vacuum expectation values (VEVs) of time- ordered ﬁeld-operator products such as the Feynman propagator
DF (x y) = 0 T φ(x)φ(y) 0 . (1.2) − h | | i In QM the analog of (1.2) is simply1
xf , tf T xˆ(t1)ˆx(t2) xi, ti . (1.3) h | | i 1In this part of the lecture, we will use hats to distinguish operators from their classical counterparts which appear in the path integral.
1 Focusing on the case t1 > t2 and inserting complete sets of states we can write the latter expression as
−iH(tf −t1) −iH(t1−t2) −iH(t2−ti) xf , tf xˆ(t1)ˆx(t2) xi, ti = xf e xˆS e xˆS e xi h | | i h i Z (1.4) −iH(tf −t1) −iH(t1−t2) −iH(t2−ti) = dx1dx2 xf e x1 x1 xˆS e x2 x2 xˆS e xi . h | ih | | ih | i Using now thatx ˆ x = x x , replacing all three expectation values by (1.1) and combining | i | i the three path integrals with the integrations over x1 and x2 into a single path integral the result (1.4) simpliﬁes further. One ﬁnds that
Z Z tf xf , tf T xˆ(t1)ˆx(t2) xi, ti x x(t1)x(t2) exp i dt (x, x˙) . (1.5) h | | i ∝ D ti L
For t2 > t1 the same result holds, because time ordering is automatic in the path-integral formulation. It should also be clear that results similar to (1.5) apply for a product of an arbitrary number of operatorsx ˆ. Furthermore, it can be shown that
lim xf , tf T (ˆx(t1) ... xˆ(tn)) xi, ti 0 T (ˆx(t1) ... xˆ(tn)) 0 . (1.6) ti,f →∓∞h | | i ∝ h | | i Therefore one arrives at Z iS[x] 0 T (ˆx(t1) ... xˆ(tn)) 0 x x(t1) . . . x(tn) e , (1.7) h | | i ∝ D with S[x] the action functional.
1.2 Basics of QFT Path Integrals In order to keep the following discussion as simple as possible we will focus on the real scalar ﬁeld φ. An extension to more complicated theories would however be straightforward. As we saw the Green’s functions of the form
(n) ˆ ˆ (x1, . . . , xn) = 0 T φ(x1) ... φ(xn) 0 , (1.8) G play an important role in QFT. In analogy to (1.7) these objects can be written as Z (n) iS[x] (x1, . . . , xn) = φ φ(t1) . . . φ(tn) e , (1.9) G N D where is a normalization constant. LikeN in QM, we introduce the generating functional Z Z 4 W [J] = φ exp i d x (φ, ∂µφ) + J(x)φ(x) , (1.10) N D L
2 for the Green’s functions such that −n n (n) i δ W [J] (x1, . . . , xn) = . (1.11) G δJ(x1) . . . δJ(xn) J=0 The value of is ﬁxed by requiring that N
W [J] = W  = 0 0 = 1 . (1.12) J=0 h | i Recalling that in QM a second generating functional called Z[J] has been introduced, we also deﬁne Z[J] = i ln W [J] . (1.13) − By apply n functional diﬀerentiations to Z[J] we get another type of Green’s functions2
1−n n (n) i δ Z[J] G (x1, . . . , xn) = , (1.14) δJ(x1) . . . δJ(xn) J=0 which correspond to connected Feynman diagrams, so that it makes sense to call Z[J] the generating functional for connected Green’s functions. Notice that all information on the QFT is now encoded in the generating functionals, which are hence the primary objects to calculate. We will do this below for the simplest case of a free real scalar ﬁeld.
1.3 Generating Functionals For Free Real Scalar For the purpose of explicit calculations it turns out to be useful to introduce a Euclidean or Wick-rotated version WE[J] of the generating functional. To do this we deﬁne Euclidean ¯µ 4-vectorsx ¯ = (¯x0, x¯) = (ix0, x), associated derivatives ∂ = ∂/∂x¯µ, and an Euclidean version ¯ of the Lagrangian, E = E(φ, ∂µφ). To give an example, L L 1 2 1 2 2 1 ¯ 2 1 2 2 = (∂µφ) m φ E = (∂µφ) m φ . (1.15) L 2 − 2 → L −2 − 2 Starting from (1.10) it is easy to see that the Euclidean version of W [J] is Z Z 4 ¯ WE[J] = φ exp i d x¯ E(φ, ∂µφ) + J(¯x)φ(¯x) . (1.16) N D L The corresponding Euclidean Green’s function are then obtained by
−n n (n) i δ WE[J] E (¯x1,..., x¯n) = . (1.17) G δJ(¯x1) . . . δJ(¯xn) J=0 We now want to derive W [J] for the real Klein-Gordon theory. We start by writing Z Z 4 ¯ ¯µ 4 4 ¯y ¯µ (4) d x¯ (∂µφ(¯x))(∂ φ(¯x)) = d xd¯ y¯ φ(¯y) ∂ ∂ δ (¯x y¯)φ(¯x) . (1.18) µ x −
2Realize that these Green’s functions are the ones we meet already in (1.65) and (1.66) of the script “Interacting Quantum Fields”.
3 It follows that Z Z Z 1 4 4 4 WE[J] = φ exp d xd¯ y¯ φ(¯y)A(¯y, x¯)φ(¯x) + d x¯ J(¯x)φ(¯x) , (1.19) N D −2 with A(¯y, x¯) = ∂¯y∂¯µ + m2 δ(4)(¯x y¯) . (1.20) µ x − This is a Gaussian path integral with a source J of precisely the type you have discussed in the QM context in the solid-state part of this lecture. Given this analogy we perform a variable transformation to ﬁnd an explicit expression for (1.19). Skipping over the details of the actual calculation, one obtains Z ¯ 1 4 4 E WE[J] = exp d xd¯ y¯ J(¯y)D (¯y x¯)J(¯x) , (1.21) N 2 F − with DE(¯y x¯) = A−1(¯y x¯) , (1.22) F − − and ¯ an appropriate normalization. Fine,N but how do we calculate the inverse of the operator A? The idea is to use Fourier transformations and then to go back to Minkowski space. We ﬁrst recall that Z d4p¯ δ(4)(¯x y¯) = eip¯(¯x−y¯) , (1.23) − (2π)4 which we use to write Z d4p¯ A(¯y, x¯) = ∂¯y∂¯µ + m2 δ(4)(¯x y¯) = p¯2 + m2 eip¯(¯x−y¯) . (1.24) µ x − (2π)4 Now we invert A by taking the inverse inside the Fourier transformation, i.e., Z d4p¯ 1 A−1(¯y x¯) = DE(¯y x¯) = eip¯(¯x−y¯) . (1.25) − F − (2π)4 p¯2 + m2
To go to Minkowski space we introduce p = (p0, p) = (ip¯0, p¯). Putting things together one ﬁnds for the generating functional in Minkowski space Z 1 4 4 W [J] = exp d xd y J(y)DF (y x)J(x) , (1.26) −2 − where Z 4 d p i −ip(x−y) DF (x y) = e , (1.27) − (2π)4 p2 m2 + i − is our Feynman propagator. Notice that we have chosen ¯ = 1 so that (1.12) holds. From (1.26) we can now derive Green’s functions eﬀortless.N E.g., for the 2-point function we get from (1.11) 2 (2) δ W [J] (x, y) = = DF (x y) , (1.28) G −δJ(x)δJ(y) J=0 −
4 which agrees with the result that we got using canonical quantization. From (1.13) and (1.26) we also ﬁnd Z i 4 4 Z[J] = d xd y J(y)DF (y x)J(x) , (1.29) 2 − for the generating functional Z[J] of connected Green’s functions. It is important to bear in mind that the results (1.26) and (1.29) hold for the free ﬁeld theory only.
1.4 Eﬀective Action Path integrals also provide an intuitive picture for the transition between classical and quantum physics. In order to illustrate this property we deﬁne the classical ﬁeld φc by δZ[J] φ (x) = . (1.30) c δJ(x)
We have ˆ δ i δW [J] 0 φ(x) 0 J φc(x) = ( i ln W [J]) = = h | | i . (1.31) δJ(x) − −W [J] δJ(x) 0 0 J h | i Here we have deﬁned VEVs in the presence of J as follows
ˆ δW [J] 0 0 J = W [J] , 0 φ(x) 0 J = i . (1.32) h | i h | | i − δJ(x)
The ﬁnal result for φc shows that the classical ﬁeld is the suitably normalized VEV of the ﬁeld operator φˆ, which from a physical standpoint sounds quite reasonable. Recall also that W [J] = exp (iZ[J]) which suggest that the generating functional of connected Green’s functions is something like the action in our path integrals R φ . . . eiS[φ]. This suggest that Z[J] is some sort of eﬀective action. D To remove the eﬀect of the source term that is present in Z[J] we use a Legendre transform. We deﬁne the eﬀective action as Z 4 Γ[φc] = Z[J] d x J(x)φc(x) . (1.33) − In fact, with this deﬁnition one has Z Z δΓ[φc] δZ[J] 4 δJ(x) 4 (4) = d x φc(x) = φc(y) d x δ (x y)φc(x) = 0 , (1.34) δJ(y) δJ(y) − δJ(y) − − so Γ[φc] is independent of the source J.
5 To further see that the deﬁnition (1.33) is meaningful, let us discuss the free ﬁeld case. We begin by deriving an explicit expression for the classic ﬁeld: Z δ i 4 4 φc(x) = d yd z J(y)DF (y z)J(z) δJ(x) 2 − Z i 4 4 (4) (4) = d yd z δ (x y) DF (y z)J(z) + δ (x z) DF (y z)J(y) (1.35) 2 − − − − Z 4 = i d y DF (x y)J(y) . − Since 2 (4) x + m DF (x y) = iδ (x y) , (1.36) − − − we arrive at 2 x + m φc(x) = J(x) , (1.37) which means that φc(x) is a solution to the Klein-Gordon equation with source J(x). This is exactly what one would expect for a classical ﬁeld coupled to J. Furthermore, inserting (1.29) and (1.35) into the eﬀective action (1.33) it follows that Z Z i 4 4 4 Γ[φc] = d xd y J(x)DF (x y)J(y) d x J(x)φc(x) 2 − − Z Z 1 4 1 4 2 = d x φc(x)J(x) = d x φc(x) x + m φc(x) (1.38) −2 −2 Z 4 1 2 1 2 2 = d x (∂µφc(x)) m φ (x) , 2 − 2 c where in the last step we have used integration by parts. In the free ﬁeld case the eﬀective action hence coincides with the classic action of the real scalar ﬁeld. For interacting theories, the generating functional can typically not be calculated exactly. Yet, one can evaluate the path integral (1.10) in the saddle point approximation. The solu- tion φ0 to the classical equations of motions (EOMs) is determined from
δS[φ] δS = [φ0] = J(x) . (1.39) δφ(x) δφ(x) φ=φ0 − Then to leading order in the saddle point approximation one has Z 4 W [J] = exp iS[φ0] + i d x J(x)φ0(x) , N (1.40) Z 4 Z[J] = S[φ0] + d x J(x)φ0(x) .
Comparing these results to (1.30) and (1.33) it is readily seen that
φc(x) = φ0(x) , Γ[φc] = S[φ0] . (1.41)
6 Hence to lowest order the eﬀective action Γ[φc] is simply the classic action S[φ0]. Beyond the leading order the eﬀective action will however receive quantum corrections and as a result one has generically Γ[φc] = S[φ0]. Let me add that6 the above formalism allows one to shed some light on a point that we glossed over in our discussion of spontaneous symmetry breaking in Section 1.4 of the script “Classical Field Theory”. In this discussion we only talked about classic theories. However, one should ask whether the same or similar results would be obtained in the corresponding quantum theories. In fact, as it turns out spontaneous symmetry breaking should be analyzed with the eﬀective action Γ[φc] (or more precisely the eﬀective potential) rather than the classic action S[φ0]. If this is done, one can convince oneself that our discussion based on the classic theory makes sense even in the quantum theory. Yet, the classical analysis has to be viewed as a leading-order approximation. Since in weakly-coupled theory quantum corrections are always suppressed, the classic analysis of spontaneous symmetry breaking is therefore typically a good approximation to the full story.
1.5 Feynman Integrals from Path Integrals In order to develop perturbation theory the path integral formalism, we split the Lagrangian as = 0 + int , (1.42) L L L where 0 is the Lagrangian of the free theory, while int contains all interactions. E.g., in L 4 4 L the case of φ theory one has int = λ/4! φ . The generating functional associated to 0 is L − L called W0[J], while we will denote the full generating functional by W [J]. Explicitly, one has Z Z 4 W0[J] = 0 φ exp i d x 0 + J(x)φ(x) , N D L (1.43) Z Z 4 W [J] = φ exp i d x 0 + int + J(x)φ(x) . N D L L It follows that we can write W [J] as Z 4 δ W [J] = exp i d x int i W0[J] N L − δJ(x) (1.44) " ∞ # X in Z δ δ = 1 + d4x . . . d4x i ... i W [J] . n! 1 n int δJ(x ) int δJ(x ) 0 N n=1 L − 1 L − n Here Z −1 4 δ = exp i d x int i W0[J] , (1.45) N L − δJ(x) J=0 to ensure that (1.12) is satisﬁed. The result (1.44) shows that W [J] is a perturbative series in terms of W0[J]. But from (1.26) we know that Z 1 4 4 W0[J] = exp d xd y J(y)DF (y x)J(x) , (1.46) −2 −
7 so in fact all functional derivatives can be carried out explicitly and lead to Feynman diagrams. It is straightforward to see that the Green’s functions (1.11) can be written as
" ∞ δ δ X im Z (n)(x , . . . , x ) = ... 1 + d4y . . . d4y 1 n δJ(x ) δJ(x ) m! 1 m G N 1 n m=1 (1.47)
δ δ int i ... int i W0[J] . × L − δJ(y1) L − δJ(ym) J=0 While this result looks kind of horrible it can in fact be worked out order by order in pertur- bation theory using Wick’s theorem. This results in a sum over products of propagators DF (suitably integrated) and each term can be associated to a Feynman graph. This is exactly what we have obtained before using canonical quantization, so the two approaches give at the end the same result. The path integral formalism is however more elegant.
1.6 A Simple Application To become more familiar with the path integral formalism in QFT it seems worthwhile to consider a simple but educated example. In the following we will study a theory with a massless real scalar ﬁeld described by the classic Lagrangian
1 2 λ 3 = (∂µφ) φ . (1.48) L 2 − 3! Remember that the dimension of the coupling λ is [λ] = 1. In this case, W0[J] is given by (1.46) employing the massless Feynman propagator of a scalar ﬁeld. The full generating functional takes the following form
" Z 3# 4 λ δ W [J] = exp i d x i W0[J] N − 3! − δJ(x)
∞ " 3#V X 1 iλ Z δ = d4x i (1.49) N V ! − 3! − δJ(x) V =0
( ∞ Z P ) X 1 1 4 4 d yd z J(y)DF (y z)J(z) , × P ! −2 − P =0 where V and P count the number of vertices and propagators. In order to evaluate this expression we can use the tools of Feynman diagrams. First, we determine the number of surviving sources which is equal to the number of external legs L of the graph. Since each propagator connects two points (external or internal) and our scalar theory (1.48) has only 3-point vertices, the number of external legs is given in terms of V and P by the simple formula L = 2P 3V. (1.50) − 8 So instead of using V and P to order the perturbative series, we can also use V and L, and this is what we will do below. For instance, let us consider V = L = 1. In this case we have
Z λ 4 4 = d xd y J(x)DF (x y)DF (y y) , (1.51) x y − 2 − − where the dot in the diagram denotes a vertex while the cross indicates a source. The factor of 2 appearing in the denominator is the symmetry factor of the diagram (the ends of the line meeting at y can be interchanged without altering the result). In the case of V = 2 and L = 0, we get on the other hand
2 Z λ 4 4 3 = d xd y DF (x y) , (1.52) x y 12 − and 2 Z λ 4 4 = d xd y DF (x x)DF (x y)DF (y y) . (1.53) 8 − − − x y
The value of the symmetry factor is in the ﬁrst case S = 2 3! = 12, where the factor of 2 arises from the exchange of x with y and the factor 3! stems from· the possible ways to interchange the lines joining x and y. In the second case similar arguments lead to S = 2 2 2 = 8. Dropping the multiplicative overall factors it is also easy to give a pictorial· representation· of the generating functional Z[J] as deﬁned in (1.13). One has
Z[J] = + ... (λ0) O
+ + ... (λ) O (1.54)
+ + + ... (λ2) O
9 Notice that I have ordered the individual diagrams in the perturbative series of Z[J] corre- sponding to their number of external legs and powers of λ (or equivalent number of vertices). Let us now try to calculate the classic ﬁeld φc as deﬁned in (1.30). Up to and including terms of order λ2, we get
Z 2 Z 4 λ 4 4 4 2 φc(x) = i d y DF (x y)J(y) + d yd zd u DF (x y)D (y z)DF (z u)J(u) − 4 − F − −
(1.55) = + , x y x y z u where the black square indicates the position x of the classic ﬁeld φc(x). The factor of 1/4 in the (λ2) term arises again from the symmetry factor of the associated graph. O We now compute φc(x). Recalling that DF fulﬁls the Klein-Gordon equation (1.36), we obtain 2 Z iλ 4 2 φc(x) = J(x) d yD (x y)DF (y z)J(z) , (1.56) − 4 F − − meaning that 2 Z iλ 4 2 J(x) = φc(x) + d yD (x y)DF (y z)J(z) . (1.57) 4 F − − We solve this equation recursively by making the ansatz
2 J(x) = J0(x) + J2(x)λ . (1.58)
We get J0(x) = φc(x) , (1.59) and Z i 4 4 2 J2(x) = d yd z D (x y)DF (y z) zφc(z) 4 F − − Z i 4 4 2 = d yd z D (x y) zDF (y z) φc(z) (1.60) 4 F − − Z 1 4 2 = d y D (x y)φc(y) . 4 F −
To obtain the ﬁnal result we have employed integration by parts twice, then used that DF satisﬁes the Klein-Gordon equation with delta function source, which in turn allowed us to integrate over z. Combining (1.59) and (1.60), it follows that
2 Z λ 4 2 J(x) = φc(x) + d y D (x y)φc(y) . (1.61) 4 F −
10 We now move our attention to the eﬀective action (1.33). At (λ0) one ﬁnds in terms of O J0(x) = φc(x) the following expression Z Z i 4 4 4 Γ0[φc] = d xd y φc(x)DF (x y) φc(y) d x ( φc(x)) φc(x) 2 − − (1.62) Z 1 = d4x (∂ φ (x))2 . 2 µ c
Here I have again used integration by parts and the fact that DF is a Green’s function of the Klein-Gordon equation. This result looks quite familiar. In fact, we have already derived it in (1.38). It is the kinetic term of the classic action. There is of course no mass term, because we have set m = 0 by hand in the classic theory (1.48). By inserting (1.61) into the deﬁnition (1.33), one can also show that the (λ) part of the eﬀective action takes the form O Z 4 λ 3 λ Γ1[φc] = d x φ (x) DF (0)φc(x) . (1.63) −3! c − 2
The term λ/3! φ3 is again part of the classic action or Lagrangian (1.48). But the term − c
λ DF (0)φc(x) = , (1.64) − 2 is a new contribution that arises from quantum corrections. These so-called tadpole contribu- tions can be removed (by a proper renormalization) and therefore not aﬀect physical processes. So let’s forget about them and press on. 2 At (λ ) one gets a contribution to Γ[φc] from the graph O
, (1.65) i.e., the third diagram in the second line of Z[J] as given in (1.54) as well as a similar R 4 contribution from d x J(x)φc(x). Using the expansion (1.61) of the source J, one ﬁnds after some algebra −
Z 2 Z 4 λ 4 2 Γ2[φc] = d x φc(x) d y D (x y)φc(y) = . (1.66) − 4 F −
2 This is an interesting result. Since (1.66) is bilinear in φc the (λ ) term of the eﬀective action corresponds to a loop-induced mass term for the classic ﬁeld.O So our scalar ﬁeld will get a mass from radiative corrections even if we start with m = 0. The general lesson to learn here is that if there is no symmetry that forbids a speciﬁc term in the Lagrangian, one better includes it in the theory. If one does not do this, one will always get it back in the quantum theory.