Path Integrals in Quantum Field Theory C6, HT 2014 Uli Haischa aRudolf Peierls Centre for Theoretical Physics University of Oxford OX1 3PN Oxford, United Kingdom Please send corrections to [email protected]. 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 j i h j j 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 learnedj i in the lecturesj i onj interactingi j 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 j j i In QM the analog of (1.2) is simply1 xf ; tf T x^(t1)^x(t2) xi; ti : (1.3) h j j 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 x^(t1)^x(t2) xi; ti = xf e x^S e x^S e xi h j j i h i Z (1.4) −iH(tf −t1) −iH(t1−t2) −iH(t2−ti) = dx1dx2 xf e x1 x1 x^S e x2 x2 x^S e xi : h j ih j j ih j i Using now thatx ^ x = x x , replacing all three expectation values by (1.1) and combining j i j 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 x^(t1)^x(t2) xi; ti x x(t1)x(t2) exp i dt (x; x_) : (1.5) h j j 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 ^. Furthermore, it can be shown that lim xf ; tf T (^x(t1) ::: x^(tn)) xi; ti 0 T (^x(t1) ::: x^(tn)) 0 : (1.6) ti;f →∓∞h j j i / h j j i Therefore one arrives at Z iS[x] 0 T (^x(t1) ::: x^(tn)) 0 x x(t1) : : : x(tn) e ; (1.7) h j j 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) ^ ^ (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 0 = 1 : (1.12) J=0 h j 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 ^ δ i δW [J] 0 φ(x) 0 J φc(x) = ( i ln W [J]) = = h j j i : (1.31) δJ(x) − −W [J] δJ(x) 0 0 J h j i Here we have defined VEVs in the presence of J as follows ^ δW [J] 0 0 J = W [J] ; 0 φ(x) 0 J = i : (1.32) h j i h j j i − δJ(x) The final result for φc shows that the classical field is the suitably normalized VEV of the field operator φ^, which from a physical standpoint sounds quite reasonable.