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Quantum Field Theory II

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Quantum Field Theory II Quantum Field Theory II ETH Zurich FS 2011 Prof. Dr. Matthias R. Gaberdiel Prof. Dr. Aude Gehrmann-De Ridder typeset: Felix Hahl¨ and Prof. M. R. Gaberdiel (chapter 1) August 21, 2011 This document contains lecture notes taken during the lectures on quantum field theory by Profs. M. R. Gaberdiel and Aude Gehrmann-de Ridder in spring 2011. I'd like to thank Profs. Gaberdiel and Gehrmann-de Ridder for looking at these notes, correcting and improving them. In case you still find mistakes, feel free to report them to Felix Haehl ([email protected]). 1 Contents 1 Path Integral Formalism4 1.1 Path Integrals in Quantum Mechanics.....................4 1.1.1 Feynman-Kac Formula.........................5 1.1.2 The Quantum Mechanical Path Integral...............6 1.1.3 The Interpretation as Path Integral..................7 1.1.4 Amplitudes...............................9 1.1.5 Generalisation to Arbitrary Hamiltonians............... 10 1.2 Functional Quantization of Scalar Fields................... 11 1.2.1 Correlation Functions.......................... 11 1.2.2 Feynman Rules............................. 13 1.2.3 Functional Derivatives and the Generating Functional........ 18 1.3 Fermionic Path Integrals............................ 21 1.3.1 The Dirac Propagator......................... 24 2 Functional Quantization of Gauge Fields 26 2.1 Non-Abelian Gauge Theories.......................... 26 2.1.1 U(1) Gauge Invariance......................... 26 2.1.2 SU(N) Gauge Invariance........................ 29 2.1.3 Polarisation Vectors for the Gauge Fields............... 34 2.2 Quantization of the QED Gauge Field Aµ .................. 36 2.2.1 The Green's Function Approach.................... 37 2.2.2 Functional Method........................... 38 2.3 Quantization of the non-Abelian Gauge Field................ 41 2.3.1 Feynman Rules for QCD........................ 44 2.4 Ghosts and Gauge Invariance......................... 47 2.4.1 QCD Ward Identity........................... 47 2.4.2 Physical States and Ghosts: Polarisation Sums Revisited...... 49 2.5 BRST Symmetry................................ 51 2.5.1 The Definition of BRST-Symmetry.................. 51 2.5.2 Implications of the BRST Symmetry................. 52 2 CONTENTS 3 Renormalisation Group 56 3.1 Wilson's Approach to Renormalisation.................... 56 3.2 Renormalisation Group Flows......................... 60 3.3 Callan-Symanzik Equation........................... 63 3.3.1 The φ4-Theory............................. 63 3.3.2 The General Structure......................... 68 3.4 Asymptotic Freedom.............................. 69 4 Spontaneous Symmetry Breaking and the Weinberg-Salam Model of the Electroweak Interactions 75 4.1 Electroweak Interactions............................ 75 4.1.1 Characteristics of Weak Interactions................. 75 4.1.2 Electroweak Interactions........................ 77 4.2 Spontaneous Symmetry Breaking....................... 80 4.2.1 Discrete Symmetries (2 Examples).................. 80 4.2.2 Spontaneous Symmetry Breaking of a Global Gauge Symmetry.. 82 4.2.3 Spontaneous Symmetry Breaking of a Local Gauge Symmetry and the Abelian Higgs Mechanism..................... 85 4.2.4 Spontaneous Symmetry Breaking of a Local SU(2) U(1) Gauge Symmetry: Non-Abelian Higgs Mechanism..............× 87 4.2.5 The Electroweak Standard Model Lagrangian............ 89 5 Quantization of Spontaneously Broken Gauge Theories 92 5.1 The Abelian Model............................... 92 5.1.1 ξ-dependence in Physical Processes.................. 95 5.2 Quantization of Sp. Broken non-Abelian Gauge Theories.......... 98 5.3 Rξ Gauge Dependence in Perturbation Theory................ 104 5.3.1 QFT with Spontaneous Symmetry Breaking and finite ξ ...... 105 5.3.2 QFT with Spontaneous Symmetry Breaking for ξ ....... 105 ! 1 6 Renormalizability of Broken and Unbroken Gauge Theories: Main Cri- teria 107 6.1 A Renormalization Program (UV Divergences only)............. 107 6.2 Overall Renormalization of QED....................... 108 6.3 Renormalizability of QCD........................... 111 6.3.1 One Loop Renormalization of QCD.................. 116 6.4 Renormalizability of Spontaneously Broken Gauge Theories......... 117 6.4.1 The Linear σ-Model.......................... 118 3 Chapter 1 Path Integral Formalism For the description of advanced topics in quantum field theory, in particular the quanti- zation of non-abelian gauge theories, the formulation of quantum field theory in the path integral formulation is important. We begin by explaining the path integral formulation of quantum mechanics. 1.1 Path Integrals in Quantum Mechanics In the Schr¨odingerpicture the dynamics of quantum mechanics is described by the Schr¨odinger equation d i~ (t) = H (t) : (1.1) dtj i j i If the Hamilton operator is not explicitly time-dependent, the solution of this equation is simply − it H (t) = e ~ (0) : (1.2) j i j i Expanding the wave function in terms of position states, i.e. doing wave mechanics, we then have −itH=~ (t; q) q (t) = dq0 q e q0 q0 (0) = dq0 K(t; q; q0) (0; q0) ; (1.3) ≡ h j i h j j i h j i Z Z where we have used the completeness relation 1 = dq0 q0 q0 (1.4) j i h j Z and introduced the propagator kernel −itH=~ K(t; q; q0) = q e q0 : (1.5) h j j i It describes the probability for a particle at q0 at time t = 0 to propagate to q at time t and will play an important role in the following. 4 1.1. PATH INTEGRALS IN QUANTUM MECHANICS By construction, the propagator satisfies the time-dependent Schr¨odingerequation d i~ K(t; q; q0) = HK(t; q; q0) ; (1.6) dt where H acts on q. It is furthermore characterised by the initial condition lim K(t; q; q0) = δ(q q0) : (1.7) t!0 − For a free particle in one dimension with Hamilton operator 2 2 1 2 ~ d H0 = p = (1.8) 2m −2m dx2 the propagator, which is uniquely determined by (1.6) and (1.7), is 1 2 2 −itH0=~ m (q q0) K0(t; q; q0) = q e q0 = exp im − : (1.9) h j j i 2πi~t 2~t To derive this formula, one can for example use a complete momentum basis; then −itH0=~ 1 −itH0=~ q e q0 = dp q p p e q0 h j j i 2π h j i h j j i ~ Z 1 2 = dp eiqp=~ e−itp =2m~ e−ipq0=~ 2π ~ Z 2 2 1 (q q0) it m(q q0) = exp im − dp exp p − ; 2π 2 t −2m − t ~ ~ Z " ~ # which leads after Gaussian integration to (1.9). Here we have used that i qp 1 q p = e ~ ; 1 = dp p p : (1.10) h j i 2π j ih j ~ Z The path integral is a method to calculate the propagator kernel for a general (non-free) quantum mechanical system. In order to derive it we need a small mathematical result. 1.1.1 Feynman-Kac Formula The path integral formulation of quantum mechanics was first developed by Richard Feyn- man; the underlying mathematical technique had been previously developed by Marc Kac in the context of statistical physics. The key formula underlying the whole formalism is the product formula of Trotter. In its simplest form (in which it was already proven by Lie) it states n eA+B = lim eA=n eB=n ; (1.11) n!1 5 1.1. PATH INTEGRALS IN QUANTUM MECHANICS where A and B are bounded operators on a Hilbert space. To prove it, we define (A + B) A B S = exp ;T = exp exp : (1.12) n n n n n Then we calculate A+B A=n B=n n n n e (e e ) = Sn Tn (1.13) jj − jj jj n−−1 jj n−2 n−1 = S (Sn Tn) + S (Sn Tn)Tn + + (Sn Tn)T : jj n − n − ··· − n jj Since the norm of a product is always smaller or equal to the products of the norms, it follows (after using the triangle inequality X + Y X + Y ) jj jj ≤ jj jj jj jj exp(X) exp( X ) : (1.14) jj jj ≤ jj jj Using the triangle inequality again it follows that (jjAjj+jjBjj)=n 1=n (jjAjj+jjBjj)=n 1=n Sn e a ; Tn e a : (1.15) jj jj ≤ ≡ jj jj ≤ ≡ Plugging into (1.13) leads, again after using the triangle inequality, to n n (n−1)=n S T n a Sn Tn : (1.16) jj n − n jj ≤ jj − jj Finally, because of the Baker-Campbell-Hausdorff formula [A; B] −3 Sn Tn = + (n ) ; (1.17) − − 2n2 O and the product formula (1.11) follows. If A and B are not bounded operators, the analysis is more difficult. If both A and B are self-adjoint (as is usually the case for the operators appearing in quantum mechanics), one can still prove that n e−it(A+B) = lim e−itA=n e−itB=n (1.18) n!1 where the convergence is in the strong topology, i.e. the result holds when applied to any vector that lies in the domain of both A and B. 1.1.2 The Quantum Mechanical Path Integral With these preparations we can now derive the path integral formulation of quantum mechanics. Let us assume that the Hamilton operator is of the form p2 H = H + V (q) H = ; (1.19) 0 0 2m 6 1.1. PATH INTEGRALS IN QUANTUM MECHANICS where H0 is the Hamilton operator of the free particle, and V (q) is the potential. Applying the product formula (1.11) with A = H0=~ and B = V=~ to (1.5) we obtain −itH=~ K(t; q; q0) = q e q0 h j j i n −itH0=~n −itV=~n = lim q e e q0 n!1 j j D j=n−1 E −itH0=~n −itV=~n = lim dq1 dqn−1 qj+1 e e qj ; (1.20) n!1 ··· j j Z j=0 Y where q qn, and we have, after each application of the exponential, introduced a partition of unity ≡ 1 = dqj qj qj : (1.21) j i h j Z Since the potential acts diagonally in the position representation, we now have −itH0=~n −itV=~n −itV (qj )=~n −itH0=~n qj+1 e e qj = e qj+1 e qj : (1.22) j j j j Thus we can use the propagator kernel of the free particle (1.9) to get, with t=n = 1 2 2 −itH0=~n −itV=~n mn i m qj+1 qj qj+1 e e qj = exp − V (qj) : (1.23) j j 2πi t 2 − ~ " ~ !# Hence we have for the complete propagator kernel the Feynman-Kac formula n n−1 2 m 2 i m qj+1 qj K(t; q; q0) = lim dq1 dqn−1 exp − V (qj) : n!1 ··· 2πi~ " ~ 2 − !# Z j=0 X (1.24) q q q n 0 1 Figure 1.1: Interpretation as path integral.
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