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Quantum Field Theory I

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Quantum Field Theory I Quantum Field Theory I HS 2010 Prof. Dr. Thomas Gehrmann typeset and revision: Felix Hahl¨ April 30, 2011 Contents 0 Introduction4 0.1 Natural Units....................................4 1 Relativistic Quantum Mechanics5 1.1 Klein-Gordon Equation..............................6 1.2 Dirac Equation...................................6 1.3 The Dirac Equation in an Electromagnetic Field................9 1.4 Lorentz Covariance of the Dirac Equation....................9 1.4.1 The Lorentz Group and the Poincar´eGroup............... 10 1.4.2 Transformation of Dirac-spinors...................... 11 1.4.3 Spin of Dirac-Spinor............................ 12 1.4.4 Parity and Time-reversal......................... 13 1.4.5 Transformation of ¯ ............................ 13 1.4.6 Bilinear Covariants............................. 14 1.5 Charge Conjugation................................ 15 1.6 Solutions of Dirac Equation for free particles.................. 16 1.6.1 Energy Projection Operators....................... 19 1.6.2 Helicity and Chirality........................... 19 2 Quantization of the Scalar Field 23 2.1 Classical Field Theory............................... 23 2.1.1 Symmetries and Noether-Theorem.................... 24 2.1.2 Energy-Momentum Tensor........................ 26 2.2 The Klein-Gordon Field.............................. 26 2.2.1 Quantization of Classical Systems.................... 27 2.2.2 Fourier Decomposition of the Klein-Gordon Field............ 28 2.2.3 Klein-Gordon Propagator......................... 32 3 Quantization of the Free Dirac Field 38 3.1 Field Operator of the Free Dirac Field...................... 38 3.2 Single-Particle States in the Dirac Theory.................... 41 3.3 Conserved Quantities in the Dirac Theory.................... 42 3.4 The Fermion Propagator.............................. 44 1 CONTENTS 4 Quantization of the Electromagnetic Field 46 4.1 Gauge Invariance of the Electromagnetic Field................. 46 4.2 Quantization in Lorenz Gauge........................... 47 4.3 Fock Space States................................. 51 4.4 The Photon Propagator.............................. 52 5 Quantum Electrodynamics (QED) 53 5.1 The Interaction Picture and the Time Evolution Operator........... 53 5.1.1 The Ground State............................. 56 5.2 Wick's Theorem.................................. 57 5.3 Feynman Diagrams................................. 59 5.3.1 λφ4-Theory................................. 59 5.3.2 Connected and Disconnected Diagrams................. 63 5.4 Scattering Matrix.................................. 65 5.5 Feynman Rules of QED.............................. 66 5.5.1 S-operator to First Order......................... 67 5.5.2 S-Operator to Second Order........................ 68 5.5.3 Contraction with External Momentum Eigenstates........... 70 5.6 Scattering Cross Section.............................. 73 5.6.1 Decay Kinematics............................. 74 5.6.2 Scattering Cross Sections......................... 75 5.7 Kinematics of Particle Scattering......................... 76 5.7.1 Angular Distribution............................ 78 5.7.2 Two-particle Phase Space......................... 78 5.8 Trace Techniques.................................. 79 5.9 Møller Scattering.................................. 81 5.10 Compton Scattering................................ 83 6 Renormalization 87 6.1 Divergences in Feynman Diagrams........................ 87 6.1.1 Divergences in λφ4-Theory........................ 87 6.1.2 Divergences in QED............................ 89 6.2 Dimensional Regularization............................ 90 6.2.1 Dimensional Analysis in d Dimensions.................. 91 6.2.2 Lorentz and Dirac Algebras in d Dimensions.............. 92 6.2.3 Integration in d Dimensions........................ 92 6.3 Divergent Diagrams in QED............................ 95 6.4 Renormalization of QED.............................. 100 6.5 Renormalization Conditions: The On-Shell Scheme (OS)............ 100 6.5.1 Corrections to the Fermion Propagator.................. 100 6.5.2 Corrections to the Photon Propagator.................. 103 6.5.3 Electron Vertex Corrections........................ 104 6.6 Renormalization Conditions: Minimal Subtraction............... 108 6.6.1 Renormalized Coupling Constant..................... 109 6.7 Running Coupling Constant............................ 109 6.8 Ward-Takahashi Identity.............................. 112 6.9 Spectral Representation.............................. 117 2 CONTENTS 6.10 Lehmann-Symanzik-Zimmermann (LSZ) Reduction Formula.......... 121 6.11 Infrared Singularities................................ 123 3 Chapter 0 Introduction What is the aim of quantum field theory (QFT)? In one word, it is the (successful) attempt to describe matter and fields in a symmetric, consistent and quantum mechanical as well as relativistic way. In electrodynamics, matter is identified with point charges and electromagnetic fields are induced by external sources. Quantum mechanics is reigned by the Schr¨odingerequation and matter is treated as point charges (or masses) which carry mass, momentum and spin. In quantum mechanics, electromagnetic fields appear as external potentials. This point of view allows us to describe the dynamics of matter in the presence of external fields but not the generation of these fields by matter. An improved ansatz with a quantized photon field treats matter still as wave functions but the electromagnetic field is then described by Fock space states. In this way, one can describe matter in the presence of external fields and also oscillating fields (waves) and the creation or annihilation of field quanta. But still the creation and annihilation of matter is not included. The aim of QFT is to overcome this asymmetry and to describe both matter and fields as Fock space states which can be created and annihilated. This kind of unification is one major step on the way to unify apparently different physical concepts. 0.1 Natural Units Planck's constant and the velocity of light are −34 ~ = 1:0546 10 Js (0.1) · m c = 2:998 108 : (0.2) · s In the following we will use natural units with ~ = c = 1. As a consequence, length and time will have the same units, for example. The unit of electrical charge e is based on the dimensionless fine structure constant, which reads in SI-units e2 α = = 7:2972 10−3 (0.3) 4π0~c · In natural units, one sets 0 = 1, such that the electrical charge is dimensionless: e = p4πα : (0.4) 4 Chapter 1 Relativistic Quantum Mechanics In courses on quantum mechanics we worked in a non-relativistic regime. We used the Schr¨odingerequation, @ ∆ i (x; t) = H (x; t) with H = + V (x); (1.1) @t −2m with free solutions (\free\ meaning V (x) = 0) of the form 2 i(p·x− p t) free(x; t) = Ce 2m : (1.2) The free Schr¨odingerequation can immediately be obtained from the classical energy-momentum relation p2 E = (1.3) 2m by using the correspondence principle @ E i ; p ir: (1.4) ! @t ! − We make a first attempt towards a relativistic generalization of the Schr¨odingerequation by using the relativistic energy-momentum relation E = p2 + m2: (1.5) The correspondence principle yields p @ i (x; t) = ∆ + m2 (x; t) (1.6) @t − p 1 ∆2 = m ∆ + ::: (x; t): (1.7) − 2m − 8m2 This equation immediately raises some problems: the temporal and spatial derivatives are treated asymmetrically which cannot be true in a Lorentz covariant theory. Moreover, the fact that there appear arbitrarily high derivatives of (x; t), does not only imply that our theory becomes highly non-linear, but also that we need infinitely many boundary conditions in oder to determine the solution. 5 1.1. KLEIN-GORDON EQUATION 1.1 Klein-Gordon Equation A more elaborate attempt to find a relativistic Schr¨odingerequation takes as a starting point the equation E2 = p2 + m2: (1.8) Applying the correspondence principle, we obtain the Klein-Gordon equation: @2 (x; t) = ( ∆ + m2) (x; t) (1.9) −@t2 − µ 2 @µ@ + m (x; t) = 0: (1.10) , Solutions are (x; t) = Cei(p·x−Et) with E = p2 + m2: (1.11) ± Note that there are positive and negative energy eigenvaluespE = p2 + m2. Apart from ± second order time derivatives, this is another reason why we cannot interpret the Klein- p Gordon equation as a usual Schr¨odinger equation. From non-relativistic quantum mechanics we are used to treat a spectrum which is symmetric around zero and extends to infinity as being unphysical: no state would be stable because it could always release more and more energy in order to reach a lower energy state. A third reason is given by the continuity equation which is implied by the Klein-Gordon equation. To find the continuity equation, we consider ∗ µ 2 (@µ@ + m ) = 0 ∗ µ µ ∗ µ 2 ∗ @µ@ @µ@ = 0: (1.12) (@µ@ + m ) = 0) ) − 1 Multiplying this equation with 2mi yields @ 1 @ @ ∗ 1 ∗ +r ( ∗r r ∗) = 0: (1.13) @t −2mi @t − @t 2mi − =:ρ =:j Note that ρ is not| positive definite{z which reflects} the| fact that{z there appear} second order time derivatives in the equations of motion. 1.2 Dirac Equation We want to find a relativistic wave equation in which only first order derivatives appear. We hope that this will solve at least some of the problems of the Klein-Gordon equation. We make an ansatz for the first order equation (whose square will yield the second order Klein-Gordon equation) in covariant notation: µ ( iγ @µ + m) (x; t) = 0 (1.14) − @ iγ0 iγ r + m (x; t) = 0 (1.15) , − @t − · 6 1.2. DIRAC EQUATION with yet unknown objects γµ. Using the correspondence principle in the form @ i E (1.16) @t $ ir p (1.17) − $
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