9. Interacting Relativistic Field Theories 9.1. Asymptotic States and the Scattering Operator 9.2. Reduction Formulae 9.3. Path Integrals 9.4. Perturbation Theory 9.5. Quantization of Gauge Fields 9.6. Renormalization 9.7. Quantum Electrodynamics

9.1. Asymptotic States and the Scattering Operator Non-relativistic quantum mechanics: Problem of single particle in external potential is always solvable. System of N non-relativistic, non-interacting particles: Creation and annihilation operators for stable single particle eigenstates. System of relativistic non-interacting particles: Number of particles not fixed even in closed, isolated systems. System of interacting particles: Ground state = vacuum state; other eigenstates: usually not known. States/particles created by field operators usually not eigenstates/stable. Exceptions: interaction removable by canonical transformation, e.g., elementary excitations such as phonons, magnon, excitons, polarons, Cooper pairs, etc. Description of free particles : Plane wave ~ wave packet. Interacting particles: Plane waves still useful unless bound states are formed.

Scattering Problem Short-ranged interaction → free particles outside interaction region. Tricks justifying the use of plane waves instead of wave packet : 1. Replace incoming particle with a steady stream of particles. 2. Adiabatic switching: replace H = H 0 + H int with → Field operators for asymptotic states: Z = wavefunction renormalization constant ~ phase shift due to scattering.

Incoming asymptotic state of N particles: ‘Out’ state of scattered particles: Probability amplitude = See appendix D for calculating other physical quantities of interest. If ‘in’ and ‘out’ states belong to the same multi-particle (Fock) space, S = scattering operator Conservation of probability → Normalization of asymptotic states preserved. → S is unitary. →

9.2. Reduction Formulae

9.3. Path Integrals

9.4. Perturbation Theory

9.5. Quantization of Gauge Fields

9.6. Renormalization

9.7. Quantum Electrodynamics