Relativistic Field Theory

Abstract

This chapter starts with Wigner’s classification of particles based on the unitary representations of the Poincare group and the classification of field variants as enforced by Lorentz symmetry. Generic requirements of action functionals in a (quantum) field theory are formulated. Next appropriate actions are derived for the various field variants. By building Lorentz scalars from the fields and their derivatives it is shown that the free-field part for a spin-0 and spin-1/2 field is fixed from Poincare symmetry and arguments of dimensional renormalizability. The quantization of these scalar and spinor field theories is sketched. The central theme in fundamental physics is the idea of a gauge theory – or – Yang-Mills theory. Gauge fields are introduced in order to render global phase symmetries of wave functions spacetime-dependent. Both the coupling of these spin-1 fields to the spinors and their kinetic energy term in a Lagrangian are largely “dictated” by the local internal symmetry. It is shown how the Klein-Noether identities fix the action for any Yang-Mills theory (with mild conditions on the symmetry group). Field equations for higher-spin fields are mentioned. A further subsection deals with spontaneous symmetry0 breaking which arises in a (quantum) field theory if a symmetry of an action is no longer a symmetry of the “ground state”. The mechanism by which in a Yang-Mills gauge theory would-be Goldstone bosons disappear at the expense of massive gauge bosons is explained. In a further chapter the discrete symmetries related to space inversion, time reversal, and charge conjugation, as well as the CPT theorem are dealt with. The last section refers to effective field theories, a notion which is becoming more and more accepted. Among other things, it allows to relate theories which apply to phenomena on different levels of granularity or on different length scales. An important technical means is the approach of the renormalization group flow and the notion of \(\beta \) functions.