# Symmetries IV: Local symmetries in field theory

# Symmetries IV: Local symmetries in field theory

This chapter examines the additional rich structure introduced when a local quantum field theory displays a local gauge symmetry. It shows how such symmetries require a generalization of the canonical Lagrangian/Hamiltonian formalism discussed in Section 12.3 of Chapter 12 in order to handle the presence of constraints entailed by the presence of local symmetries. The chapter is organized as follows. Section 15.1 introduces the concept of a local symmetry with a simple example from classical mechanics. Section 15.2 describes the Dirac constrained Hamiltonian theory, and the Faddeev–deWitt functional quantization method for such systems. The quantization of gauge theories using this functional (path-integral) method is then explained, first using abelian gauge theory in Section 15.3, where the technical complications are minimal. In Section 15.4 the extension to non-abelian gauge theories is performed, again using path-integral methods applied to the constrained Hamiltonian, leading to the Feynman rules for general (unbroken) non-abelian gauge theories. Section 15.5 explores the existence of quantum anomalies in the chiral currents of internal global symmetries. It shows that the classical current conservation implied by Noether's theorem may be violated by quantum effects, yielding a non-vanishing divergence of the Noether current explicitly proportional to Planck's constant. Section 15.6 focuses on the features of spontaneous symmetry breaking in the presence of local gauge symmetry. The chapter then explains the famous ‘Higgs phenomenon’ in the context of the electroweak sector of the Standard Model and outlines the derivation of the Feynman rules for a general spontaneously broken local gauge theory.

*Keywords:*
local quantum field theory, gauge symmetry, local symmetry, Dirac constrained Hamiltonian theory, Faddeev–deWitt functional quantization, global symmetries, Noether's, Higgs phenomenon

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