Quantum field theory and its symmetry reduction

TL;DR

This paper explores how symmetry reduction interacts with quantization in quantum field theories, using a toy model to identify the most physically consistent notion of quantum symmetry and extending it to loop quantum gravity.

Contribution

It introduces and compares three notions of quantum symmetry, identifying the most suitable one that preserves physical criteria and commutes with quantization, with implications for loop quantum gravity.

Findings

The most appropriate quantum symmetry notion satisfies physical criteria.

Invariance under the group action is not suitable for symmetry in quantum theories.

The approach is generalized to loop quantum gravity, offering new insights.

Abstract

The relation between symmetry reduction before and after quantization of a field theory is discussed using a toy model: the axisymmetric Klein-Gordon field. We consider three possible notions of symmetry at the quantum level: invariance under the group action, and two notions derived from imposing symmetry as a system of constraints a la Dirac, reformulated as a first class system. One of the latter two turns out to be the most appropriate notion of symmetry in the sense that it satisfies a number of physical criteria, including the commutativity of quantization and symmetry reduction. Somewhat surprisingly, the requirement of invariance under the symmetry group action is not appropriate for this purpose. A generalization of the physically selected notion of symmetry to loop quantum gravity is presented and briefly discussed.