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Quantum field theory and its symmetry reduction

Jonathan Engle
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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.

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Jonathan Engle. 2006-04-01. Quantum field theory and its symmetry reduction. https://doi.org/10.1088/0264-9381%2F23%2F9%2F007

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