Quantum field theory and its symmetry reduction

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.

38 pages; minor clarifications added, typos corrected, contact with Klauder's `universal procedure' for imposing constraints incorporated, and discussion of application to LQG elaborated a bit; minor clarifications added, minor phraseology changes, and 8 bibliographic entries corrected/updated

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• Quantum geometry and the Schwarzschild singularity
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• A black hole mass threshold from non-singular quantum gravitational collapse
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• Gauge invariant perturbations around symmetry reduced sectors of general relativity: applications to cosmology
• Relating loop quantum cosmology to loop quantum gravity: Symmetric sectors and embeddings
• Quantization of Midisuperspace Models
• The Dark Side of a Patchwork Universe
• Perturbative Degrees of Freedom in Loop Quantum Gravity: Anisotropies
• Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models
• Coherent states, constraint classes, and area operators in the new spin-foam models
• Minisuperspace models as infrared contributions
• Loop Quantization of Polarized Gowdy Model on $T^3$: Kinematical States and Constraint Operators
• Embedding loop quantum cosmology without piecewise linearity
• Invariant Connections in Loop Quantum Gravity
• Diffeomorphism invariant cosmological sector in loop quantum gravity
• A Cosmological Sector in Loop Quantum Gravity
• Piecewise linear loop quantum gravity
• Quantum isotropy and the reduction of dynamics in Bianchi I
• A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time
• Symmetry Reduction of Quasi-Free States
• On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC