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Strict deformation quantization of abelian lattice gauge fields

Abstract

This paper shows how to construct classical and quantum field C*-algebras modeling a \(U(1)^n\)-gauge theory in any dimension using a novel approach to lattice gauge theory, while simultaneously constructing a strict deformation quantization between the respective field algebras. The construction starts with quantization maps defined on operator systems (instead of C*-algebras) associated with the lattices, in a way that quantization commutes with all lattice refinements, therefore giving rise to a quantization map on the continuum (meaning ultraviolet and infrared) limit. Although working with operator systems at the finite level, in the continuum limit we obtain genuine C*-algebras. We also prove that the C*-algebras (classical and quantum) are invariant under time evolutions related to the electric part of abelian Yang–Mills. Our classical and quantum systems at the finite level are essentially the ones of van Nuland and Stienstra (Classical and quantized resolvent algebras for the cylinder, 2020. arXiv:2003.13492 [math-ph]), which admit completely general dynamics, and we briefly discuss ways to extend this powerful result to the continuum limit. We also briefly discuss reduction, and how the current setup should be generalized to the non-abelian case.

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Acknowledgements

I am grateful to Peter Hochs, Klaas Landsman, Walter van Suijlekom, and the anonymous referee, for proofreading and providing highly valuable feedback. Research supported by NWO Physics Projectruimte (680-91-101).

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Correspondence to Teun D. H. van Nuland.

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van Nuland, T.D.H. Strict deformation quantization of abelian lattice gauge fields. Lett Math Phys 112, 34 (2022). https://doi.org/10.1007/s11005-022-01525-2

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  • DOI: https://doi.org/10.1007/s11005-022-01525-2

Keywords

  • Operator systems
  • Limit C*-algebra
  • Quantum lattice gauge theory
  • Quantum electrodynamics
  • C*-algebraic quantization
  • Resolvent algebra

Mathematics Subject Classification

  • 46L65 81T27
  • 46L60