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Strict deformation quantization of abelian lattice gauge fields


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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  1. Arici, F., Stienstra, R., van Suijlekom, W.D.: Quantum lattice gauge fields and groupoid C*-algebras. Ann. Henri Poincaré 19(11), 3241–3266 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  2. Buchholz, D., Grundling, H.: The resolvent algebra: a new approach to canonical quantum systems. J. Funct. Anal. 254(11), 2725–2779 (2008)

    MathSciNet  Article  Google Scholar 

  3. Binz, E., Honegger, R., Rieckers, A.: Field-theoretic Weyl quantization as a strict and continuous deformation quantization. Ann. Henri Poincaré 5(2), 327–346 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  4. Brothier, A., Stottmeister, A.: Canonical quantization of 1+1-dimensional Yang-Mills theory: an operator-algebraic approach. arXiv:1907.05549 [math-ph] (2019)

  5. Brothier, A., Stottmeister, A.: Operator-algebraic construction of gauge theories and Jones’ actions of Thompson’s groups. Commun. Math. Phys. 376(2), 841–891 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  6. Grundling, H., Rudolph, G.: Dynamics for QCD on an infinite lattice. Commun. Math. Phys. 349(3), 1163–1202 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  7. Guth, A.H.: Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory. Phys. Rev. D 21(8), 2291–2307 (1980)

    ADS  MathSciNet  Article  Google Scholar 

  8. Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Springer (2012)

  9. Kogut, J., Susskind, L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11(2), 395–408 (1975)

    ADS  Article  Google Scholar 

  10. Landsman, N.P.: Strict deformation quantization of a particle in external gravitational and Yang-Mills fields. J. Geom. Phys. 12, 93–132 (1993)

    ADS  MathSciNet  Article  Google Scholar 

  11. Landsman, N.P.: Mathematical topics between classical and quantum mechanics. Springer Monographs in Mathematics. Springer, New York, xx+529 pp (1998)

  12. Landsman, N.P., Moretti, V., van de Ven, C.J.F.: Strict deformation quantization of the state space of \(M_k(\mathbb{C})\) with applications to the Curie-Weiss model. Rev. Math. Phys. 32(10), 2050031 (2020)

    MathSciNet  Article  Google Scholar 

  13. Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531–562 (1989)

    ADS  MathSciNet  Article  Google Scholar 

  14. Rieffel, M.A.: Deformation quantization for actions of \(\mathbb{R}^{d}\). Mem. Am. Math. Soc. 106, x+93 (1993)

    MATH  Google Scholar 

  15. Stienstra, R.: Quantisation versus lattice gauge theory. PhD thesis, Radboud University (2019)

  16. Stottmeister, A., Thiemann, T.: Coherent states, quantum gravity, and the Born-Openheimer approximation. III.: applications to loop quantum gravity. J. Math. Phys. 57(8), 083509 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  17. van Nuland, T.D.H.: Quantization and the resolvent algebra. J. Funct. Anal. 277(8), 2815–2838 (2019)

    MathSciNet  Article  Google Scholar 

  18. van Nuland, T.D.H., Stienstra, R.: Classical and quantized resolvent algebras for the cylinder. arXiv:2003.13492 [math-ph] (2020)

  19. Wilson, K.G.: Confinement of quarks. Phys. Rev. D 10(8), 2445–2459 (1974)

    ADS  Article  Google Scholar 

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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).

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

Mathematics Subject Classification

  • 46L65 81T27
  • 46L60