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Annales Henri Poincaré

, Volume 19, Issue 11, pp 3241–3266 | Cite as

Quantum Lattice Gauge Fields and Groupoid \(\hbox {C}^{*}\)-Algebras

  • Francesca Arici
  • Ruben Stienstra
  • Walter D. van Suijlekom
Open Access
Article
  • 65 Downloads

Abstract

We present an operator-algebraic approach to the quantization and reduction of lattice field theories. Our approach uses groupoid \(\hbox {C}^{*}\)-algebras to describe the observables. We introduce direct systems of Hilbert spaces and direct systems of (observable) \(\hbox {C}^{*}\)-algebras, and, dually, corresponding inverse systems of configuration spaces and (pair) groupoids. The continuum and thermodynamic limit of the theory can then be described by taking the corresponding limits, thereby keeping the duality between the Hilbert space and observable \(\hbox {C}^{*}\)-algebra on the one hand, and the configuration space and the pair groupoid on the other. Since all constructions are equivariant with respect to the gauge group, the reduction procedure applies in the limit as well.

References

  1. 1.
    Aastrup, J., Grimstrup, J.M.: Intersecting quantum gravity with noncommutative geometry: a review. SIGMA 8, 018 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aastrup, J., Grimstrup, J.M., Nest, R.: On spectral triples in quantum gravity. II. J. Noncommut. Geom. 3, 47–81 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ashtekar, A., Lewandowski, J.: Representation theory of analytic holonomy \(\text{C}^{\ast }\)- algebras. Knots Quantum Gravity 21–61 (1994)Google Scholar
  4. 4.
    Araki, H., Woods, E.J.: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Math. Phys. 4, 637–662 (1963)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., Thiemann, T.: Coherent state transforms for spaces of connections. J. Funct. Anal. 135, 519–551 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baez, J.C.: Generalized measures in gauge theory. Lett. Math. Phys. 31, 213–223 (1994)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Baez, J.C.: Spin networks in gauge theory. Adv. Math. 117, 253–272 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Buneci, M.R.: Groupoid \(\text{ C }^{\ast }\)-algebras. Surv. Math. Appl. 1, 71–98 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fischer, E., Rudolph, G., Schmidt, M.: A lattice gauge model of singular Marsden–Weinstein reduction. Part I. Kinematics. J. Geom. Phys. 57, 1193–1213 (2007)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Grundling, H., Rudolph, G.: QCD on an infinite lattice. Commun. Math. Phys. 318, 717–766 (2013)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Grundling, H., Rudolph, G.: Dynamics for QCD on an infinite lattice. Commun. Math. Phys. 349, 1163–1202 (2017)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Hall, B.: The Segal–Bargmann “coherent state” transform for compact lie groups. J. Funct. Anal. 122, 103–151 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huebschmann, J.: Singular Poisson-Kähler geometry of stratified Kähler spaces and quantization. Trav. Math. 19, 27–63 (2011)zbMATHGoogle Scholar
  15. 15.
    Huebschmann, J., Rudolph, G., Schmidt, M.: A gauge model for quantum mechanics on a stratified space. Commun. Math. Phys. 286, 459–494 (2009)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kijowski, J.: Symplectic geometry and second quantization. Rep. Math. Phys. 11, 97–109 (1977)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kijowski, J., Rudolph, G.: On the Gauss law and global charge for QCD. J. Math. Phys. 43, 1796–1808 (2002)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kijowski, J., Rudolph, G.: Charge superselection sectors for QCD on the lattice. J. Math. Phys. 46, 032303 (2004)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kijowski, J., Okołów, A.: A modification of the projective construction of quantum states for field theories. J. Math. Phys. 58, 062303 (2017)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kogut, J., Susskind, L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11, 395–408 (1975)ADSCrossRefGoogle Scholar
  21. 21.
    Landsman, N.P.: Rieffel induction as generalized quantum Marsden–Weinstein reduction. J. Geom. Phys. 15, 285–319 (1995)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Landsman, N.P.: Mathematical Topics Between Classical and Quantum Mechanics. Springer, Berlin (1998)CrossRefGoogle Scholar
  23. 23.
    Lanéry, S., Thiemann, T.: Projective limits of state spaces I. Classical formalism. J. Geom. Phys. 111, 6–39 (2017)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Lanéry, S., Thiemann, T.: Projective limits of state spaces II. Quantum formalism. J. Geom. Phys. 116, 10–51 (2017)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Lanéry, S., Thiemann, T.: Projective limits of state spaces III. Toy-models. J. Geom. Phys. 123, 98–126 (2018)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Lanéry, S., Thiemann, T.: Projective limits of state spaces IV. Fractal label sets. J. Geom. Phys. 123, 127–155 (2018)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Lanéry, S.: Projective limits of state spaces: quantum field theory without a vacuum. EJTP 14, 1–20 (2018)Google Scholar
  28. 28.
    Lewandowski, J.: Topological measure and graph-differential geometry on the quotient space of connections. Int. J. Theoret. Phys. 3, 207–211 (1994)ADSMathSciNetGoogle Scholar
  29. 29.
    Maclane, S.: Categories for the Working Mathematician. Springer, Berlin (1998)zbMATHGoogle Scholar
  30. 30.
    Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130 (1974)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Marolf, D., Mourão, J.M.: On the support of the Ashtekar–Lewandowski measure. Comm. Math. Phys. 170, 583–605 (1995)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Muhly, P.S., Renault, J.N., Williams, D.P.: Equivalence and isomorphism for groupoid \(\text{ C }^{\ast }\)- algebras. J. Oper. Theory 17, 3–22 (1987)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Okołów, A.: Construction of spaces of kinematic quantum states for field theories via projective techniques. Class. Quantum Grav. 30, 195003 (2013)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Paterson, A.: Groupoids, Inverse Semigroups, and their Operator Algebras. Birkhäuser, Basel (1999)CrossRefGoogle Scholar
  35. 35.
    Renault, J.: A Groupoid Approach to C\(^\ast \)-Algebras. Springer, Berlin (1980)CrossRefGoogle Scholar
  36. 36.
    Rendall, A.: Comment on a paper of A. Ashtekar and C. J. Isham. Class. Quantum Grav. 10, 605–608 (1993)ADSCrossRefGoogle Scholar
  37. 37.
    Ribes, L., Zalesskii, P.: Profinite Groups. Springer, Berlin (2010)CrossRefGoogle Scholar
  38. 38.
    Rieffel, M.A.: Induced representation of C\(^\ast \)-algebras. Adv. Math. 13, 176–257 (1974)CrossRefGoogle Scholar
  39. 39.
    Rieffel, M.A.: Quantization and operator algebras. XIIth International Congress of Mathematical Physics (ICMP ’97) (Brisbane), 254–260, Int. Press, Cambridge, MA (1999)Google Scholar
  40. 40.
    Rudin, W.: Functional Analysis. McGraw-hill, Inc., NY (1991)zbMATHGoogle Scholar
  41. 41.
    Schwartz, L.: Radon Measures. Oxford University Press, Oxford (1973)zbMATHGoogle Scholar
  42. 42.
    Yngvason, J.: The role of type III factors in quantum field theory. Rep. Math. Phys. 55, 135–147 (2005)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Wilson, K.G.: Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974)ADSCrossRefGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Francesca Arici
    • 1
  • Ruben Stienstra
    • 1
  • Walter D. van Suijlekom
    • 1
  1. 1.Institute for Mathematics, Astrophysics and Particle PhysicsRadboud University NijmegenNijmegenThe Netherlands

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