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 SuijlekomEmail author
Open Access


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.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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
    Email author
  1. 1.Institute for Mathematics, Astrophysics and Particle PhysicsRadboud University NijmegenNijmegenThe Netherlands

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