Quantum lattice gauge fields and groupoid \(\mathrm{C}^{\ast}\)-algebras. (English) Zbl 1402.81202
Summary: We present an operator-algebraic approach to the quantization and reduction of lattice field theories. Our approach uses groupoid \(\mathrm{C}^{\ast}\)-algebras to describe the observables. We introduce direct systems of Hilbert spaces and direct systems of (observable) \(\mathrm{C}^{\ast}\)-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 \(\mathrm{C}^{\ast}\)-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.
MSC:
| 81T25 | Quantum field theory on lattices |
| 81T70 | Quantization in field theory; cohomological methods |
| 46L05 | General theory of \(C^*\)-algebras |
| 18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |
| 81T27 | Continuum limits in quantum field theory |
| 46L60 | Applications of selfadjoint operator algebras to physics |
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