Canonical Momenta in Digitized SU(2) Lattice Gauge Theory: Definition and Free Theory

• URL: http://arxiv.org/abs/2304.02322v2
• Date: Fri, 28 Jul 2023 13:21:17 GMT
• Title: Canonical Momenta in Digitized SU(2) Lattice Gauge Theory: Definition and Free Theory
• Authors: Timo Jakobs and Marco Garofalo and Tobias Hartung and Karl Jansen and Johann Ostmeyer and Dominik Rolfes and Simone Romiti and Carsten Urbach
• Abstract summary: Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space H. Here we give a prescription for gauge links and canonical momenta of an SU(2) gauge theory. We show that the fundamental commutation relations are fulfilled up to discretisation artefacts.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space H. Here we give a prescription for gauge links and canonical momenta of an SU(2) gauge theory, such that the matrix representation of the former is diagonal in H. This is achieved by discretising the sphere $S_3$ isomorphic to SU(2) and the corresponding directional derivatives. We show that the fundamental commutation relations are fulfilled up to discretisation artefacts. Moreover, we directly construct the Casimir operator corresponding to the Laplace-Beltrami operator on $S_3$ and show that the spectrum of the free theory is reproduced again up to discretisation effects. Qualitatively, these results do not depend on the specific discretisation of SU(2), but the actual convergence rates do.

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