Canonical Momenta in Digitized SU(2) Lattice Gauge Theory: Definition and Free Theory
Jakobs, Timo, Garofalo, Marco, Hartung, Tobias, Jansen, Karl, Ostmeyer, Johann, Rolfes, Dominik, Romiti, Simone and Urbach, Carsten (2023) Canonical Momenta in Digitized SU(2) Lattice Gauge Theory: Definition and Free Theory. European Physical Journal C.
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 S3 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 S3 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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