Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory

Abstract

Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space mathcal {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 mathcal {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.