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Hamiltonian simulations of quantum systems require a finite-dimensionalrepresentation of the operators acting on the Hilbert space H. Here we give aprescription 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 isachieved by discretising the sphere $S_3$ isomorphic to SU(2) and thecorresponding directional derivatives. We show that the fundamental commutationrelations are fulfilled up to discretisation artefacts. Moreover, we directlyconstruct the Casimir operator corresponding to the Laplace-Beltrami operatoron $S_3$ and show that the spectrum of the free theory is reproduced again upto discretisation effects. Qualitatively, these results do not depend on thespecific discretisation of SU(2), but the actual convergence rates do.

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