Speaker: Timo Jakobs (HISKP)
Title: 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 discretizing 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 ot the Laplace-Beltrami operator on $S_3$ and show that the spectrum of the free theory is reproduced again up to discretisation effects.