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

Timo Jakobs; Marco Garofalo; Tobias Hartung; Karl Jansen; Johann Ostmeyer; Dominik Rolfes; Simone Romiti; Carsten Urbach

doi:10.1140/epjc/s10052-023-11829-9

2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2023) 83: 669
https://doi.org/10.1140/epjc/s10052-023-11829-9

Regular Article - Theoretical Physics

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

Timo Jakobs¹^,2, Marco Garofalo¹^,2, Tobias Hartung³, Karl Jansen⁴^,5, Johann Ostmeyer⁶, Dominik Rolfes¹^,2, Simone Romiti¹^,2 and Carsten Urbach¹^,2^h

¹ Helmholtz-Institut für Strahlen- und Kernphysik, University of Bonn, Nussallee 14-16, 53115, Bonn, Germany
² Bethe Center for Theoretical Physics, University of Bonn, Nussallee 12, 53115, Bonn, Germany
³ Northeastern University-London, Devon House, St Katharine Docks, E1W 1LP, London, UK
⁴ CQTA, DESY Zeuthen, Platanenallee 6, 15738, Zeuthen, Germany
⁵ Computation-Based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Street, 2121, Nicosia, Cyprus
⁶ Department of Mathematical Sciences, University of Liverpool, L69 7ZL, Liverpool, UK

^h urbach@hiskp.uni-bonn.de

Received: 18 April 2023
Accepted: 11 July 2023
Published online: 27 July 2023

Abstract

Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space $H$ $\mathcal {H}$ . 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 $H$ $\mathcal {H}$ . This is achieved by discretising the sphere $S_{3}$ $$S_3$$ 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 $S_{3}$ $$S_3$$ 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.

© The Author(s) 2023

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