Gauss’s law, duality, and the Hamiltonian formulation of U(1) lattice gauge theory

David B Kaplan,Jesse R Stryker

doi:10.1103/physrevd.102.094515

Abstract

Quantum computers have the potential to explore the vast Hilbert space of entangled states that play an important role in the behavior of strongly interacting matter. This opportunity motivates reconsidering the Hamiltonian formulation of gauge theories, with a suitable truncation scheme to render the Hilbert space finite-dimensional. Conventional formulations lead to a Hilbert space largely spanned by unphysical states; given the current inability to perform large scale quantum computations, we examine here how one might restrict wave function evolution entirely or mostly to the physical subspace. We consider such constructions for the simplest of these theories containing dynamical gauge bosons -- U(1) lattice gauge theory without matter in $d=2,3$ spatial dimensions -- and find that electric-magnetic duality naturally plays an important role. We conclude that this approach is likely to significantly reduce computational overhead in $d=2$ by a reduction of variables and by allowing one to regulate magnetic fluctuations instead of electric. The former advantage does not exist in $d=3$, but the latter might be important for asymptotically-free gauge theories.

Highlights

Wilson’s path integral construction provides a nonperturbative definition of lattice gauge theory and an efficient computational tool for some types of calculations
There are a couple of drawbacks to this approach which we address here: (i) It appears preferable to work entirely in Hphys if possible, in order to require fewer qubits and to avoid computational errors causing states initially in Hphys to evolve into the much larger space of unphysical states; (ii) A cutoff on electric fields is appropriate for strong coupling, for which electric fluctuations are suppressed, but is not ideal for weak coupling, such as one would encounter in the continuum limit for any gauge theory in d < 3, or asymptotically-free theories in d 1⁄4 3, where d is the spatial dimension
In this paper we examine these issues in two of the simplest gauge theories—U(1) theories without matter in d 1⁄4 2 and d 1⁄4 3—and find that both concerns lead directly to a formulation of the electromagnetic dual theory

Summary

Published by the American Physical Society

Operator Ei is the conjugate momentum for the vector potential Ai. Here we consider compact U(1) gauge theory formulated on a spatial lattice L with lattice spacing as, periodic boundary conditions, and coordinates fn; l; p; cg for sites, links, plaquettes, and cubes, respectively. Most states in H violate Eq (5) and are unphysical, and simulating Hamiltonian evolution in H will use more qubits on a quantum computer than physically necessary To better understand this constraint, consider the lattice L with periodic boundary conditions in d 1⁄4 2, 3 dimensions with n sites, and l 1⁄4 nd links, p 1⁄4 ndðd−1Þ=2 plaquettes, and c 1⁄4 ndðd − 1Þðd − 2Þ=6 cubes. By ignoring the redundancy in our definition of jAiν in Eq (7), we can treat An⋆ as an independent integervalued variable on each site and use product states ⊗n⋆ jAn⋆i as a basis for a Hilbert space H⋆ In terms of these states, we can define the two local coordinate and shift operators, Un⋆ and Qn⋆, living on sites of the dual lattice as. We can define link operators Ul⋆ and Ql⋆ operators exactly as in Eq (8), and the dual

Hamiltonian is computed to be

Findings

CONCLUSIONS