Harmonic Oscillator Prepotentials in SU(2) Lattice Gauge Theory

TL;DR

This paper reformulates SU(2) lattice gauge theory using harmonic oscillator prepotentials, revealing new gauge-invariant operators and clarifying the role of U(1) confinement in the strong coupling limit.

Contribution

It introduces a novel prepotential formulation of SU(2) lattice gauge theory, simplifying the Hamiltonian and characterizing the physical Hilbert space with gauge-invariant integers.

Findings

Reformulation of SU(2) Hamiltonian in terms of harmonic oscillators.

Identification of gauge-invariant operators in the prepotential framework.

Clarification of U(1) gauge invariance and confinement mechanisms.

Abstract

We write the SU(2) lattice gauge theory Hamiltonian in (d+1) dimensions in terms of prepotentials which are the SU(2) fundamental doublets of harmonic oscillators. The Hamiltonian in terms of prepotentials has $SU(2) \otimes U(1)$ local gauge invariance. In the strong coupling limit, the color confinement in this formulation is due to the U(1) gauge group. We further solve the $SU(2) \otimes U(1)$ Gauss law to characterize the physical Hilbert space in terms of a set of gauge invariant integers. We also obtain certain novel gauge invariant operators in terms of the above oscillators. The corresponding prepotential formulation of SU(N) lattice gauge theory is also simple and discussed.