Canonical Transformations and Loop Formulation of SU(N) Lattice Gauge Theories

TL;DR

This paper reformulates SU(N) lattice gauge theories using canonical transformations to fundamental loop and string flux operators, revealing new insights into their structure, dynamics, and spectrum, especially for SU(2).

Contribution

It introduces a novel loop formulation of SU(N) lattice gauge theories via canonical transformations, decoupling string degrees of freedom and providing a new perspective on gauge invariance and dynamics.

Findings

SU(N) string degrees of freedom become cyclic and decouple from physical space.

The Hamiltonian in loop variables exhibits global SU(N) invariance without gauge fields.

In the weak coupling limit, the dynamics map to an SU(N) spin Hamiltonian with nearest neighbor interactions.

Abstract

We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space ${\cal H}^p$. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators over the entire lattice are worked out in a self consistent manner. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the $(1/g^2)$ magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the $(g^2)$ electric field terms describe all their interactions. In the weak coupling ($g^2 \rightarrow 0$) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group $SO(4,2)$ in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.