Gauge-invariant Hamiltonian formulation of lattice Yang-Mills theory and the Heisenberg double

TL;DR

This paper extends the Hamiltonian formulation of lattice Yang-Mills theory by incorporating the Heisenberg double, a fundamental structure in Lie-Poisson and quantum group theory, providing a more general framework.

Contribution

It introduces a generalized Hamiltonian formulation of lattice Yang-Mills theory using the Heisenberg double, broadening the mathematical foundation of the theory.

Findings

Generalization of Hamiltonian formulation to Heisenberg double

Connection between lattice gauge theory and quantum groups

Potential new approaches to quantization

Abstract

It it known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge $A_{0}=0$ one should place on every link the cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the paper that there is a generalization of the usual Hamiltonian formulation to the case of the Heisenberg double.