Defining < A^2 > in the finite volume hamiltonian formalism

TL;DR

This paper explores how to define and compute the gauge-invariant expectation value of the squared gauge field in non-abelian gauge theories within the finite volume Hamiltonian formalism, addressing non-locality and Gribov copies.

Contribution

It introduces a method to implement the Hamiltonian on the fundamental domain with boundary conditions to handle gauge invariance and Gribov copies in finite volume non-abelian gauge theories.

Findings

Proposes a framework for gauge-invariant ||A||^2 in finite volume

Addresses the role of the fundamental domain and boundary conditions

Discusses the relation to Gribov copies

Abstract

It is shown how in principle for non-abelian gauge theories it is possible in the finite volume hamiltonian framework to make sense of calculating the expectation value of ||A||^2=\int d^3x(A^a_i(x))^2. Gauge invariance requires one to replace ||A||^2 by its minimum over the gauge orbit, which makes it a highly non-local quantity. We comment on the difficulty of finding a gauge invariant expression for ||A||^2_{min} analogous to that found for the abelian case, and the relation of this question to Gribov copies. We deal with these issues by implementing the hamiltonian on the so-called fundamental domain, with appropriate boundary conditions in field space, essential to correctly represent the physics of the problem.