Expectation values of coherent states for SU(2) Lattice Gauge Theories

Abstract

This article investigates properties of semiclassical Gauge Field Theory Coherent States for general quantum gauge theories. Useful, e.g., for the canonical formulation of Lattice Gauge Theories these states are labelled by a point in the classical phase space and constructed such that the expectation values of the canonical operators are sharply peaked on said phase space point.For the case of the non-abelian gauge group SU(2), we will explicitly compute the expectation value of general polynomials including the first order quantum corrections. This allows asking more precise questions about the quantum fluctuations of any given semiclassical system. I. INTRODUCTIONOne of the astonishing facts of modern physics is that many of its most powerful theories can be described using gauge symmetries. To understand their quantisation, a promising avenue comes in the study of Lattice Gauge Theories (LGT) [1][2][3][4][5][6]. On the one hand, high performance computations in lattice quantum chromodynamics are the main tool to aid the experiments in particle and nuclear physics [7,8]. On the other hand, LGT provides a theoretical framework that is ideally suited to make an impact on models trying to explore new physics beyond and within the standard model [9,10]. In the recent decades, methods from LGT have been further developed in the emergent field of quantum gravity, as it transpired that general relativity can be understood as an SU(2) gauge theory as well [11][12][13]. Further progress is much needed, since despite the active research on the quantisation of gauge theories not a single interacting 4-dimensional quantum Yang-Mills theory obeying the Wightman axioms has been constructed as of today. It remains one of the open millennium problems of the Clay Mathematical Institute [14]. A possible route for attacking this caveat with the needed mathematical rigour might come in the Hamiltonian formulation to gauge theories. The latter has originally been developed by Kogut and Susskind for pure quantum Yang-Mills theories [15] and enabled the construction of a well-defined kinematical Hilbert space, where the natural Haar measure * Electronic address: liegener1@lsu.edu † Electronic address: ernst-albrecht.zwicknagel@fau.deon the compact gauge group can be used in order to define the Hilbert space measure. Nonetheless, implementing the dynamics of the theory poses a challenge: while a regularised Hamiltonian in presence of a finite ultraviolet cutoff is well defined, the necessary continuum limit is in general problematic. This caveat is hoped to be overcome in the renormalisation group program [16][17][18][19][20], of which extensions to the Hamiltonian sector are currently under development [21][22][23].The present paper, however, will focus its analysis on the kinematical Hilbert space and the question of how semi-classical field configurations can be recovered. In the canonical setting this is envisioned to be achieved by using so-called coherent states. By this we mean states in the Hilbert space that are sharply peak...

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