On U(1) Gauge Theory Transfer-Matrix in Fourier Basis

TL;DR

This paper analyzes the transfer-matrix of U(1) lattice gauge theory in the Fourier basis, revealing its block structure, properties of eigenvectors, and the ground-state characteristics, supported by explicit expressions and numerical tests.

Contribution

It provides a detailed analysis of the transfer-matrix structure and properties in U(1) gauge theory within the Fourier basis, including explicit formulas and numerical validation.

Findings

Transfer-matrix is block-diagonal.

Eigenvectors within a block can be derived from any known vector.

Ground-state resides in the zero-mode's block.

Abstract

The properties of the transfer-matrix of U(1) lattice gauge theory in the Fourier basis are explored. Among other statements it is shown: 1) the transfer-matrix is block-diagonal, 2) all consisting vectors of a block are known based on an arbitrary block vector, 3) the ground-state belongs to the zero-mode's block. The emergence of maximum-points in matrix-elements as functions of the gauge coupling is clarified. Based on explicit expressions for the matrix-elements we present numerical results as tests of our statements.