Eigenvector and eigenvalue problem for 3D bosonic model

TL;DR

This paper reformulates free field theories on lattices as evolution models, constructs formal eigenvectors for the evolution operator, and applies this to analyze the spectrum and partition function of a 3D bosonic model.

Contribution

It introduces a new approach to analyze lattice free field theories via eigenvector construction of the evolution operator, exemplified on a 3D bosonic model.

Findings

Constructed formal eigenvectors for the evolution operator.

Derived spectrum and partition function from eigenvectors.

Applied method to Bazhanov--Baxter's free bosonic model.

Abstract

In this paper we reformulate free field theory models defined on the rectangular $D+1$ dimensional lattices as $D+1$ evolution models. This evolution is in part a simple linear evolution on free (``creation'' and ``annihilation'') operators. Formal eigenvectors of this linear evolution can be directly constructed, and them play the role of the ``physical'' creation and annihilation operators. These operators being completed by a ``physical'' vacuum vector give the spectrum of the evolution operator, as well as the trace of the evolution operator give a correct expression for the partition function. As an example, Bazhanov -- Baxter's free bosonic model is considered.