Construction of a non-standard quantum field theory through a generalized Heisenberg algebra

TL;DR

This paper develops a novel quantum field theory based on a generalized Heisenberg algebra derived from the Klein-Gordon equation on a finite interval, leading to a unique mass spectrum for physical excitations.

Contribution

It introduces a new algebraic framework for quantum fields using a generalized Heisenberg algebra tailored to a finite interval, extending standard quantum field theory methods.

Findings

Derived a Heisenberg-like algebra from the Klein-Gordon equation on a finite interval.

Constructed a 3+1 dimensional quantum field theory using this algebra.

Obtained a discrete mass spectrum for excitations in a finite potential well.

Abstract

We construct a Heisenberg-like algebra for the one dimensional quantum free Klein-Gordon equation defined on the interval of the real line of length $L$. Using the realization of the ladder operators of this type Heisenberg algebra in terms of physical operators we build a 3+1 dimensional free quantum field theory based on this algebra. We introduce fields written in terms of the ladder operators of this type Heisenberg algebra and a free quantum Hamiltonian in terms of these fields. The mass spectrum of the physical excitations of this quantum field theory are given by $\sqrt{n^2 \pi^2/L^2+m_q^2}$, where $n= 1,2,...$ denotes the level of the particle with mass $m_q$ in an infinite square-well potential of width $L$.