A non covariant fermionic determinant and its connection to Luttinger systems

TL;DR

This paper develops a non covariant fermionic determinant approach for (1+1)-dimensional non relativistic systems, successfully connecting path-integral and operator bosonization methods for Luttinger liquids.

Contribution

It introduces a heat-kernel regularization method tailored for non covariant fermionic determinants, aligning bosonization results with physical dispersion relations.

Findings

Derived the correct dispersion relations for bosonic excitations.

Established a precise heat-kernel regularization for non covariant systems.

Achieved full agreement between path-integral and operator bosonization approaches.

Abstract

We consider a fermionic determinant associated to a non covariant Quantum Field Theory used to describe a non relativistic system in (1+1) dimensions. By exploiting the freedom that arises when Lorentz invariance is not mandatory, we determine the heat-kernel regulating operator so as to reproduce the correct dispersion relations of the bosonic excitations. We also derive the Hamiltonian of the functionally bosonized model and the corresponding currents. In this way we were able to establish the precise heat-kernel regularization that yields complete agreement between the path-integral and operational approaches to the bosonization of the Tomonaga-Luttinger model.