Myopic Bosonization

TL;DR

This paper explores bosonization of fermions in multiple dimensions by attempting to reformulate the fermionic problem using bosonic density fluctuations, revealing both limitations and partial successes in capturing free and interacting theories.

Contribution

It introduces a novel approach to bosonizing fermions across dimensions, addressing inconsistencies and successfully computing the propagator for interacting systems.

Findings

Reproduces key features of free fermionic theory

Identifies inconsistency in kinetic energy operator construction

Calculates the full propagator for interacting fermions

Abstract

As the title suggests, this is an attempt at bosonizing fermions in any number of dimensions without paying attention to the fact that the Fermi surface is an extended object. One is tempted to introduce the density fluctuation and its conjugate and recast the interacting problem in terms of these canonical Bose fields. However, we find that the attempt is short-sighted figuratively as well for the same reason. But surprisingly, this flaw, which manifests itself as an inconsistency between Menikoff-Sharp's construction of the kinetic energy operator in terms of currents and densities, and our ansatz for this operator, is nevertheless able to reproduce(although reluctantly) many salient features of the free theory. Buoyed by this success, we solve the interacting problem and compute the full propagator.