Fermions in the harmonic potential and string theory

TL;DR

This paper derives an exact bosonization of fermions in a harmonic potential, revealing a coupled scalar and Liouville field theory that resembles string theory formulations and offers insights into space-time dual descriptions.

Contribution

It provides a novel collective field theory for fermions in harmonic potential, connecting it to string theory and D-brane representations, with potential generalizations to c=1 string theory.

Findings

Exact bosonization of fermions in harmonic potential

Identification of a coupled scalar and Liouville field theory

Representation of fermion operators as vertex operators in string theory

Abstract

We explicitly derive collective field theory description for the system of fermions in the harmonic potential. This field theory appears to be a coupled system of free scalar and (modified) Liouville field. This theory should be considered as an exact bosonization of the system of non-relativistic fermions in the harmonic potential. Being surprisingly similar to the world-sheet formulation of c=1 string theory, this theory has quite different physical features and it is conjectured to give space-time description of the string theory, dual to the fermions in the harmonic potential. A vertex operator in this theory is shown to be a field theoretical representation of the local fermion operator, thus describing a D0 brane in the string language. Possible generalization of this result and its derivation for the case of c=1 string theory (fermions in the inverse harmonic potential) is discussed.