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arXiv:hep-th/9109013

Collective Field Representation of Nonrelativistic Fermions in (1+1) Dimensions

Dimitra Karabali A

Published Sep 10, 1991• 2 citations
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Abstract

A collective field formalism for nonrelativistic fermions in (1+1) dimensions is presented. Applications to the D=1 hermitian matrix model and the system of one-dimensional fermions in the presence of a weak electromagnetic field are discussed.
DOI
10.48550/arxiv.hep-th/9109013

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We present a collective field formalism for nonrelativistic fermions in one spatial dimension. A bosonization technique is used to convert the quantum mechanical fermionic problem to a bosonic one, which is further described as a second quantized Schrödinger field theory. A formulation in terms of current and density variables gives rise to the collective field representation. Applications of our formalism to the D=1 Hermitian matrix model and the system of one-dimensional fermions in the presence of a weak electromagnetic field are discussed.
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The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.
We first present a general method for extracting collective variables out of nonrelativistic fermions by extending the gauge theory of collective coordinates to fermionic systems.We then apply the method to a system of non-interacting flavored fermions confined in a one-dimensional flavor-independent potential.In the limit of a large number of particles we obtain a Lagrangian with the Wess-Zumino-Witten term, which is the wellknown Lagrangian describing the non-Abelian bosonization of chiral fermions on a circle.The result is universal and does not depend on the details of the confining potential.