XII - Correlation Function of Fields

Summary

Introduction

The aim of this chapter is to obtain the determinant representation for the equal-time correlation function 〈Ψ^†(x)Ψ(0)〉 in the nonrelativistic one-dimensional Bose gas (NS model). To apply the approach of the previous chapters, one has to introduce local fields Ψ^†(x), Ψ(0) into the two-site generalized model. This is done in section 1. In section 2, the formula for the matrix elements of the operator Ψ^†(x)Ψ(0) in the generalized model are derived. In terms of the auxiliary quantum fields introduced in section 3, this formula is transformed to the vacuum mean value of the determinant with respect to the vacuum in the auxiliary Fock space. The determinant representation for the mean value of the operator Ψ^†(x)Ψ(0) with respect to the N-particle Bethe eigenstate is derived in section 4. The derivation is similar to that of the previous chapter. It is also similar to that of section 6 of Chapter IX. The derivation is based on the representation of the determinant of the sum of two matrices in terms of the minors of the individual matrices. The explicit formula is given in the Appendix to Chapter IX. In section 5 expressions for the correlation function in the ground state of the one-dimensional Bose gas (zero and finite temperature) are obtained and the thermodynamic limit is also considered.

Local fields in the generalized model

To consider local quantum fields Ψ^†(x)Ψ(0) in the frame of the two-site generalized model, one has to clarify the structure of the monodromy matrices T(1|λ), T(2|λ) (see section XI.1, formulæe (XI.1.6)–(XI.1.12)).