Correlation functions of charged free boson and fermion systems

TL;DR

This paper explores the correlation functions of charged free boson and fermion systems using the quantum inverse scattering method, revealing their inverse relationship and providing an alternative approach to integrable hierarchies.

Contribution

It introduces operators for charged free bosons and fermions and demonstrates their correlation functions are inverses, offering new insights into integrable systems and hierarchies.

Findings

Correlation functions of fermionic and bosonic systems are inverses.

Operators analogous to those in fermionic systems are constructed for bosons.

Provides an alternative framework for KP and BKP hierarchies.

Abstract

Using the idea of the quantum inverse scattering method, we introduce the operators $\mathbf{B}(x), \mathbf{C}(x)$ and $\mathbf{\tilde{B}}(x), \mathbf{\tilde{C}}(x)$ corresponding to the off-diagonal entries of the monodromy matrix $T$ for the phase model and $i$-boson model in terms of bc fermions and neutral fermions respectively, thus giving alternative treatment of the KP and BKP hierarchies. We also introduce analogous operators $\mathbf{B}^{*}(x)$ and $\mathbf{C}^{*}(x)$ for the charged free boson system and show that they are in complete analogy to those of $bc$ fermionic fields. It is proved that the correlation function $\langle 0|\mathbf{C}(x_N)\cdots\mathbf{C}(x_1)\mathbf{B}(y_1)\cdots $ $\mathbf{B}(y_N)|0\rangle$ in the $bc$ fermionic fields is the inverse of the correlation function $\langle 0|\mathbf{C}^{*}(x_N)\cdots\mathbf{C}^{*}(x_1)\mathbf{B}^{*}(y_1)\cdots \mathbf{B}^{*}(y_N)|0\rangle$ in the charged free bosons.