Correlation functions of charged free boson and fermion systems

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 $\mathbf{B}<sup>{*}(x)$ and $\mathbf{C}<sup>{*}(x)$ 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 $$\langle 0|\mathbf{C}^{{<em>}(x_N)\cdots\mathbf{C}^{</em>}(x_1)\mathbf{B}^{{*}(y_1)\cdots}} \mathbf{B}<sup>{*}(y_N)|0\rangle$$ in the charged free bosons.