Particle-Hole Transformation in the Continuum and Determinantal Point Processes

Abstract

Let X be an underlying space with a reference measure sigma . Let K be an integral operator in L^2(X,sigma ) with integral kernel K(x, y). A point process mu on X is called determinantal with the correlation operator K if the correlation functions of mu are given by k^{(n)}(x_1,dots ,x_n)={text {det}}[K(x_i,x_j)]_{i,j=1,dots ,n}. It is known that each determinantal point process with a self-adjoint correlation operator K is the joint spectral measure of the particle density rho (x)=mathcal A^+(x)mathcal A^-(x) (xin X), where the operator-valued distributions mathcal A^+(x), mathcal A^-(x) come from a gauge-invariant quasi-free representation of the canonical anticommutation relations (CAR). If the space X is discrete and divided into two disjoint parts, X_1 and X_2, by exchanging particles and holes on the X_2 part of the space, one obtains from a determinantal point process with a self-adjoint correlation operator K the determinantal point process with the J-self-adjoint correlation operator widehat{K}=KP_1+(1-K)P_2. Here P_i is the orthogonal projection of L^2(X,sigma ) onto L^2(X_i,sigma ). In the case where the space X is continuous, the exchange of particles and holes makes no sense. Instead, we apply a Bogoliubov transformation to a gauge-invariant quasi-free representation of the CAR. This transformation acts identically on the X_1 part of the space and exchanges the creation operators mathcal A^+(x) and the annihilation operators mathcal A^-(x) for xin X_2. This leads to a quasi-free representation of the CAR, which is not anymore gauge-invariant. We prove that the joint spectral measure of the corresponding particle density is the determinantal point process with the correlation operator widehat{K}.