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 $k^{(n)}(x_1,\dots ,x_n)=\det [K(x_i,x_j){]}_{i,j=1,\dots ,n}$. It is known that each determinantal point process with a self-adjoint correlation operator $K$ is the joint spectral measure of the particle density $\rho (x)={\mathcal{A}}^{+}(x){\mathcal{A}}^{-}(x)$ ($x\in X$), 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 $\hat{K}=K{P}_1+(1-K)P_2$. 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 $x\in X_2$. 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 $\hat{K}$.