Particle-Hole Transformation in the Continuum and Determinantal Point Processes

Abstract

AbstractLet X be an underlying space with a reference measure $$\sigma $$ σ . Let K be an integral operator in $$L^2(X,\sigma )$$ L 2 ( X , σ ) with integral kernel K(x, y). A point process $$\mu $$ μ on X is called determinantal with the correlation operator K if the correlation functions of $$\mu $$ μ are given by $$k^{(n)}(x_1,\dots ,x_n)={\text {det}}[K(x_i,x_j)]_{i,j=1,\dots ,n}$$ k ( n ) ( x 1 , ⋯ , x n ) = det [ K ( x i , x j ) ] i , j = 1 , ⋯ , 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 $$\rho (x)=\mathcal A^+(x)\mathcal A^-(x)$$ ρ ( x ) = A + ( x ) A - ( x ) ($$x\in X$$ x ∈ X ), where the operator-valued distributions $$\mathcal A^+(x)$$ A + ( x ) , $$\mathcal A^-(x)$$ 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, $$X_1$$ X 1 and $$X_2$$ X 2 , by exchanging particles and holes on the $$X_2$$ 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 $$\widehat{K}=KP_1+(1-K)P_2$$ K ^ = K P 1 + ( 1 - K ) P 2 . Here $$P_i$$ P i is the orthogonal projection of $$L^2(X,\sigma )$$ L 2 ( X , σ ) onto $$L^2(X_i,\sigma )$$ L 2 ( X i , σ ) . 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 $$X_1$$ X 1 part of the space and exchanges the creation operators $$\mathcal A^+(x)$$ A + ( x ) and the annihilation operators $$\mathcal A^-(x)$$ A - ( x ) for $$x\in X_2$$ x ∈ 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 $$\widehat{K}$$ K ^ .