论文标题
连续和确定点过程中的粒子孔转换
Particle-hole transformation in the continuum and determinantal point processes
论文作者
论文摘要
令$ x $为带有参考度量$σ$的基础空间。让$ k $是$ l^2（x，σ）$的积分运算符，带有积分内核$ k（x，y）$。 $ x $上的点过程$μ$在$μ$的相关函数由$ k^{（n）}（x_1，x_1，\ dots，x_n）= \ propatatOrnAme {det} [k（x_i，x_j jj）_ = 1 = 1 = 1，称为n.众所周知，每个具有自动化相关操作员$ k $的确定点过程是粒子密度$ρ（x）= \ MATHCAL A^+（x）\ MATHCAL A^ - （x）$（$ x）$（in x $中）的联合光谱度量，在pertial-valued分发$ \ Mathcal a^$ a^$ a^+（x）$（x $）中规范反通信关系（CAR）的规格不变的准无准代表。如果Space $ x $是离散的，并通过交换$ x_2 $部分的粒子和孔的空间部分分为两个不同的零件，$ x_1 $和$ x_2 $ k = kp_1+（1-k）p_2 $。这里$ p_i $是$ l^2（x，σ）$的正交投影到$ l^2（x_i，σ）$。在空间$ x $连续的情况下，颗粒和孔的交换没有意义。取而代之的是，我们将Bogoliubov转换应用于汽车的无不分因素准代表。此转换在空间的$ x_1 $一部分上相同，并交换创建运营商$ \ mathcal a^+（x）$和an灭操作员$ \ mathcal a^ - （x）$ in x_2 $中的$ x \。这导致了汽车的准无准代表，这已经不再是衡量的。我们证明，相应粒子密度的联合光谱度量是具有相关算子$ \ widehat k $的确定点过程。
Let $X$ be an underlying space with a reference measure $σ$. Let $K$ be an integral operator in $L^2(X,σ)$ with integral kernel $K(x,y)$. A point process $μ$ on $X$ is called determinantal with the correlation operator $K$ if the correlation functions of $μ$ are given by $k^{(n)}(x_1,\dots,x_n)=\operatorname{det}[K(x_i,x_j)]_{i,j=1,\dots,n}$. It is known that each determinantal point process with a self-adjoint correlation operator $K$ is the joint spectral measure of the particle density $ρ(x)=\mathcal A^+(x)\mathcal A^-(x)$ ($x\in X$), 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 $\widehat K=KP_1+(1-K)P_2$. Here $P_i$ is the orthogonal projection of $L^2(X,σ)$ onto $L^2(X_i,σ)$. 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 $X_1$ 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 $\widehat K$.
