论文标题
CCR代数上的Hafnian点过程和不准确的状态
Hafnian point processes and quasi-free states on the CCR algebra
论文作者
论文摘要
令$ x $为本地紧凑的波兰空间，而$σ$ $ x $上的非原子参考度量（通常为$ x = \ mathbb r^d $，$σ$是lebesgue措施）。令$ x^2 \ ni（x，y）\ mapsto \ mathbb k（x，y）\ in \ mathbb c^{2 \ times 2} $ be a $ 2 \ times 2 $ -matrix-valued内核，可满足$ \ mathbb k^t（x，y）= \ mathbb k（x，y）我们说，如果每个$ n $ in \ mathbb n $，$ n $ th $ n $ th的相关函数（相对于$σ^^{\ otimies n} $，则为$ n $ th相关函数，$ n $ th的相关函数， $ k^{（n）}（x_1，\ dots，x_n）= \ operatotorname {haf} \ big [\ mathbb k（x_i，x_j）\ big] _ {i，j = 1，j = 1，\ dots，\ dots，n} \，$。这里$ \ operatorname {haf}（c）$表示对称矩阵$ c $的hafnian。 HAFNIAN点过程包括特殊案例的永久性和2-永久点过程。 COX流程$π_r$是$ x $的泊松点过程，随机强度$ r（x）$。令$ g（x）$为满足$ \int_Δ\ mathbb e（| g（x）|^2）σ（dx）<\ infty $的$ x $上的复杂高斯字段。然后，带有$ r（x）= | g（x）|^2 $的COX进程$π_r$是一个hafnian点过程。本文的主要结果是，每个这样的过程$π_r$是在对称的fock空间中代表的严格定义粒子密度（CCR）的粒子密度的联合光谱度量，为此，CCR代数上相应的真空状态是无用的。
Let $X$ be a locally compact Polish space and $σ$ a nonatomic reference measure on $X$ (typically $X=\mathbb R^d$ and $σ$ is the Lebesgue measure). Let $X^2\ni(x,y)\mapsto\mathbb K(x,y)\in\mathbb C^{2\times 2}$ be a $2\times 2$-matrix-valued kernel that satisfies $\mathbb K^T(x,y)=\mathbb K(y,x)$. We say that a point process $μ$ in $X$ is hafnian with correlation kernel $\mathbb K(x,y)$ if, for each $n\in\mathbb N$, the $n$th correlation function of $μ$ (with respect to $σ^{\otimes n}$) exists and is given by $k^{(n)}(x_1,\dots,x_n)=\operatorname{haf}\big[\mathbb K(x_i,x_j)\big]_{i,j=1,\dots,n}\,$. Here $\operatorname{haf}(C)$ denotes the hafnian of a symmetric matrix $C$. Hafnian point processes include permanental and 2-permanental point processes as special cases. A Cox process $Π_R$ is a Poisson point process in $X$ with random intensity $R(x)$. Let $G(x)$ be a complex Gaussian field on $X$ satisfying $\int_Δ\mathbb E(|G(x)|^2)σ(dx)<\infty$ for each compact $Δ\subset X$. Then the Cox process $Π_R$ with $R(x)=|G(x)|^2$ is a hafnian point process. The main result of the paper is that each such process $Π_R$ is the joint spectral measure of a rigorously defined particle density of a representation of the canonical commutation relations (CCR), in a symmetric Fock space, for which the corresponding vacuum state on the CCR algebra is quasi-free.
