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 $\rho (x)=\mathcal A^+(x)\mathcal A^-(x)$ ($x\in X$), 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 $L^2(X,\sigma )$ 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 $\widehat{K}$.

1 Introduction

Let X be an underlying space with a reference measure $\sigma$. (Typically $X=\mathbb {R}^d$ and $\sigma$ is the Lebesgue measure.) Let $\Gamma _X$ denote the space of configurations in X, i.e., locally finite subsets of X. A point process in X is a probability measure on $\Gamma _X$, see e.g. [13]. A point process $\mu$ is called determinantal if the correlation functions of $\mu$ are given by

$$\begin{aligned} k_\mu ^{(n)}(x_1,\dots ,x_n)=\det \big [K(x_i,x_j)\big ]_{i,j=1}^n,\quad n\in \mathbb N,\end{aligned}$$

(1)

see e.g. [6, 14, 23, 26]. The function K(x, y) is called the correlation kernel of $\mu$. To study $\mu$, one usually considers a (bounded) integral operator K in the (complex) space $\mathcal H:=L^2(X,\sigma )$ with integral kernel K(x, y). One calls K the correlation operator of $\mu$.

Assume that the correlation kernel K(x, y) is Hermitian, i.e., $K(x,y)=\overline{K(y,x)}$, equivalently the operator K is self-adjoint. Then, the Macchi–Soshnikov theorem [23, 26] gives a necessary and sufficient condition of existence of a determinantal point process $\mu$ with the correlation kernel K(x, y).

It was shown in [18, 20] (see also [6, 27]) that this point process $\mu$ is the joint spectral measure of the particle density of a gauge-invariant quasi-free representation of the canononical anticommutation relations (CAR). More precisely, assume that operators $\mathcal A^+(\varphi )$ and $\mathcal A^-(\varphi )$ $(\varphi \in \mathcal H)$ satisfy the CAR:

$$\begin{aligned}{} & {} \mathcal A^-(\varphi )=\left( \mathcal A^+(\varphi )\right) ^*, \end{aligned}$$

(2)

$$\begin{aligned}{} & {} \{ \mathcal A^+(\varphi ), \mathcal A^+(\psi )\}=\{ \mathcal A^-(\varphi ), \mathcal A^-(\psi )\}=0,\qquad \{ \mathcal A^-(\varphi ), \mathcal A^+(\psi )\}=(\psi ,\varphi )_{\mathcal H}. \end{aligned}$$

(3)

Here $\{A,B\}=AB+BA$ is the anticommutator. Assume that the operators $\mathcal A^+(\varphi )$ and $\mathcal A^-(\varphi )$ act in $\mathcal{A}\mathcal{F}(\mathcal H\oplus \mathcal H)$, the antisymmetric Fock space over $\mathcal H\oplus \mathcal H$. Let $\textbf{A}$ be the CAR $*$-algebra generated by these operators, and let $\tau$ be the vacuum state on $\textbf{A}$. Furthermore, assume that the operators $\mathcal A^+(\varphi )$ and $\mathcal A^-(\varphi )$ are such that the state $\tau$ is gauge-invariant quasi-free, i.e.,

$$\begin{aligned} \tau \big ( \mathcal A^+(\varphi _m)\cdots \mathcal A^+(\varphi _1) \mathcal A^-(\psi _1)\cdots \mathcal A^-(\psi _n)\big )=\delta _{m,n}\det \big [(K\varphi _i,\psi _j)_{\mathcal H})\big ]_{i,j=1}^n, \end{aligned}$$

(4)

where $\delta _{m,n}$ is the Kronecker symbol and K is the self-adjoint bounded linear operator in $\mathcal H$ satisfying

$$\begin{aligned} (K\varphi ,\psi )_{\mathcal H}=\tau \big ( \mathcal A^+(\varphi ) \mathcal A^-(\psi )\big ),\quad \varphi ,\psi \in \mathcal H. \end{aligned}$$

(5)

See [2] or [12, Subsection 5.2.3].

Define operator-valued distributions $\mathcal A^+(x)$ and $\mathcal A^-(x)$ on X by

$$\begin{aligned} \mathcal A^+(\varphi )=\int _X \varphi (x) \mathcal A^+(x)\sigma (dx),\quad \mathcal A^-(\varphi )=\int _X\overline{\varphi (x)} \mathcal A^-(x)\sigma (dx),\quad \varphi \in \mathcal H.\end{aligned}$$

(6)

The corresponding particle density is formally defined as the operator-valued distribution $\rho (x):= \mathcal A^+(x) \mathcal A^-(x)$, and in the smeared form,

$$\begin{aligned} \rho (\Delta )=\int _\Delta \rho (x)\sigma (dx)=\int _\Delta \mathcal A^+(x) \mathcal A^-(x)\sigma (dx), \end{aligned}$$

where $\Delta \subset X$ is measurable and pre-compact. Note that, at least formally, each operator $\rho (\Delta )$ is Hermitian and for any sets $\Delta _1$ and $\Delta _2$, the operators $\rho (\Delta _1)$ and $\rho (\Delta _2)$ commute. The main result of [20] was that, if K is an integral operator and its integral (Hermitian) kernel K(x, y) is such that the corresponding determinantal point process $\mu$ exists, then the operators $\rho (\Delta )$ are well-defined, essentially self-adjoint, commuting, and furthermore

$$\begin{aligned} \tau \big (\rho (\Delta _1)\cdots \rho (\Delta _n)\big )=\int _{\Gamma _X}\gamma (\Delta _1)\cdots \gamma (\Delta _n)\,\mu (d\gamma ), \end{aligned}$$

(7)

where $\gamma (\Delta ):=|\gamma \cap \Delta |$, the number of points of the configuration $\gamma$ that belong to $\Delta$. Formula (7) states that the moments of the operators $\rho (\Delta )$ under the gauge-invariant quasi-free state state $\tau$ are equal to the moments of the determinantal point process $\mu$. It should be stressed that the set of all monomials $\gamma (\Delta _1)\cdots \gamma (\Delta _n)$ is total in $L^2(\Gamma _X,\mu )$.

For each measurable, pre-compact set $\Delta \subset X$, we denote by $\tilde{\rho }(\Delta )$ the closure of the operator $\rho (\Delta )$. (Note that the operators $\tilde{\rho }(\Delta )$ are self-adjoint.) According to [4, Chapter 3], formula (7) means that the determinantal point process $\mu$ is the joint spectral measure of the operators $\tilde{\rho }(\Delta )$.

Let us now assume that the underlying space X is divided into two disjoint parts, $X_1$ and $X_2$. For $i=1,2$, let $P_i$ denote the orthogonal projection of $\mathcal H$ onto $\mathcal H_i:=L^2(X_i,\sigma )$, and let $J:=P_1-P_2$. An (indefinite) J-scalar product in $\mathcal H$ is defined by

$$\begin{aligned}{}[f, g]:= (Jf, g)_{\mathcal H} = ({P}_{1}f, {P}_{1}g)_{\mathcal H}- ({P}_{2}f, {P}_{2}g)_{\mathcal H},\quad f,g\in \mathcal H, \end{aligned}$$

see e.g. [3]. A bounded linear operator operator $\mathbb K\in \mathcal L(\mathcal H)$ is called J-self-adjoint if $[\mathbb Kf, g] = [f, \mathbb Kg]$ for all $f, g \in H$. If an integral operator $\mathbb K\in \mathcal L(\mathcal H)$ is J-self-adjoint, then its integral kernel $\mathbb K(x,y)$ is called J-Hermitian. The integral kernel $\mathbb K(x,y)$ of an integral operator $\mathbb K\in \mathcal L(\mathcal H)$ is J-Hermitian if and only if $\mathbb K(x,y)=\overline{\mathbb K(y,x)}$ for x and y that belong to the same part $X_i$ and $\mathbb K(x,y)=-\overline{\mathbb K(y,x)}$ if $x\in X_i$, $y\in X_j$ with $i\ne j$.

For an arbitrary bounded linear operator $K\in \mathcal L(\mathcal H)$, we define

$$\begin{aligned} \widehat{K}:=KP_1+(1-K)P_2. \end{aligned}$$

Note that the transformation $K\mapsto \widehat{K}$ is an involution. If an operator K is self-adjoint, then $\widehat{K}$ is J-self-adjoint, and if an operator $\mathbb K\in \mathcal L(\mathcal H)$ is J-self-adjoint, then $\widehat{\mathbb K}$ is self-adjoint.

A necessary and sufficient condition of existence of a determinantal point process with a J-Hermitian correlation kernel $\mathbb K(x,y)$ was given in [19]. We note that J-Hermitian correlation kernels naturally arise in the context of asymptotic representation theory of classical groups, such as symmetric or unitary groups of growing rank, see [8,9,10,11, 22].

Assume, for a moment, that the underlying space X is discrete and $\sigma$ is the counting measure. Then any determinantal point process with a J-Hermitian correlation kernel can be obtained from a determinantal point process with a Hermitian correlation kernel through a particle-hole transformation. More precisely, let $\gamma$ be a configuration in X, i.e., $\gamma \subset X$. We define a new configuration

$$\begin{aligned} I\gamma :=(\gamma \cap X_1)\cup (X_2\setminus \gamma ), \end{aligned}$$

(8)

i.e., $I\gamma$ coincides with $\gamma$ in $X_1$, and with the holes of $\gamma$ (the points unoccupied by $\gamma$) in $X_2$. Note that $I:\Gamma _X\rightarrow \Gamma _X$ is an involution. It was shown in [7] that, if $\mu$ is a determinantal point process with a correlation operator K, then $I_*\mu$ , the pushforward of $\mu$ induced by I, is the determinatal point process with the correlation operator $\widehat{K}$. Therefore, $\mu$ is a determinantal point process with a J-Hermitian correlation kernel $\mathbb K(x,y)$ if and only if $\mu =I_*\nu$, where $\nu$ is the determinantal point process with the Hermitian correlation kernel $\widehat{\mathbb K}(x,y)$. (Note that $\nu =I_*\mu$.)

If the underlying space X is continuous and $\sigma$ is a non-atomic measure, then a direct generalization of the above result is impossible. Indeed, for a configuration $\gamma$ in X, the set $I\gamma$ defined by (8) is uncountable, hence it is not anymore a configuration in X. Furthermore, the Macchi–Soshnikov theorem and [19, Theorem 3] imply that, if K(x, y), the correlation kernel of a determinantal point process, is either Hermitian or J-Hermitian, then the operator $\widehat{K}$ is not even an integral operator, i.e., $\widehat{K}(x,y)$ does not exist.

The aim of this paper is to prove the following main result, which involves a Bogoliubov transformation of the CAR $*$-algebra $\textbf{A}$. We refer the reader to e.g. [12, Section 5.2.2] for the definition of a Bogoliubov transformation and a discussion of its properties.

Main result. Let $\mathbb K\in \mathcal L(\mathcal H)$ be a J-self-adjoint integral operator, and assume that the J-Hermitian integral kernel $\mathbb K(x,y)$ of the operator $\mathbb K$ is the correlation kernel of a determinantal point process $\mu$. For the self-adjoint operator $K:=\widehat{\mathbb K}$, consider the corresponding gauge-invariant quasi-free representation of the CAR, i.e., the $*$-algebra $\textbf{A}$ is generated by operators $\mathcal A^+(\varphi )$, $\mathcal A^-(\varphi )$ in $\mathcal{A}\mathcal{F}(\mathcal H\oplus \mathcal H)$ that satisfy (2), (3) and, for the vacuum state $\tau$ on $\textbf{A}$, formulas (4), (5) hold. Define a Bogoliubov transformation of the CAR $*$-algebra $\textbf{A}$ by

$$\begin{aligned} \big ( \mathcal A^+(\varphi ),\, \mathcal A^-(\varphi )\big )\mapsto \big (A^+(\varphi ),\, A^-(\varphi )\big ),\end{aligned}$$

(9)

where for each $\varphi \in \mathcal H$,

$$\begin{aligned} A^+(\varphi ):= \mathcal A^+(P_1\varphi )+ \mathcal A^-(P_2\mathcal C\varphi ),\quad A^-(\varphi ):= \mathcal A^-(P_1\varphi )+ \mathcal A^+(P_2\mathcal C\varphi ). \end{aligned}$$

(10)

Here $(\mathcal C\varphi )(x):=\overline{\varphi (x)}$ is the complex conjugation. (Note that the vacuum state $\tau$ on the CAR $*$-algebra $\mathbb A$ generated by the operators $A^+(\varphi )$, $A^-(\varphi )$ ($\varphi \in \mathcal H$) is still quasi-free but not anymore gauge-invariant.) Let operator-valued distributions $A^+(x)$ and $A^-(x)$ be determined by $A^+(\varphi )$, $A^-(\varphi )$ similarly to (6). Then the corresponding particle density

$$\begin{aligned} \rho (\Delta )=\int _\Delta A^+(x)A^-(x)\,\sigma (dx)\quad (\Delta \subset X\quad \text {measurable and pre-compact}) \end{aligned}$$

(11)

is a family of well-defined, essentially self-adjoint, commuting operators in $\mathcal{A}\mathcal{F}(\mathcal H\oplus \mathcal H)$. Furthermore, for these operators $\rho (\Delta )$ and the determinantal point process $\mu$ with the correlation kernel $\mathbb K(x,y)$, formula (7) holds. In other words, the determinantal point process $\mu$ is the joint spectral measure of the family of self-adjoint operators $\tilde{\rho }(\Delta )$.

Note that formula (10) implies that

$$\begin{aligned} A^+(x)={\left\{ \begin{array}{ll} \mathcal A^+(x),&{} \quad \text {if }x\in X_1,\\ \mathcal A^-(x),&{} \quad \text {if }x\in X_2 \end{array}\right. },\quad A^-(x)={\left\{ \begin{array}{ll} \mathcal A^-(x),&{} \quad \text {if }x\in X_1,\\ \mathcal A^+(x),&{} \quad \text {if }x\in X_2. \end{array}\right. }\end{aligned}$$

(12)

In words, on the $X_1$ part of X we use the creation and annihilation operators of the original representation of the CAR, while on the $X_2$ part of X we exchange the creation and annihilation operators of the original representation. Hence, the Bogoliubov transformation (9), (10) can be thought of as a counterpart of the involution I defined by (8) in the case where the space X is discrete.

In fact, in our derivation of the main result, the existence of the determinantal point process $\mu$ follows from a general theorem regarding the joint spectral measure of a family of self-adjoint commuting operators, see [20, Theorem 1]. Hence, as a by-product of our considerations, we obtain a new proof of existence of a determinantal point process with a J-Hermitian correlation kernel.

In the case of a discrete space X, Koshida [16] proved that each pfaffian point process appears (in the terminology of the present paper) as the joint spectral measure of the particle density of a quasi-free representation of the CAR. Koshida notes: ‘it seems highly nontrivial if our construction can be extended to the case of continuous systems.’ While the present paper does not provide a solution to this problem, it still solves it for a particular class of (non-gauge-invariant) quasi-free states.

The paper is organized as follows. In Sect. 2, we discuss necessary preliminaries regarding determinantal point processes, J-Hermitian correlation kernels, the correlation measures and the joint spectral measure of a family of commuting self-adjoint operators, and quasi-free states on the CAR algebra.

In Sect. 3, we employ heuristic considerations, involving formula (11), in order to give a rigorous definition of Hermitian operators $\rho (\Delta )$. We also prove that these operators (algebraically) commute.

In Sect. 4, we derive rigorous formulas for the Wick (normal) product

$$\begin{aligned} {:}\rho (\Delta _1)\cdots \rho (\Delta _n){:}=\int _{\Delta _1\times \cdots \times \Delta _n}A^+(x_n)\cdots A^+(x_1)A^-(x_1)\cdots A^-(x_n)\,\sigma (dx_1)\cdots \sigma (dx_n).\nonumber \\ \end{aligned}$$

(13)

In Sect. 5, we formulate the main theorem of the paper (Theorem 5.2), which states that the the determinantal point process $\mu$ with the J-Hermitian correlation kernel $\mathbb K(x,y)$ is the joint spectral measure of the family of the commuting self-adjoint operators $\tilde{\rho }(\Delta )$. We start proving this result in Sect. 5.

Finally, in Sect. 6, we prove that the operators $\rho (\Delta )$ possess correlation functions and these are given by the right-hand side of formula (1) in which K(x, y) is replaced by $\mathbb K(x,y)$. This concludes the proof of our main result.

2 Preliminaries

2.1 Determinantal point processes

Let X be a locally compact Polish space, let $\mathcal B(X)$ be the Borel $\sigma$-algebra on X, and let $\mathcal B_0(X)$ denote the collection of all sets from $\mathcal B(X)$ which are pre-compact. The configuration space over X is defined as the set of all locally finite subsets of X:

$$\begin{aligned} \Gamma _X:=\{\gamma \subset X\mid \text{ for } \text{ all } \Delta \in \mathcal B_0(X)\ |\gamma \cap \Delta |<\infty \}. \end{aligned}$$

Here, for a set $\Lambda$, $|\Lambda |$ denotes its capacity. Elements $\gamma \in \Gamma _X$ are called configurations. One identifies each configuration $\gamma =\{x_i\}_{i\ge 1}$ with the measure $\gamma =\sum _i\delta _{x_i}$ on X. Here, for $x\in X$, $\delta _{x}$ denotes the Dirac measure with mass at x. Through this identification, one gets the embedding of $\Gamma _X$ into the space of all Radon (i.e., locally finite) measures on X.

The space $\Gamma _X$ is endowed with the vague topology, i.e., the weakest topology on $\Gamma _X$ with respect to which all maps $\Gamma _X\ni \gamma \mapsto \langle \gamma ,f\rangle =\sum _{x\in \gamma } f(x)$, $f\in C_0(X)$, are continuous. Here $C_0(X)$ is the space of all continuous real-valued functions on X with compact support. We will denote by $\mathcal B(\Gamma _X)$ the Borel $\sigma$-algebra on $\Gamma _X$. A probability measure $\mu$ on $(\Gamma _X,\mathcal B(\Gamma _X))$ is called a point process on X. For more detail, see e.g. [13, 15].

A point process $\mu$ can be described with the help of its correlation measures. Denote $X^{(n)}:=\{(x_1,\dots ,x_n)\in X^n\mid x_i\ne x_j\text { if }i\ne j\}$. The n-th correlation measure of $\mu$ is the symmetric measure $\theta ^{(n)}$ on $X^{(n)}$ that satisfies

$$\begin{aligned} \int _{\Gamma _X} \sum _{\{x_1,\dots ,x_n\}\subset \gamma }f^{(n)}(x_1,\dots ,x_n)\,\mu (d\gamma ) = \int _{X^{(n)}}f^{(n)}(x_1,\dots ,x_n)\,\theta ^{(n)}(dx_1\cdots dx_n) \end{aligned}$$

for all measurable symmetric functions $f^{(n)}:X^{(n)}\rightarrow [0,\infty )$. Let $\sigma$ be a reference Radon measure on $(X,\mathcal B(X))$. If the correlation measure $\theta ^{(n)}$ has density $k^{(n)}:X^{(n)}\rightarrow [0,\infty )$ with respect to $\frac{1}{n!}\,\sigma ^{\otimes n}$, then $k^{(n)}$ is called the n-th correlation function of $\mu$. Under a mild condition on the growth of correlation measures as $n\rightarrow \infty$, they determine a point process uniquely [17].

Recall that a point process $\mu$ is called determinantal if there exists a complex-valued function K(x, y) on $X^2$, called the correlation kernel, such that (1) holds, see e.g. [6, 26]. The integral operator K in the complex $L^2$-space $\mathcal H=L^2(X,\sigma )$ which has integral kernel K(x, y) is called the correlation operator of $\mu$.

Note that, for a given integral operator K in $\mathcal H$, the integral kernel K(x, y) is defined up to a set of zero measure $\sigma ^{\otimes 2}$. When calculating the value of $\det \big [K(x_i,x_j)\big ]_{i,j=1}^n$, one has to use the values of $K(\cdot ,\cdot )$ on the diagonal $\{(x,x)\in X^2\mid x\in X\}$. However, the latter set is of zero measure $\sigma ^{\otimes 2}$ if the measure $\sigma$ is non-atomic, i.e., $\sigma (\{x\})=0$ for all $x\in X$. Hence, when speaking about the correlation operator K of a determinantal point process, one has to properly choose the values of the integral kernel of K on the diagonal in $X^2$.

2.2 J-Hermitian correlation kernels

Assume that the underlying space X is split into two disjoint measurable parts, $X_1$ and $X_2$, of positive measure $\sigma$. Just as in Introduction, we denote by $P_i$ the orthogonal projection of $\mathcal H$ onto $\mathcal H_i=L^2(X_i,\sigma )$, and we let $J=P_1-P_2$.

According to the orthogonal sum $\mathcal H=\mathcal H_1\oplus \mathcal H_2$, each operator $A\in \mathcal L(\mathcal H)$ can be represented in the block form,

$$\begin{aligned} A=\left[ \begin{matrix} A^{11}&{} \quad A^{21}\\ A^{12}&{} \quad A^{22} \end{matrix}\right] ,\end{aligned}$$

(14)

where $A^{ij}:\mathcal H_j\rightarrow \mathcal H_i$, $i,j=1,2$. Here $A^{ij}:=P_iAP_j$. Then the operator A being J-self-adjoint means that $(A^{ii})^*=A^{ii}$ ($i=1,2$) and $(A^{21})^*=-A^{12}$.

We denote by $\mathcal S_1(\mathcal H)$ the set of all trace-class operators in $\mathcal H$, and by $\mathcal S_2(\mathcal H)$ the set of all Hilbert–Schmidt operators in $\mathcal H$. For $\Delta \in \mathcal B_0(X)$, we denote by $P_\Delta$ the orthogonal projection of $\mathcal H$ onto $L^2(\Delta ,\sigma )$. For $i=1,2$, we denote $\mathcal B_0(X_i):=\{\Delta \in \mathcal B_0(X)\mid \Delta \subset X_i\}$.

We say that an operator $K\in \mathcal L(\mathcal H)$ is locally trace-class on $X_i$ ($i=1,2$) if, for each $\Delta _i\in \mathcal B_0(X_i)$, we have $K_{\Delta _i}:=P_{\Delta _i}KP_{\Delta _i}\in \mathcal S_1(\mathcal H)$.

The following lemma can be easily checked by using basic properties of trace-class and Hilbert–Schmidt operators, see e.g. [25].

Lemma 2.1

Let $K\in \mathcal L(\mathcal H)$ satisfy $\textbf{0}\le K\le \textbf{1}$. Define $K_1:=\sqrt{K}$, $K_2:=\sqrt{\textbf{1}- K}$. Then the following statements are equivalent.

• (i) The operator K is locally trace-class on $X_1$ and the operator $\textbf{1}- K$ is locally trace-class on $X_2$.

• (ii) For any $\Delta _i\in \mathcal B_0(X_i)$ ($i=1,2$), we have $K_iP_{\Delta _i}\in \mathcal S_2(\mathcal H)$, or equivalently $P_{\Delta _i}K_i\in \mathcal S_2(\mathcal H)$.

Note that, in Lemma 2.1, $K_iP_i$ and $P_i K_i$ ($i=1,2$) are integral operators and their respective integral kernels $K_i(x, y)$ with $(x,y)\in (X\times X_i)\cup (X_i\times X)$ satisfy

$$\begin{aligned} \int _{(X\times \Delta _i)\cup (\Delta _i\times X)}| K_i(x, y)|^2\,\sigma (dx)\,\sigma (dy) <\infty , \end{aligned}$$

(15)

for any $\Delta _i\in \mathcal B_0(X_i)$. Without loss of generality, we may assume that $K_i(x,y)=\overline{K_i(y,x)}$ for all $(x,y)\in (X\times X_i)\cup (X_i\times X)$, and $\int _X | K(x,y)|^2\,\sigma (dy)<\infty$ for all $x\in X_i$.

Now consider a J-self-adjoint operator $\mathbb K\in \mathcal L(\mathcal H)$ and denote $K:=\widehat{\mathbb K}=\mathbb KP_1+(\textbf{1}-\mathbb K)P_2$. Note that $\mathbb K^{11}= K^{11}$, $\mathbb K^{22}= (\textbf{ 1}-K)^{22}$, $\mathbb K^{21}=K^{21}$, and $\mathbb K^{12}=- K^{12}=-(K^{21})^*$. Let us assume that the operator K satisfies the assumptions of Lemma 2.1, equivalently the operator $\mathbb K$ is locally trace-class on both $X_1$ and $X_2$ and $\textbf{0}\le \widehat{\mathbb K}\le \textbf{1}$.

Let us show that $\mathbb K$ is an integral operator, and let us present an integral kernel of $\mathbb K$. For $i=1,2$, we set

$$\begin{aligned} \mathbb K(x,y)=\int _X K_i(x,z) K_i(z,y)\, \sigma (dz),\quad (x,y)\in X_i^2, \end{aligned}$$

(16)

which is an integral kernel of $\mathbb K^{ii}$. Note that, for all $(x,y)\in X_i^2$, we have $\mathbb K(y,x)=\overline{\mathbb K(x,y)}$. Next, for any $\Delta _i\in \mathcal B_0(X_i)$ ($i=1,2$), we have $P_{\Delta _2}\mathbb KP_{\Delta _1}= P_{\Delta _2}\mathcal \mathbb KP_{\Delta _1}\in \mathcal S_2(\mathcal H)$. Hence, $\mathbb K^{21}$ is an integral operator. We choose an arbitrary integral kernel of $\mathbb K^{21}$, denoted by $\mathbb K(x,y)$ with $x\in X_2$ and $y\in X_1$. Finally, we set $\mathbb K(x,y)=-\overline{\mathbb K(y,x)}$ for $x\in X_1$ and $y\in X_2$. Thus, we have constructed an integral kernel of the operator $\mathbb K$.

The following theorem is shown in [19, Theorem 2].

Theorem 2.2

Let a J-self-adjoint operator $\mathbb K\in \mathcal L(\mathcal H)$ be locally trace-class on both $X_1$ and $X_2$ and such that $\textbf{0}\le \widehat{\mathbb K}\le \textbf{1}$. Let the integral kernel $\mathbb K(x,y)$ of the integral operator $\mathbb K$ be chosen as above. Then there exists a unique determinantal point process with the correlation kernel $\mathbb K(x,y)$.

Remark 2.3

In fact, the conditions of Theorem 2.2 are necessary for the existence of a determinantal point process with a J-Hermitian correlation kernel, see [19, Theorem 3].

Remark 2.4

While Theorem 2.2 will serve as a motivation for our studies, we will not actually use it and will derive the existence of a determinantal point process as in Theorem 2.2 by methods different to [19]. Nevertheless, we will use the choice of the integral kernel $\mathbb K(x,y)$ as described above.

2.3 Joint spectral measure of a family of commuting self-adjoint operators

Let us now present a result from [20] on the joint spectral measure of a family of commuting self-adjoint operators. Our brief presentation essentially follows [1, Section 4].

Let $\mathcal F$ be a separable Hilbert space and let $\mathcal D$ be a dense subspace of $\mathcal F$. For each $\Delta \in \mathcal B_0(X)$, let $\rho (\Delta ):\mathcal D\rightarrow \mathcal D$ be a linear Hermitian operator in $\mathcal F$. We further assume:

• for any $\Delta _1,\Delta _2\in \mathcal B_0(X)$ with $\Delta _1\cap \Delta _2=\varnothing$, we have $\rho (\Delta _1\cup \Delta _2)=\rho (\Delta _1)+\rho (\Delta _2)$;

• the operators $\rho (\Delta )$ commute, i.e., $[\rho (\Delta _1),\rho (\Delta _2)]=0$ for any $\Delta _1,\Delta _2\in \mathcal B_0(X)$.

Let $\mathcal A$ denote the (commutative) $*$-algebra generated by $(\rho (\Delta ))_{\Delta \in \mathcal B_0(X)}$. Let $\Omega$ be a fixed vector in $\mathcal D$ with $\Vert \Omega \Vert _{\mathcal F}=1$, and let a state $\tau :\mathcal A\rightarrow \mathbb C$ be defined by $\tau (a):=(a\Omega ,\Omega )_{\mathcal F}$ for $a\in \mathcal A$.

We define Wick polynomials in $\mathcal A$ by the following recurrence formula:

$$\begin{aligned} {:}\rho (\Delta ){:}&=\rho (\Delta ),\nonumber \\ {:}\rho (\Delta _1)\cdots \rho (\Delta _{n+1}){:}&=\rho (\Delta _{n+1})\, {:}\rho (\Delta _1)\cdots \rho (\Delta _{n}){:}\nonumber \\&\quad -\sum _{i=1}^n {:}\rho (\Delta _1)\cdots \rho (\Delta _{i-1})\rho (\Delta _i\cap \Delta _{n+1})\rho (\Delta _{i+1})\cdots \rho (\Delta _n){:}\,, \end{aligned}$$

(17)

where $\Delta ,\Delta _1,\dots ,\Delta _{n+1}\in \mathcal B_0(X)$ and $n\in \mathbb N$. It is easy to see that, for each permutation $\pi \in S_n$,

$$\begin{aligned} {:}\rho (\Delta _1)\cdots \rho (\Delta _n){:} = {:}\rho (\Delta _{\pi (1)})\cdots \rho (\Delta _{\pi (n)}){:}\,. \end{aligned}$$

We assume that, for each $n\in \mathbb N$, there exists a symmetric measure $\theta ^{(n)}$ on $X^n$ that is concentrated on $X^{(n)}$ (i.e., $\theta ^{(n)}(X^n\setminus X^{(n)})=0$) and such that

$$\begin{aligned} \theta ^{(n)}\big (\Delta _1\times \dots \times \Delta _n\big )=\frac{1}{n!}\, \tau \big ({:}\rho (\Delta _1)\cdots \rho (\Delta _{n}){:}\big ),\quad \Delta _1,\dots ,\Delta _{n}\in \mathcal B_0(X).\end{aligned}$$

(18)

Note that, if the measure $\theta ^{(n)}$ exists, then it is unique. The $\theta ^{(n)}$ is called the n-th correlation measure of the operators $\rho (\Delta )$. If $\theta ^{(n)}$ has a density $k^{(n)}$ with respect to $\frac{1}{n!}\sigma ^{\otimes n}$, then $k^{(n)}$ is called the nth correlation function of the operators $\rho (\Delta )$.

Theorem 2.5

[20] Let $(\rho (\Delta ))_{\Delta \in \mathcal B_0(X)}$ be a family of Hermitian operators in $\mathcal F$ as above. In particular, these operators have correlation measures $(\theta ^{(n)})_{n=1}^\infty$ respective the state $\tau$. Furthermore, we assume that the following two conditions are satisfied.

(LB1) For each $\Delta \in \mathcal B_0(X)$, there exists a constant $C_\Delta >0$ such that

$$\begin{aligned} \theta ^{(n)}(\Delta ^n)\le C_\Delta ^n,\quad n\in \mathbb N. \end{aligned}$$

(19)

(LB2) For any sequence $\{\Delta _{l}\}_{l\in \mathbb {N}}\subset \mathcal {B}_{0}(X)$ such that $\Delta _{l}\downarrow \varnothing$ (i.e., $\Delta _1\supset \Delta _2\supset \Delta _3\supset \cdots$ and $\bigcap _{l=1}^\infty \Delta _l=\varnothing$), we have $C_{\Delta _{l}}\rightarrow 0$ as $l\rightarrow \infty$.

Then the following statements hold.

• (i) Let $\mathfrak D:=\{a\Omega \mid a\in \mathcal A\}$ and let $\mathfrak F$ denote the closure of $\mathfrak D$ in $\mathcal F$. Each operator $(\rho (\Delta ),\mathfrak D)$ is essentially self-adjoint in $\mathfrak F$, i.e., the closure of $\rho (\Delta )$, denoted by $\widetilde{\rho }(\Delta )$, is a self-adjoint operator in $\mathfrak F$.

• (ii) For any $\Delta _1,\Delta _2\in \mathcal B_0(X)$, the projection-valued measures (resolutions of the identity) of the operators $\widetilde{\rho }(\Delta _1)$ and $\widetilde{\rho }(\Delta _2)$ commute.

• (iii) There exist a unique point process $\mu$ on X and a unique unitary operator$U:\mathfrak F\rightarrow L^2(\Gamma _X,\mu )$ satisfying $U\Omega =1$ and

  $$\begin{aligned} U(\rho (\Delta _1)\cdots \rho (\Delta _{n})\Omega )=\gamma (\Delta _1)\cdots \gamma (\Delta _n)\end{aligned}$$

  (20)

  for any $\Delta _1,\dots ,\Delta _{n}\in \mathcal B_0(X)$ ($n\in \mathbb N$). In particular,

  $$\begin{aligned} \tau \big (\rho (\Delta _1)\cdots \rho (\Delta _{n})\big )= \int _{\Gamma _X}\gamma (\Delta _1)\cdots \gamma (\Delta _n)\,\mu (d\gamma ).\end{aligned}$$

  (21)

• (iv) The correlations measures of the point process $\mu$ are $(\theta ^{(n)})_{n=1}^\infty$.

According to [4, Chapter 3], the point process $\mu$ from Theorem 2.5 is the joint spectral measure of the family of commuting self-adjoint operators $(\widetilde{\rho }(\Delta ))_{\Delta \in \mathcal B_0(X)}$.

2.4 Quasi-free states on the CAR algebra

Let $\mathcal F$ be a separable Hilbert space, and let $a^+(\varphi )$ and $a^-(\varphi )$ ($\varphi \in \mathcal H$) be bounded linear operators in $\mathcal F$ such that $a^+(\varphi )$ linearly depends on $\varphi$ and $a^-(\varphi )=\big (a^+(\varphi )\big )^*$. Let $a^+(\varphi )$ and $a^-(\varphi )$ satisfy the CAR, i.e., formula (3) holds in which the operators $\mathcal A^+(\varphi )$, $\mathcal A^-(\varphi )$ are replaced by $a^+(\varphi )$, $a^-(\varphi )$. Let $\mathbb A$ be the $*$-algebra generated by these operators. We define field operators $b(\varphi ):=a^+(\varphi )+a^-(\varphi )$ ($\varphi \in \mathcal H$). As easily seen, these operators also generate $\mathbb A$.

Let $\tau :\mathbb A\rightarrow \mathbb C$ be a state on $\mathbb A$. The state $\tau$ is completely determined by the functionals $T^{(n)}:\mathcal H^n\rightarrow \mathbb C$ $(n\in \mathbb N)$ defined by

$$\begin{aligned} T^{(n)}(\varphi _1,\dots ,\varphi _n):=\tau \big (b(\varphi _1)\cdots b(\varphi _n)\big ). \end{aligned}$$

(22)

The state $\tau$ is called quasi-free if

$$\begin{aligned}{} & {} T^{(2n-1)}=0, \end{aligned}$$

(23)

$$\begin{aligned}{} & {} T^{(2n)}(\varphi _1,\dots ,\varphi _{2n})=\sum (-1)^{{\text {Cross}}(\nu )}\, T^{(2)}(\varphi _{i_1},\varphi _{j_1})\cdots T^{(2)}(\varphi _{i_n},\varphi _{j_n}),\quad n\in \mathbb N,\nonumber \\ \end{aligned}$$

(24)

where the summation is over all partitions $\nu =\big \{\{i_1,j_1\},\dots ,\{i_n,j_n\}\big \}$ of $\{1,\dots ,2n\}$ with $i_k<j_k$ ($k=1,\dots ,n$) and ${\text {Cross}}(\nu )$ denotes the number of all crossings in $\nu$, i.e., the number of all choices of $\{i_k,j_k\},\{i_l,j_l\}\in \nu$ such that $i_k<i_l<j_k<j_l$, see e.g. [12, Section 5.2.3].

The state $\tau$ is called gauge-invariant if, for each $q\in \mathbb C$ with $|q|=1$, we have $T^{(n)}(q\varphi _1,\dots ,q\varphi _n)=T^{(n)}(\varphi _1,\dots ,\varphi _n)$ for all $\varphi _1,\dots ,\varphi _n\in \mathcal H$, $n\in \mathbb N$. The state $\tau$ can also be uniquely characterized by the n-point functions $S^{(m,n)}:\mathcal H^{m+n}\rightarrow \mathbb C$ ($m+n\ge 1$) defined by

$$\begin{aligned} S^{(m,n)}(\varphi _1\dots ,\varphi _m,\psi _1,\dots ,\psi _n):=\tau \big (a^+(\varphi _1)\cdots a^+(\varphi _m)a^-(\psi _1)\cdots a^-(\psi _n)\big ). \end{aligned}$$

(25)

The state $\tau$ is gauge-invariant quasi-free if and only if

$$\begin{aligned} S^{(m,n)}(\varphi _m\dots ,\varphi _1,\psi _1,\dots ,\psi _n)=\delta _{m,n}{\text {det}}\left[ S^{(1,1)}(\varphi _i,\psi _j)\right] _{i,j=1,\dots ,n}\,. \end{aligned}$$

(26)

Let us briefly recall the Araki–Wyss [2] construction of the gauge-invariant quasi-free states. Let $\mathcal G$ denote a separable complex Hilbert space. Let $\mathcal{A}\mathcal{F}(\mathcal G):=\bigoplus _{n=0}^\infty \mathcal G^{\wedge n}n!$ denote the antisymmetric Fock space of $\mathcal G$. Here $\wedge$ denotes the antysymmetric tensor product and elements of the Hilbert space $\mathcal{A}\mathcal{F}(\mathcal G)$ are sequences $g=(g^{(n)})_{n=0}^\infty$ with $g^{(n)}\in \mathcal G^{\wedge n}$ ($\mathcal G^{\wedge 0}:=\mathbb C$) and $\Vert g\Vert ^2_{\mathcal{A}\mathcal{F}(\mathcal G)}=\sum _{n=0}^\infty \Vert g^{(n)}\Vert ^2_{\mathcal G^{\wedge n}}\,n!<\infty$. The vector $\Omega =(1,0,0,\dots )$ is called the vacuum.

For $\varphi \in \mathcal G$, we define a creation operator $a^+(\varphi )\in \mathcal L\big (\mathcal{A}\mathcal{F}(\mathcal G)\big )$ by $a^+(\varphi )g^{(n)}:=\varphi \wedge g^{(n)}$ for $g^{(n)}\in \mathcal G^{\wedge n}$. For each $\varphi \in \mathcal G$, we define an annihilation operator $a^-(\varphi ):=a^+(\varphi )^*$. Then,

$$\begin{aligned} a^-(\varphi )g_1\wedge \cdots \wedge g_n=\sum _{i=1}^n (-1)^{i+1}(g_i,\varphi )_{\mathcal G}\,g_1\wedge \cdots g_{i-1}\wedge g_{i+1}\cdots g_n \end{aligned}$$

for all $g_1,\dots ,g_n\in \mathcal G$. Note that

$$\begin{aligned} \Vert a^+(\varphi )\Vert _{\mathcal L(\mathcal{A}\mathcal{F}(\mathcal G))}=\Vert a^-(\varphi )\Vert _{\mathcal L(\mathcal{A}\mathcal{F}(\mathcal G))}=\Vert \varphi \Vert _\mathcal G\,. \end{aligned}$$

(27)

The operators $a^+(\varphi )$, $a^-(\varphi )$ satisfy the CAR (over $\mathcal G$).

Let now $\mathcal G=\mathcal H\oplus \mathcal H$, and for $\varphi \in \mathcal H$ and $\Diamond \in \{+,-\}$, we denote $a^\Diamond _1(\varphi ):=a^\Diamond (\varphi ,0)$, $a^\Diamond _2(\varphi ):=a^\Diamond (0,\varphi )$.

We fix any $K\in \mathcal L(\mathcal H)$ such that $\textbf{0}\le K\le \textbf{1}$, and define the operators $K_1$ and $K_2$ as in Lemma 2.1. We define operators

$$\begin{aligned} \mathcal A^+(\varphi ):=a^+_2( K_2\varphi )+a_1^-(\mathcal C K_1\varphi ),\quad \mathcal A^-(\varphi ):=a_2^-( K_2\varphi )+a_1^+(\mathcal C K_1\varphi ), \end{aligned}$$

(28)

where $\mathcal C$ is the complex conjugation in $\mathcal H$. The operators $\mathcal A^+(\varphi )$, $\mathcal A^-(\varphi )$ satisfy the CAR (2), (3). Let $\textbf{A}$ denote the corresponding CAR $*$-algebra. The vacuum state on $\textbf{A}$ is defined by $\tau (a):=(a\Omega ,\Omega )_{\mathcal{A}\mathcal{F}(\mathcal G)}$ ($a\in \textbf{A}$). This state is gauge-invariant quasi-free. More exactly, setting $\mathcal F=\mathcal{A}\mathcal{F}(\mathcal G)$ and $a^\pm (\varphi )=\mathcal A^\pm (\varphi )$, one shows that formulas (25), (26) hold, with $S^{(1,1)}(\varphi ,\psi )=(K\varphi ,\psi )_{\mathcal H}$ ($\varphi ,\psi \in \mathcal H$). In fact, each gauge-invariant quasi-free state on the CAR algebra over $\mathcal H$ can be constructed in such a way [2].

Next, just as in Sect. 2.2, we assume that the space X is divided into two disjoint parts, $X_1$ and $X_2$. We define operators $A^+(\varphi )$, $A^-(\varphi )$ ($\varphi \in \mathcal H$) by formula (10). Hence, by (28),

$$\begin{aligned} A^+(\varphi )&=a^+(\mathcal CK_1\mathcal CP_2 \varphi ,\, K_2P_1\varphi )+a^-(\mathcal C K_1P_1\varphi ,\,\mathcal CK_2 P_2 \varphi ),\nonumber \\ A^-(\varphi )&=a^-(\mathcal CK_1\mathcal CP_2\varphi ,\,K_2P_1\varphi )+a^+(\mathcal CK_1P_1\varphi ,\mathcal C K_2P_2\varphi ). \end{aligned}$$

(29)

These operators also satisfy the CAR and denote by $\mathbb A$ the corresponding CAR $*$-algebra. (Note that we have the equality of $\textbf{A}$ and $\mathbb A$ as sets.) The vacuum state $\tau$ on $\mathbb A$ is not anymore gauge-invariant but it is still quasi-free. More exactly, setting $\mathcal F=\mathcal{A}\mathcal{F}(\mathcal G)$ and $a^\pm (\varphi )=A^\pm (\varphi )$, one easily shows that formulas (22)–(24) hold, with

$$\begin{aligned} T^{(2)}(\varphi ,\psi )=2i\Im ( K\mathbb J\varphi ,\mathbb J\psi )_{\mathcal H}+(\mathbb J\psi ,\mathbb J\varphi )_{\mathcal H}. \end{aligned}$$

Here $\mathbb J\varphi :=P_1\varphi +P_2\mathcal C\varphi$.

Remark 2.6

The n-point functions $S^{(m,n)}$ for the state $\tau$ on $\mathbb A$ can be calculated as follows. First, we note that

$$\begin{aligned} S^{(1,1)}(\varphi ,\psi )&=((P_1\mathbb KP_1+P_2\mathbb KP_2)\varphi ,\psi )_{\mathcal H},\\ S^{(2,0)}(\varphi ,\psi )&=(\psi ,(\mathcal CP_2KP_1+P_1K_2\mathcal CK_2P_2)\varphi )_{\mathcal H},\\ S^{(0,2)}(\varphi ,\psi )&=\overline{S^{(2,0)}(\psi ,\varphi )}. \end{aligned}$$

Next, if $m+n$ is odd, then $S^{(m,n)}=0$, and if $m+n=2k$ ($k\ge 2$), then, by using Lemma 6.1 below, we get

$$\begin{aligned} S^{(m,n)}(\varphi _1,\dots ,\varphi _{m+n})=\sum (-1)^{{\text {Cross}}(\nu )}\, S_{i_1,\,j_1}^{(2)}(\varphi _{i_1},\varphi _{j_1})\cdots S^{(2)}_{i_k,\,j_k}(\varphi _{i_k},\varphi _{j_k}), \end{aligned}$$

where the summation is over all partitions $\nu =\big \{\{i_1,j_1\},\dots ,\{i_k,j_k\}\big \}$ of $\{1,\dots ,2k\}$ with $i_l<j_l$ and $S^{(2)}_{i_l,\,j_l}:=S^{(2,0)}$ if $j_l\le m$, $S^{(2)}_{i_l,\,j_l}:=S^{(1,1)}$ if $i_l\le m<j_l$, $S^{(2)}_{i_l,\,j_l}:=S^{(0,2)}$ if $i_l\ge m+1$ ($l=1,\dots ,k$).

3 Rigorous Construction of the Particle Density

Let $\mathbb K\in \mathcal L(\mathcal H)$ be a J-self-adjoint operator satisfying the assumptions of Theorem 2.2, and let $K:=\widehat{\mathbb K}$. Let $\mathcal G=\mathcal H\oplus \mathcal H$, and let the operators $A^+(\varphi ),A^-(\varphi )\in \mathcal L\big (\mathcal{A}\mathcal{F}(\mathcal G)\big )$ ($\varphi \in \mathcal H$) be defined by (29). Let the operator-valued distributions $A^+(x)$, $A^-(x)$ ($x\in X$) be determined by $A^+(\varphi )$, $A^-(\varphi )$ similarly to (6).

Recall that $\mathcal{A}\mathcal{F}(\mathcal G)$ consists of all sequences $g=(g^{(n)})_{n=0}^\infty$ with $g^{(n)}\in \mathcal G^{\wedge n}$ satisfying $\sum _{n=0}^\infty \Vert g^{(n)}\Vert ^2_{\mathcal G^{\wedge n}}\,n!<\infty$. We denote by $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$ the dense subspace of $\mathcal{A}\mathcal{F}(\mathcal G)$ that consists of all finite sequences $g=(g^{(n)})_{n=0}^\infty$ from $\mathcal{A}\mathcal{F}(\mathcal G)$, i.e., for some $N\in \mathbb N$ (depending on g), we have $g^{(n)}=0$ for all $n\ge N$. We endow $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$ with the topology of the locally convex direct sum of the Hilbert spaces $\mathcal G^{\wedge n}$.

We denote by $\mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$ the space of continuous linear operators in $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$. Note that a linear operator A acting in $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$ is continuous if and only if, for each k there exists N such that $A\mathcal G^{\wedge k}\subset \bigoplus _{n=0}^N\mathcal G^{\wedge n}$ and A acts continuously from $\mathcal G^{\wedge k}$ into $\bigoplus _{n=0}^N\mathcal G^{\wedge n}$, see e.g. [24, Chapter II, Section 6].

Recall the heuristic definition (11) of $\rho (\Delta )$. Our aim in this section is to rigorously define $\rho (\Delta )$ for each $\Delta \in \mathcal B_0(X)$ as an operator from $\mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$ which is Hermitian in $\mathcal{A}\mathcal{F}(\mathcal G)$. To this end, we start with heuristic considerations.

Think of $K_1$ and $K_2$ as self-adjoint integral operators with integral Hermitian kernels $K_1(x,y)$ and $K_2(x,y)$, respectively. Let $\Delta \in \mathcal B_0(X_1)$, and denote by $\chi _\Delta$ the indicator function of the set $\Delta$. We have

$$\begin{aligned}&\int _\Delta A^+(x)\,\sigma (dx)=A^+(\chi _\Delta )=a_2^+(K_2\chi _\Delta )+a_1^-(\mathcal CK_1\chi _\Delta )\\&\quad =\int _X (K_2\chi _\Delta )(x)a_2^+(x)\,\sigma (dx)+\int _X (K_1\chi _\Delta )(x)a_1^-(x)\,\sigma (dx)\\&\quad =\int _X\int _\Delta K_2(x,y) a_2^+(x)\,\sigma (dy)\, \sigma (dx)+\int _X\int _\Delta K_1(x,y) a_1^-(x)\,\sigma (dy)\, \sigma (dx)\\&\quad =\int _\Delta \int _X K_2(y,x) a_2^+(y)\,\sigma (dy)\, \sigma (dx)+\int _\Delta \int _X \overline{K_1(x,y)}\, a_1^-(y)\,\sigma (dy) \,\sigma (dx)\\&\quad =\int _\Delta \big ( a_2^+(K_2(\cdot ,x))+a_1^-(K_1(x,\cdot ))\big )\,\sigma (dx). \end{aligned}$$

From here, and using similar calculations when $\Delta \in \mathcal B_0(X_i)$ ($i=1,2$), we formally conclude that

$$\begin{aligned} A^+(x)&={\left\{ \begin{array}{ll} a^+_2\big (K_2(\cdot ,x)\big )+a^-_1\big (K_1(x,\cdot )\big ) &{} \quad \text {if}\ x\in X_1, \\ a^+_1\big (K_1(x,\cdot )\big )+a^-_2\big (K_2(\cdot ,x)\big ) &{} \quad \text {if}\ x\in X_2, \end{array}\right. } \\ A^-(x)&={\left\{ \begin{array}{ll} a^-_2\big (K_2(\cdot ,x)\big )+a^+_1\big (K_1(x,\cdot )\big ) &{} \quad \text {if}\ x\in X_1, \\ a^-_1\big (K_1(x,\cdot )\big )+a^+_2\big (K_2(\cdot ,x)\big ) &{} \quad \text {if}\ x\in X_2. \end{array}\right. } \end{aligned}$$

Denoting $\rho (x):=A^+(x)A^-(x)$, we get, for $x\in X_1$,

$$\begin{aligned} \rho (x)&= a^+_2\big (K_2(\cdot ,x)\big )a^+_1\big (K_1(x,\cdot )\big ) +a^+_2\big (K_2(\cdot ,x)\big )a^-_2\big (K_2(\cdot ,x)\big ) \nonumber \\&\quad + a^-_1\big (K_1(x,\cdot )\big )a^+_1\big (K_1(x,\cdot )\big )+a^-_1\big (K_1(x,\cdot )\big )a^-_2\big (K_2(\cdot ,x)\big ), \end{aligned}$$

(30)

and for $x\in X_2$,

$$\begin{aligned} \rho (x)&= a^+_1\big (K_1(x,\cdot )\big )a^+_2\big (K_2(\cdot ,x)\big )+ a^+_1\big (K_1(x,\cdot )\big ) a^-_1\big (K_1(x,\cdot )\big )\nonumber \\&\quad + a^-_2\big (K_2(\cdot ,x)\big ) a^+_2\big (K_2(\cdot ,x)\big )+a^-_2\big (K_2(\cdot ,x)\big )a^-_1\big (K_1(x,\cdot )\big ). \end{aligned}$$

(31)

Remark 3.1

Comparing formulas (30) and (31), we see that the right-hand side of (31) can be obtained from the right-hand side of (30) by swapping places of $a_1^\Diamond (K_1(x,\cdot ))$ and $a_2^\Diamond (K_2(\cdot ,x))$ ($\Diamond \in \{+,-\}$).

For $\Delta \in \mathcal B_0(X_1)$, let us now rigorously define $\int _X \rho (x)\sigma (dx)$.

The creation and annihilation operators. We will now define

$$\begin{aligned} \int _\Delta a^+_2\big (K_2(\cdot ,x)\big )a^+_1\big (K_1(x,\cdot )\big ) \sigma (dx),\quad \int _\Delta a^-_1\big (K_1(x,\cdot )\big )a^-_2\big (K_2(\cdot ,x)\big ) \sigma (dx). \end{aligned}$$

First, we note that, for $\varphi ,\psi \in \mathcal G$ and $g^{(n)}\in \mathcal G^{\wedge n}$,

$$\begin{aligned} a^+(\varphi )a^+(\psi )g^{(n)}=\varphi \wedge \psi \wedge g^{(n)}=\mathcal A_{n+2}\big (\varphi \otimes \psi \otimes g^{(n)}\big ). \end{aligned}$$

Here, for each $n\in \mathbb N$, we denote by $\mathcal A_n$ the antisymmetrization operator in $\mathcal G^{\otimes n}$, i.e., the orthogonal projection of $\mathcal G^{\otimes n}$ onto $\mathcal G^{\wedge n}$.

For $\varphi ^{(2)}\in \mathcal G^{\otimes 2}$, we define a creation operator $a^+(\varphi ^{(2)})$ in $\mathcal A\mathcal F_{\textrm{fin}}(\mathcal G)$ by

$$\begin{aligned} a^+(\varphi ^{(2)})g^{(n)}:= \mathcal A_{n+2}(\varphi ^{(2)}\otimes g^{(n)}),\quad g^{(n)}\in \mathcal G^{\wedge n}. \end{aligned}$$

Obviously $a^+(\varphi ^{(2)})\in \mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$. We also denote

$$\begin{aligned} a^-(\varphi ^{(2)}):= a^+(\varphi ^{(2)})^*\upharpoonright \mathcal A\mathcal F_{\textrm{fin}}(\mathcal G), \end{aligned}$$

which also belongs to $\mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$. One can easily derive explicit formulas for the action of this operator.

Thus, we formally have

$$\begin{aligned} \int _\Delta a^+_2\big (K_2(\cdot ,x)\big )a^+_1\big (K_1(x,\cdot )\big ) \sigma (dx)=a^+\bigg (\int _\Delta \big (0,K_2(\cdot ,x)\big )\otimes \big (K_1(x,\cdot ),0\big )\,\sigma (dx)\bigg ).\end{aligned}$$

(32)

To give the above operator a rigorous meaning, we will first identify $\int _\Delta K_2(\cdot ,x)\otimes K_1(x,\cdot ) \,\sigma (dx)$ as an element of $\mathcal H^{\otimes 2}$.

For a linear operator $B\in \mathcal L(\mathcal H)$, denote $\overline{B}:=\mathcal CB\mathcal C$, i.e., $\overline{B}$ is the complex conjugate of B. If B is an integral operator with integral kernel B(x, y), then $\overline{B}$ is the integral operator with integral kernel $\overline{B(x,y)}$. In particular, if B is self-adjoint, then the integral kernel of $\overline{B}$ is B(y, x).

Now, for any $\varphi ,\psi \in \mathcal H$, we formally calculate

$$\begin{aligned}&\bigg (\int _\Delta K_2(\cdot ,x)\otimes K_1(x,\cdot )\,\sigma (dx) , \varphi \otimes \psi \bigg )_{\mathcal H^{\otimes 2}} \nonumber \\&\quad = \int _\Delta \big (K_2(\cdot ,x)\otimes K_1(x,\cdot ),\varphi \otimes \psi \big )_{\mathcal H^{\otimes 2}}\, \sigma (dx)\nonumber \\&\quad =\int _\Delta \int _X K_2(y,x) \overline{\varphi (y)}\,\sigma (dy) \int _X K_1(x,z)\overline{\psi (z)}\, \sigma (dz)\, \sigma (dx) \nonumber \\&\quad =\int _X\chi _\Delta (x) \big (K_1 \overline{\psi }\big )(x) \big (\overline{K_2}\,\bar{\varphi }\big )(x)\, \sigma (dx)= \int _X\chi _\Delta (x)\big (K_1 \overline{\psi }\big )(x) \overline{\big ( K_2 \varphi \big )(x)}\, \sigma (dx)\nonumber \\&\quad =\big (P_\Delta K_1 \overline{\psi },K_2 \varphi \big )_\mathcal H=\big (K_2P_\Delta K_1 \overline{\psi }, \varphi \big )_\mathcal H\,. \end{aligned}$$

(33)

Since $P_\Delta K_1\in \mathcal S_2(H)$ (see Lemma 2.1 (ii)), we get $K_2P_\Delta K_1\in \mathcal S_2(H)$. Therefore, $K_2P_\Delta K_1$ is an integral operator and we denote its integral kernel by $\big (K_2P_\Delta K_1\big )(x,y)$. Note that $K_2P_\Delta K_1(\cdot ,\cdot )\in \mathcal H^{\otimes 2}$. Thus, we continue (33) as follows:

$$\begin{aligned} =\int _X\int _X \big (K_2P_\Delta K_1\big )(x,y)\overline{\psi (y)}\,\sigma (dy)\, \overline{\varphi (x)}\,\sigma (dx) =\big (K_2P_\Delta K_1(\cdot ,\cdot ),\varphi \otimes \psi \big )_{\mathcal H^{\otimes 2}}. \end{aligned}$$

Hence, we rigorously define

$$\begin{aligned} \int _\Delta K_2(\cdot ,x)\otimes K_1(x,\cdot ) \, \sigma (dx) := (K_2P_\Delta K_1)(\cdot ,\cdot )\in \mathcal H^{\otimes 2}. \end{aligned}$$

(34)

We define an isometry

$$\begin{aligned}{} & {} \mathcal I_{21}:\mathcal H^{\otimes 2}\rightarrow \mathcal G^{\otimes 2}=(\mathcal H\oplus \mathcal H)\otimes (\mathcal H\oplus \mathcal H),\\{} & {} \mathcal I_{21}\varphi \otimes \psi =(0,\varphi )\otimes (\psi ,0),\quad \varphi ,\psi \in \mathcal H. \end{aligned}$$

We denote $\big (K_2P_\Delta K_1\big )_{2,1}:=\mathcal I_{21}(K_2P_\Delta K_1)(\cdot ,\cdot )$. Then, in view of (32) and (34), we define

$$\begin{aligned} \int _\Delta a^+_2\big (K_2(\cdot ,x)\big )a^+_1\big (K_1(x,\cdot )\big ) \sigma (dx):= a^+\big (\big (K_2P_\Delta K_1\big )_{2,1}\big ),\end{aligned}$$

(35)

and so

$$\begin{aligned} \int _\Delta a^-_1\big (K_1(x,\cdot )\big )a^-_2\big (K_2(\cdot ,x)\big ) \sigma (dx):= a^-\big (\big (K_2P_\Delta K_1\big )_{2,1}\big ).\end{aligned}$$

(36)

The neutral operator. Our next aim is to rigorously define operators

$$\begin{aligned} \int _\Delta a^+_2\big (K_2(\cdot ,x)\big )a^-_2\big (K_2(\cdot ,x)\big )\sigma (dx),\quad \int _\Delta a^-_1\big (K_1(x,\cdot )\big ) a^+_1\big (K_1(x,\cdot )\big )\sigma (dx). \end{aligned}$$

For a linear operator $B\in \mathcal G$, the differential second quantization of B is defined as a linear operator $d\Gamma (B)\in \mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$ satisfying $d\Gamma (B)\Omega :=0$ (recall that $\Omega$ is the vacuum), and for any $g_1,\dots ,g_n\in \mathcal G$,

$$\begin{aligned} d\Gamma (B)g_1\wedge \cdots \wedge g_n=\sum _{i=1}^n g_1\wedge \cdots \wedge g_{i-1}\wedge (Bg_i)\wedge g_{i+1}\wedge \cdots \wedge g_n. \end{aligned}$$

As easily seen, for any $\varphi ,\psi \in \mathcal G$, we have $a^+(\varphi )a^-(\psi )=d\Gamma \big ((\cdot ,\psi )_{\mathcal G}\,\varphi \big )$, i.e., the differential second quantization of the operator $\mathcal G\ni g\mapsto (g,\psi )_{\mathcal G}\,\varphi$. Hence, we may formally write

$$\begin{aligned} \int _\Delta a^+_2\big (K_2(\cdot ,x)\big ) a^-_2\big (K_2(\cdot ,x)\big )\,\sigma (dx) = d\Gamma \bigg ( \int _\Delta \big (\cdot ,\mathcal I_2K_2(\cdot ,x)\big )_\mathcal G\, \mathcal I_2K_2(\cdot ,x)\,\sigma (dx)\bigg ), \end{aligned}$$

(37)

where the isometry $\mathcal I_2:\mathcal H\rightarrow \mathcal G=\mathcal H\oplus \mathcal H$ is defined by $\mathcal I_2\varphi :=(0,\varphi )$.

Let us define $\int _\Delta \big (\cdot ,K_2(\cdot ,x)\big )_\mathcal H,K_2(\cdot ,x)\,d\sigma (x)$ as a bounded linear operator in $\mathcal H$. For $\varphi ,\psi \in \mathcal H$, we formally calculate

$$\begin{aligned}&\bigg ( \int _\Delta \big (\varphi ,K_2(\cdot ,x)\big )_\mathcal H \,K_2(\cdot ,x)\,\sigma (dx),\psi \bigg )_\mathcal H \\&\quad = \int _\Delta \big (\varphi ,K_2(\cdot ,x)\big )_\mathcal H \big (K_2(\cdot ,x),\psi \big )_\mathcal H\, \sigma (dx)\\&\quad =\int _\Delta \int _X \varphi (y) \overline{K_2(y,x)}\, d\sigma (y) \int _X K_2(z,x)\overline{\psi (z)}\, \sigma (dz)\,\sigma (dx)\\&\quad =\int _\Delta \big (K_2\varphi \big )(x)\overline{(K_2\psi )(x)}\, \sigma (dx)=\big (K_2P_\Delta K_2 \varphi ,\psi \big )_\mathcal H. \end{aligned}$$

Thus, we rigorously define

$$\begin{aligned} \int _\Delta \big (\cdot ,K_2(\cdot ,x)\big )_\mathcal H\,K_2(\cdot ,x)\,\sigma (dx):=K_2P_\Delta K_2 . \end{aligned}$$

(38)

In view of (37) and (38), we define

$$\begin{aligned} \int _\Delta a^+_2\big (K_2(\cdot ,x)\big ) a^-_2\big (K_2(\cdot ,x)\big )\,\sigma (dx):= d\Gamma \big (\textbf{0}\oplus K_2P_\Delta K_2\big ). \end{aligned}$$

(39)

Next, using the CAR, we formally have

$$\begin{aligned}&\int _\Delta a^-_1\big (K_1(x,\cdot )\big ) a^+_1\big (K_1(x,\cdot )\big )\sigma (dx)\nonumber \\&\quad =- \int _\Delta a^+_1\big (K_1(x,\cdot )\big ) a^-_1\big (K_1(x,\cdot )\big )\, \sigma (dx)+ \int _\Delta \big (K_1(x,\cdot ),K_1(x,\cdot )\big )_\mathcal H\, \sigma (dx)\nonumber \\&\quad =- \int _\Delta a^+_1\big (K_1(x,\cdot )\big ) a^-_1\big (K_1(x,\cdot )\big )\, \sigma (dx)+ \int _\Delta \int _X|K_1(x,y)|^2 \sigma (dy) \, \sigma (dx). \end{aligned}$$

(40)

Since $P_\Delta K_1$ is a Hilbert–Schmidt operator (see Lemma 2.1), we have

$$\begin{aligned} \int _\Delta \int _X|K_1(x,y)|^2 \sigma (dy) \, \sigma (dx)=\Vert P_\Delta K_1\Vert ^2_{2}\,,\end{aligned}$$

(41)

where $\Vert \cdot \Vert _{2}$ denotes the Hilbert–Schmidt norm in $\mathcal S_2(\mathcal H)$. Noting that the operator $K_\Delta =P_\Delta K P_\Delta$ is self-adjoint, we easily see that

$$\begin{aligned} \Vert P_\Delta K_1\Vert ^2_{2}={\text {Tr}}(K_\Delta )={\text {Tr}}(\mathbb K_\Delta ).\end{aligned}$$

(42)

Hence, by (40)–(42) and similarly to (39), we rigorously define

$$\begin{aligned} \int _\Delta a^-_1\big (K_1(x,\cdot )\big ) a^+_1\big (K_1(x,\cdot )\big )\sigma (dx):=d\Gamma \big (\overline{-K_1P_\Delta K_1}\oplus \textbf{0}\big ) + {\text {Tr}}(\mathbb K_\Delta ).\end{aligned}$$

(43)

Thus, formulas (35), (36), (39), and (43) imply a rigorous definition of $\rho (\Delta )$ for $\Delta \in \mathcal B_0(X_1)$:

$$\begin{aligned} \rho (\Delta ):=&a^+\big (\big (K_2P_{\Delta } K_1\big )_{2,1}\big )+a^-\big (\big (K_2P_{\Delta } K_1\big )_{2,1}\big )\\ {}&+d\Gamma \big (\overline{-K_1P_{\Delta } K_1}\oplus K_2P_{\Delta } K_2\big ) + {\text{ Tr }}(\mathbb K_{\Delta }), \end{aligned}$$

Next, we note, if $\varphi ^{(2)}\in \mathcal G^{\otimes 2}$ and $\psi ^{(2)}\in \mathcal G^{\otimes 2}$ is defined by $\psi ^{(2)}(x,y):=\varphi ^{(2)}(y,x)$, then $a^+(\psi ^{(2)})=-a^+(\varphi ^{(2)})$. Using this observation and Remark 3.1, we similarly define $\rho (\Delta )$ for $\Delta \in \mathcal B_0(X_2)$:

$$\begin{aligned} \rho (\Delta ):=&- a^+\big (\big (K_2P_{\Delta } K_1\big )_{2,1}\big )- a^-\big (\big (K_2P_{\Delta } K_1\big )_{2,1}\big ) \\ {}&-d\Gamma \big (\overline{-K_1P_{\Delta } K_1}\oplus K_2P_{\Delta } K_2\big )+{\text{ Tr }}(\mathbb K_{\Delta }). \end{aligned}$$

Finally, for each $\Delta \in \mathcal B_0(X)$, we define $\rho (\Delta ):=\rho (\Delta \cap X_1)+\rho (\Delta \cap X_2)$.

We sum up our considerations in the following definition.

Definition 3.2

For each $\Delta \in \mathcal B_0(X)$, we define

$$\begin{aligned} \rho (\Delta )&:=a^+\big (\big (K_2J_{\Delta } K_1\big )_{2,1}\big )+a^-\big (\big (K_2J_{\Delta } K_1\big )_{2,1}\big )\\&\quad +d\Gamma \big (\overline{-K_1J_{\Delta } K_1}\oplus K_2J_{\Delta } K_2\big ) + {\text {Tr}}(\mathbb K_{\Delta \cap X_1})+ {\text {Tr}}(\mathbb K_{\Delta \cap X_2}), \end{aligned}$$

where $J_\Delta :=P_{\Delta \cap X_1}-P_{\Delta \cap X_2}$. Each operator $\rho (\Delta )$ belongs to $\mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$.

We note that, for each $\Delta \in \mathcal B_0(X)$, $\rho (\Delta )$ is a densely defined, Hermitian operator in $\mathcal{A}\mathcal{F}(\mathcal G)$.

Let $(e_i)_{i=1}^\infty$ be an orthonormal basis for $\mathcal H$ consisting of real-valued functions. The following proposition can be easily proved by using Definition 3.2.

Proposition 3.3

For $\Delta \in \mathcal {B}_0(X_1)$, we have

$$\begin{aligned} \rho (\Delta )&=\sum _{i,j=1}^{\infty }\Big [\big (K_2P_\Delta K_1 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)a^+_1(e_j)+ \big (K_1P_\Delta K_2 e_j,e_i\big )_\mathcal H\, a^-_1(e_i)a^-_2(e_j) \nonumber \\&\quad +\big (K_2P_\Delta K_2 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)a^-_2(e_j)+\big (K_1P_\Delta K_1 e_j,e_i\big )_\mathcal H\, a^-_1(e_i) a^+_1(e_j)\Big ], \end{aligned}$$

(44)

and for $\Delta \in \mathcal {B}_0(X_2)$, we have

$$\begin{aligned} \rho (\Delta )&=\sum _{i,j=1}^{\infty }\Big [\big (K_2P_\Delta K_1 e_i,e_j\big )_\mathcal H\, a^+_1(e_i)a^+_2(e_j)+\big (K_1P_\Delta K_2 e_i,e_j\big )_\mathcal H\, a^-_2(e_i)a^-_1(e_j) \nonumber \\&\quad + \big (K_1P_\Delta K_1 e_i,e_j\big )_\mathcal H\, a^+_1(e_i) a^-_1(e_j)+\big (K_2P_\Delta K_2 e_i,e_j\big )_\mathcal H\, a^-_2(e_i)a^+_2(e_j)\Big ]. \end{aligned}$$

(45)

In formulas (44) and (45), the series converge strongly in $\mathcal L(\mathcal {F}_{\textrm{fin}}(\mathcal G))$, i.e., for each $f\in \mathcal {F}_{\textrm{fin}}(\mathcal G)$, the series applied to the vector f converges in $\mathcal {F}_{\textrm{fin}}(\mathcal G)$ (hence also in $\mathcal {F}(\mathcal G)$).

Remark 3.4

Formulas (44) and (45) could serve as an alternative definition of $\rho (\Delta )$ for $\Delta$ from $\mathcal B_0(X_1)$ or $\mathcal B_0(X_2)$, respectively. However, if we initially accepted (44) and (45) as the definition of $\rho (\Delta )$, it would not be a priori clear if such a definition does not depend on the choice of a real orthonormal basis in $\mathcal H$.

Proposition 3.5

For any $\Delta _1,\Delta _2\in \mathcal B_0(X)$, we have $\rho (\Delta _1)\rho (\Delta _2)=\rho (\Delta _2)\rho (\Delta _1)$.

To prove Proposition 3.5, let us recall a result on strong convergence of bounded linear operators. Let $E_1$ and $E_2$ be Hilbert (or even Banach) spaces. Let $(B_n)_{n=1}^\infty$ be a sequence from $\mathcal L(E_1,E_2)$ and assume that $(B_n)_{n=1}^\infty$ converges strongly to $B\in \mathcal L(E_1,E_2)$. Then, the uniform boundedness principle states that $\sup _{n\in \mathbb N}\Vert B_n\Vert <\infty$, see e.g. [5, Chapter 8, Section 2]. This immediately implies

Lemma 3.6

Let $E_1,E_2,E_3$ be Hilbert spaces, let $\{B_n\}_{n=1}^\infty \in \mathcal L(E_1,E_2)$ and $\{C_n\}_{n=1}^\infty \in \mathcal L(E_2,E_3)$. Assume that the series $\sum _{n=1}^\infty B_n$ and $\sum _{n=1}^\infty$ strongly converge in $\mathcal L(E_1,E_2)$ and $\mathcal L(E_2,E_3)$, respectively. Then the series $\sum _{m=1}^\infty \sum _{n=1}^\infty B_mC_n$ converges strongly in $\mathcal L(E_1,E_3)$.

Proof of Proposition 3.5

The result follows from Proposition 3.3 and Lemma 3.6 by a tedious calculation that uses the CAR.

4 Wick Polynomials of the Particle Density

Recall that the Hermitian operators $\rho (\Delta )$ were heuristically defined by (11) and the corresponding Wick polynomials ${:}\rho (\Delta _1)\cdots \rho (\Delta _n){:}$ were defined by (17). Then the heuristic formula (13) holds, see e.g. [21, Section C]. For the reader’s convenience, we will now present a heuristic proof of (13).

Define the operator-valued distribution ${:}\rho (x_1)\cdots \rho (x_n){:}$ so that the following formula holds:

$$\begin{aligned} {:}\rho (\Delta _1)\cdots \rho (\Delta _n){:}=\int _{\Delta _1\times \cdots \times \Delta _n}{:}\rho (x_1)\cdots \rho (x_n){:}\,\sigma (dx_1)\cdots \sigma (dx_n) \end{aligned}$$

for all $\Delta _1,\dots ,\Delta _n\in \mathcal B_0(X)$. Then, by (17), we have ${:}\rho (x){:}=\rho (x)$ and

$$\begin{aligned} {:}\rho (x_1)\cdots \rho (x_{n+1}){:}=\rho (x_{n+1})\,{:}\rho (x_1)\cdots \rho (x_{n}){:}-\sum _{i=1}^n \delta (x_i,x_{n+1})\,{:}\rho (x_1)\cdots \rho (x_{n}){:}\,.\end{aligned}$$

(46)

Here the generalized function $\delta (x_1,x_2)$ is defined so that

$$\begin{aligned} \int _{X^2}f^{(2)}(x_1,x_2)\delta (x_1,x_2)\,\sigma (dx_1)\sigma (dx_2):=\int _Xf^{(2)}(x,x)\,\sigma (dx). \end{aligned}$$

It follows from the CAR that the operator-valued distributions $A^+(x)$, $A^-(x)$ satisfy the commutation relations:

$$\begin{aligned} \{A^+(x_1),A^+(x_2)\}=\{A^-(x_1),A^-(x_2)\}=0,\quad \{A^-(x_1),A^+(x_2)\}=\delta (x_1,x_2).\end{aligned}$$

(47)

Now (46) and (47) imply, by induction,

$$\begin{aligned} {:}\rho (x_1)\cdots \rho (x_n){:}=A^+(x_n)\cdots A^+(x_1)A^-(x_1)\cdots A^-(x_n).\end{aligned}$$

(48)

Note that formula (48) does not depend on the representation of the CAR, and just states that ${:}\rho (x_1)\cdots \rho (x_n){:}$ is the Wick (normal) ordering of the product $\rho (x_1)\cdots \rho (x_n)$.

Formula (13) can be recurrently written as

$$\begin{aligned}{} & {} {:}\rho (\Delta ){:}=\rho (\Delta ),\nonumber \\{} & {} {:}\rho (\Delta _1)\cdots \rho (\Delta _{n}){:}=\int _{\Delta _{n}}A^+(x_{n})\,{:}\rho (\Delta _1)\cdots \rho (\Delta _{n-1}){:}\,A^-(x_{n})\,\sigma (dx_{n}),\quad n\ge 2.\nonumber \\ \end{aligned}$$

(49)

The aim of this section is to derive a rigorous form of formula (49).

We start with presenting a rigorous form of the heuristic operator

$$\begin{aligned} W(\Delta ,R)={:}\rho (\Delta )R{:}=\int _\Delta A^+(X)RA^-(x)\sigma (dx),\end{aligned}$$

(50)

where $\Delta \in \mathcal B_0(X)$ and $R\in \mathcal {L}(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$. The following result is inspired by Proposition 3.3 and formula (50).

Proposition 4.1

Let $R\in \mathcal {L}(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$. For $\Delta \in \mathcal {B}_0(X_1)$, we define

$$\begin{aligned} W(\Delta ,R)&:=\sum _{i,j=1}^{\infty }\Big [\big (K_2P_\Delta K_1 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)Ra^+_1(e_j)+ \big (K_1P_\Delta K_2 e_j,e_i\big )_\mathcal H\, a^-_1(e_i)Ra^-_2(e_j)\nonumber \\&\quad +\big (K_2P_\Delta K_2 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)Ra^-_2(e_j) +\big (K_1P_\Delta K_1 e_j,e_i\big )_\mathcal H\, a^-_1(e_i) Ra^+_1(e_j)\Big ], \end{aligned}$$

(51)

and for $\Delta \in \mathcal {B}_0(X_2)$, we define

$$\begin{aligned} W(\Delta ,R)&:=\sum _{i,j=1}^{\infty }\Big [\big (K_2P_\Delta K_1 e_i,e_j\big )_\mathcal H \,a^+_1(e_i)R a^+_2(e_j)+\big (K_1P_\Delta K_2 e_i,e_j\big )_\mathcal H\, a^-_2(e_i) R a^-_1(e_j) \nonumber \\&\quad + \big (K_1P_\Delta K_1 e_i,e_j\big )_\mathcal H\, a^+_1(e_i) R a^-_1(e_j) +\big (K_2P_\Delta K_2 e_i,e_j\big )_\mathcal H a^-_2(e_i)R a^+_2(e_j)\Big ]. \end{aligned}$$

(52)

For $\Delta \in \mathcal {B}_0(X)$, we define $W(\Delta ,R):=W(\Delta \cap X_1,R)+W(\Delta \cap X_2,R)$. Then $W(\Delta ,R)\in \mathcal {L}(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$ In formulas (51) and (52), the series converge strongly in $\mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$.

Before proving Proposition 4.1, let us present the main result of this section, which gives a rigorous form of formula (49).

Proposition 4.2

For any $\Delta _1,\dots ,\Delta _{n}\in \mathcal B_0(X)$, $n\ge 2$, we have

$$\begin{aligned} {:}\rho (\Delta _1)\cdots \rho (\Delta _{n}){:}=W\big (\Delta _n,{:}\rho (\Delta _ 1)\cdots \rho (\Delta _{n-1}){:}\big ). \end{aligned}$$

(53)

Let us now prove Propositions 4.1 and 4.2.

Proof of Proposition 4.1

We will present the proof only in the case $\Delta \in \mathcal B_0(X_1)$. Below, we will denote by $\textbf{1}_k$ the identity operator in $\mathcal G^{\otimes k}$.

Step 1. Let $R_{m,n+1}\in \mathcal {L}(\mathcal G^{\wedge (n+1)},\mathcal G^{\wedge m})$. For any $\varphi ,\psi \in \mathcal G$ and $g^{(n)}\in \mathcal G^{\wedge n}$, we have

$$\begin{aligned} a^+(\varphi )R_{m,n+1}a^+(\psi )g^{(n)}=\mathcal A_{m+1}\big (\textbf{1}_1\otimes R_{m,n+1}\big ) \big (\textbf{1}_1\otimes \mathcal A_{n+1}\big ) \big (\varphi \otimes \psi \otimes g^{(n)}\big ). \end{aligned}$$

We set $e_n^{(1)}:=(e_n,0)$, $e_n^{(2)}:=(0,e_n)$ ($n\in \mathbb N$), which is an orthonormal basis for $\mathcal G$. Then, for any $M,N\in \mathbb N$,

$$\begin{aligned}&\sum _{i=1}^{M} \sum _{j=1}^{N}\big (K_2P_\Delta K_1 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)R_{m,n+1}a^+_1(e_j)g^{(n)} \\&\quad =\mathcal A_{m+1}\big (\textbf{1}_1\otimes R_{m,n+1}\big ) \big (\textbf{ 1}_1\otimes \mathcal A_{n+1}\big )\left[ \left( \sum _{i=1}^{M} \sum _{j=1}^{N}\big (K_2P_\Delta K_1 e_j,e_i\big )_\mathcal H\, e_i^{(2)}\otimes e_j^{(1)} \right) \otimes g^{(n)}\right] \\&\quad \rightarrow \mathcal A_{m+1}\big (\textbf{1}_1\otimes R_{m,n+1}\big ) \big (\textbf{ 1}_1\otimes \mathcal A_{n+1}\big ) \left[ \big (K_2P_\Delta K_1\big )_{2,1}\otimes g^{(n)}\right] \end{aligned}$$

in $\mathcal G^{\wedge (m+1)}$ as $M, N\rightarrow \infty$. This implies that the series

$$\begin{aligned} \sum _{i,j=1}^{\infty }\big (K_2P_\Delta K_1 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)Ra^+_1(e_j) \end{aligned}$$

converges strongly in $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$.

Step 2. Similarly to $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$, we define the topological vector space $\mathcal {F}_{\textrm{fin}}(\mathcal G)$ that consists of all finite sequences $g=(g^{(n)})_{n=0}^\infty$ with $g^{(n)}\in \mathcal G^{\otimes n}$ (here $\mathcal G^{\otimes 0}:=\mathbb C$).

For $\varphi \in \mathcal G$, we denote by $l^+(\varphi )$ the left creation operator and by $l^-(\varphi )$ the left annihilation operator in $\mathcal {F}_{\textrm{fin}}(\mathcal G)$:

$$\begin{aligned}&l^+(\varphi ) g_1 \otimes g_2\otimes \dots \otimes g_n= \varphi \otimes g_1\otimes \cdots \otimes g_n, \\&l^-(\varphi ) g_1 \otimes g_2\otimes \cdots \otimes g_n=(g_1,\varphi )_\mathcal G\, g_2\otimes g_3\otimes \cdots \otimes g_n, \end{aligned}$$

for $g_1,\dots ,g_n\in \mathcal G$. Obviously, $l^+(\varphi ), l^-(\varphi )\in \mathcal {L}( \mathcal {F}_{\textrm{fin}}(\mathcal G))$. Then, for each $g^{(n)}\in \mathcal G^{\wedge n}$,

$$\begin{aligned} a^+(\varphi )g^{(n)}=\mathcal A_{n+1}l^+(\varphi )g^{(n)},\quad a^-(\varphi )g^{(n)}=nl^-(\varphi )g^{(n)}. \end{aligned}$$

Let $\varphi ^{(2)}\in \mathcal G^{\otimes 2}$. We also define an operator $l^-(\varphi ^{(2)})\in \mathcal {L}(\mathcal {F}_{\textrm{fin}}(\mathcal G))$ by

$$\begin{aligned} l^-(\varphi ^{(2)}) g_1 \otimes g_2\otimes \cdots \otimes g_n=( g_1\otimes g_2,\varphi ^{(2)})_{\mathcal G^{\otimes 2}}\, g_3\otimes g_4\otimes \cdots \otimes g_n. \end{aligned}$$

In particular, if $\varphi ^{(2)}=\varphi _1\otimes \varphi _2$ with $\varphi _1, \varphi _2\in \mathcal G$, then $l^-(\varphi _1\otimes \varphi _2)=l^-(\varphi _2)l^-(\varphi _1)$.

Consider $R_{m,n-1}\in \mathcal {L}(\mathcal G^{\wedge (n-1)},\mathcal G^{\wedge m})$. Then, for $\varphi ,\psi \in \mathcal G$, $g^{(n)}\in \mathcal G^{\wedge n}$, we have

$$\begin{aligned}&a^-(\varphi )R_{m,n-1}a^-(\psi ) g^{(n)}= m\,l^-(\varphi )R_{m,n-1} n\, l^-(\psi ) g^{(n)}\\&\quad = mn\, l^-(\varphi )R_{m,n-1} \big (l^-(\psi )\restriction _\mathcal G \otimes \textbf{1}_{n-1}\big ) g^{(n)} =mn\, l^-(\varphi )\big (l^-(\psi )\restriction _\mathcal G \otimes R_{m,n-1}\big )g^{(n)}\\&\quad =mn \, l^-(\varphi ) (l^-(\psi )\restriction _\mathcal G\otimes \textbf{1}_m)(\textbf{1}_{1}\otimes R_{m,n-1})g^{(n)} =mn\,l^-(\varphi )l^-(\psi )(\textbf{1}_1\otimes R_{m,n-1}) g^{(n)}\\&\quad =mn\,l^-(\psi \otimes \varphi )(\textbf{1}_1\otimes R_{m,n-1})g^{(n)}. \end{aligned}$$

Hence, for $M,N\in \mathbb N$,

$$\begin{aligned}&\sum _{i=1}^{M} \sum _{j=1}^{N}\big (K_1P_\Delta K_2 e_j,e_i\big )_\mathcal H\, a^-_1(e_i)R_{m,n-1}a^-_2(e_j)g^{(n)} \\&\quad = mn \, l^-\Big (\sum _{i=1}^{N} \sum _{j=1}^{M} \big (K_2P_\Delta K_1 e_j,e_i\big )_\mathcal H\, e_i^{(2)}\otimes e_j^{(1)}\Big )(\textbf{1}_1\otimes R_{m,n-1})g^{(n)}\\&\quad \rightarrow mn \, l^-\big (\big (K_2P_\Delta K_1 \big )_{2,1}\big )(\textbf{1}_1\otimes R_{m,n-1})g^{(n)} \end{aligned}$$

in $\mathcal G^{\wedge n}$ as $M, N\rightarrow \infty$. This implies that the series

$$\begin{aligned} \sum _{i,j=1}^{\infty }\big (K_1P_\Delta K_2 e_j,e_i\big )_\mathcal H \,a^-_1(e_i)R_{m,n-1}a^-_2(e_j) \end{aligned}$$

converges strongly in $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$.

Step 3. Let $R_{m,n-1}\in \mathcal {L}(\mathcal G^{\wedge (n-1)}, \mathcal G^{\wedge m}).$ For $\varphi ,\psi \in \mathcal G$ and $g ^{(n)}\in \mathcal G^{\wedge n}$, we have

$$\begin{aligned} a^+(\varphi ) R_{m,n-1} a^-(\psi )g^{(n)} =n \mathcal A_{m+1}\big (\big [\big (\cdot ,\psi )_\mathcal G\, \varphi \big ]\otimes R_{m,n-1} \big )g^{(n)}. \end{aligned}$$

Hence, for any $M, N\in \mathbb {N}$,

$$\begin{aligned}&\sum _{i=1}^{M} \sum _{j=1}^{N}\big (K_2P_\Delta K_2 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)R_{m,n-1}a^-_2(e_j)g^{(n)} \\&\quad =n\mathcal A_{m+1} \bigg (\Big (\sum _{i=1}^{M} \sum _{j=1}^{N}\big (K_2P_\Delta K_2 e_j,e_i\big )_\mathcal H \, (\cdot ,e_j^{(2)})_\mathcal G\,e_i^{(2)}\Big )\otimes R_{m,n-1} \bigg )g^{(n)}\\&\quad \rightarrow n \mathcal A_{m+1} \big ( (\textbf{0}\oplus K_2P_\Delta K_2)\otimes R_{m,n-1}\big ) g^{(n)} \end{aligned}$$

in $\mathcal G^{\wedge (m+1)}$ as $M, N\rightarrow \infty$. This implies that the series

$$\begin{aligned} \sum _{i,j=1}^{\infty }\big (K_2P_\Delta K_2 e_j,e_i\big )_\mathcal H \,a^+_2(e_i)R a^-_2(e_j) \end{aligned}$$

converges strongly in $\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$.

Step 4. Recall that, for all $x\in X_1$, we have $K_1(x,\cdot )\in \mathcal H$. Therefore, for all $x\in X_1$, the operators $a_1^+\big (K_1(x,\cdot )\big )$ and $a_1^-\big (K_1(x,\cdot )\big )$ are bounded in $\mathcal{A}\mathcal{F}(\mathcal G)$ and have norm $\Vert K_1(x,\cdot )\Vert _\mathcal H$.

Let $R_{m,n+1}\in \mathcal {L}(\mathcal G^{\wedge (n+1)},\mathcal G^{\wedge m})$. We will now show that

$$\begin{aligned} \int _{\Delta } a_1^-\big (K_1(x,\cdot )\big ) R_{m,n+1} a_1^+\big (K_1(x,\cdot )\big )\,\sigma (dx) \end{aligned}$$

(54)

exists as a Bochner integral with values in $\mathcal {L}(\mathcal G^{\wedge n},\mathcal G^{\wedge (m-1)})$. For the definition and properties of a Bochner integral, see e.g. [5, Chapter 10, Section 3].

Lemma 4.3

The mappings

$$\begin{aligned} \Delta \ni x\mapsto a_1^+\big (K_1(x,\cdot )\big )\in \mathcal {L}(\mathcal{A}\mathcal{F}(\mathcal G)),\quad \Delta \ni x\mapsto a_1^-\big (K_1(x,\cdot )\big )\in \mathcal {L}(\mathcal{A}\mathcal{F}(\mathcal G)) \end{aligned}$$

are strongly measurable.

Proof

Consider the product space $\Delta \times X$ equipped with the corresponding $\sigma$-algebra. We will say that a function $g:\Delta \times X\rightarrow \mathbb {C}$ is rectangle-simple if it is of the form

$$\begin{aligned} g(x,y)=\sum _{i=1}^n c_i \chi _{A_i\times B_i}(x,y), \end{aligned}$$

where $c_i\in \mathbb C$, $A_i\in \mathcal {B}(X_1)$, $A_i\subset \Delta$, $B_i\in \mathcal {B}(X)$, $\sigma (B_i)<\infty$ ($i=1,\dots ,n$), and the sets $A_i\times B_i$ are mutually disjoint. Since $K_1(\cdot ,\cdot )\in L^2(\Delta \times X_1,\sigma ^{\otimes 2})$, there exists a sequence $(g_n)_{n=1}^\infty$ of rectangle-simple functions such that $g_n(x,y)\rightarrow K_1(x,y)$ and $|g_n(x,y)|\leqslant |K_1(x,y)|$ for $\sigma ^{\otimes 2}$-a.a. $(x,y)\in \Delta \times X$. In particular, $g_n(\cdot ,\cdot )\rightarrow K_1(\cdot ,\cdot )$ in $L^2(\Delta \times X,\sigma ^{\otimes 2})$. By (27), this implies that $\big (a^+_1(g_n(x,\cdot ))\big )_{n=1}^\infty$ and $\big (a^-_1(g_n(x,\cdot ))\big )_{n=1}^\infty$ are sequences of simple functions on $\Delta$ with values in $\mathcal {L}(\mathcal{A}\mathcal{F}(\mathcal G))$ such that, for $\sigma$-a.a. $x\in \Delta$,

$$\begin{aligned} a^+_1(g_n(x,\cdot ))\rightarrow a^+_1(K_1(x,\cdot )),\quad a^-_1(g_n(x,\cdot ))\rightarrow a^-_1(K_1(x,\cdot )), \end{aligned}$$

where convergence is in $\mathcal {L}(\mathcal{A}\mathcal{F}(\mathcal G))$. This implies that the statement of the lemma holds.    $\square$

Lemma 4.3 easily implies that the mapping

$$\begin{aligned} \Delta \ni x\mapsto a_1^-\big (K_1(x,\cdot )\big ) R_{m,n+1} a_1^+\big (K_1(x,\cdot )\big ) \in \mathcal {L}(\mathcal G^{\wedge n},\mathcal G^{\wedge (m-1)}) \end{aligned}$$

is strongly measurable. Furthermore, by (27),

$$\begin{aligned}&\int _{\Delta } \Vert a_1^-\big (K_1(x,\cdot )\big ) R_{m,n+1} a_1^+\big (K_1(x,\cdot )\big )\Vert _{\mathcal {L}(\mathcal G^{\wedge n},\mathcal G^{\wedge (m-1)})} \,\sigma (dx) \\&\quad \le \Vert R_{m,n+1} \Vert _{\mathcal {L}(\mathcal G^{\wedge (n+1)},\mathcal G^{\wedge m})} \int _{\Delta \times X} |K_1(x,y)|^2 \,\sigma (dx)\,\sigma (dy)< \infty . \end{aligned}$$

Hence, by [5, Chapter 10, Theorem 3.1], the Bochner integral (54) exists.

We have,

$$\begin{aligned} a_1^+\big (K_1(x,\cdot )\big )&= \sum _{i=1}^\infty \int _X K_1(x,y) e_i(y)\,d\sigma (y) \, a^+_1(e_i),\\ a_1^-\big (K_1(x,\cdot )\big )&= \sum _{i=1}^\infty \int _X \overline{K_1(x,y)} e_i(y)\,d\sigma (y) \, a^-_1(e_i), \end{aligned}$$

where the series converges in $\mathcal {L}(\mathcal{A}\mathcal{F}(\mathcal G))$. Note also that, for each $N\in \mathbb {N}$,

$$\begin{aligned}&\bigg \Vert \sum _{i=1}^N \int _X K_1(x,y) e_i(y)\,\sigma (dy)\, a^+_1(e_i)\bigg \Vert _{\mathcal {L}(\mathcal{A}\mathcal{F}(\mathcal G))}\leqslant \Vert K_1(x,\cdot )\Vert _\mathcal H,\\&\bigg \Vert \sum _{i=1}^N \int _X \overline{K_1(x,y)} e_i(y)\,\sigma (dy) \, a^-_1(e_i)\bigg \Vert _{\mathcal {L}(\mathcal{A}\mathcal{F}(G))}\leqslant \Vert K_1(x,\cdot )\Vert _\mathcal H. \end{aligned}$$

Using the dominated convergence theorem for a Bochner integral (see e.g. [5, Chapter 10, Exercise 3.6]), we get

$$\begin{aligned}&\int _\Delta a_1^-\big (K_1(x,\cdot )\big ) R_{m,n+1}\, a_1^+\big (K_1(x,\cdot )\big )\,\sigma (dx) \\&\quad =\sum _{i,j=1}^\infty \int _\Delta \sigma (dx)\int _X \sigma (dy)\, K_1(x,y) e_j(y) \int _X \sigma (dy^{\prime })\, \overline{K_1(x,y^{\prime })} e_i(y^{\prime }) \, a_1^-(e_i) R_{m,n+1} a_1^+(e_j)\\&\quad =\sum _{i,j=1}^\infty \big (K_1P_\Delta K_1 e_j, e_i\big )_\mathcal H\, a_1^-(e_i) R_{m,n+1} a_1^+(e_j), \end{aligned}$$

where the series converges in $\mathcal {L}(\mathcal G^{\wedge n}, \mathcal G^{\wedge (m+1)})$.

To prove Proposition 4.2, we first need the following

Lemma 4.4

Let $\Delta _1,\Delta _2\in \mathcal {B}_0(X)$ and $R\in \mathcal {L}(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$. Then

$$\begin{aligned} \rho (\Delta _1)W(\Delta _2,R)=W(\Delta _2,\rho (\Delta _1) R)+W(\Delta _1\cap \Delta _2, R). \end{aligned}$$

(55)

Proof

By linearity, we can assume that $\Delta _1\in \mathcal {B}_0(X_{i_{1}})$, $\Delta _2\in \mathcal {B}_0(X_{i_{2}})$, where $i_1,i_2\in \{1,2\}$. Consider, for example, the case where $\Delta _1,\Delta _2\in \mathcal B_0(X_1)$. By (44), (51) and Lemma 3.6, we then write down the left-hand side of (55) as

$$\begin{aligned}&\rho (\Delta _1)W(\Delta _2,R)\nonumber \\&\quad =\sum _{i,j,k,l=1}^{\infty }\Big [\big (K_2P_{\Delta _1} K_1 e_j,e_i\big )_\mathcal H\, a^+_2(e_i)a^+_1(e_j)+ \big (K_1P_{\Delta _1} K_2 e_j,e_i\big )_\mathcal H\, a^-_1(e_i)a^-_2(e_j) \nonumber \\&\qquad +\big (K_2P_{\Delta _1} K_2 e_j,e_i\big )_\mathcal H a^+_2(e_i)a^-_2(e_j)+\big (K_1P_{\Delta _1} K_1 e_j,e_i\big )_\mathcal H a^-_1(e_i) a^+_1(e_j)\Big ]\nonumber \\&\qquad \times \Big [\big (K_2P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^+_2(e_k)Ra^+_1(e_l)+ \big (K_1P_{\Delta _2} K_2 e_l,e_k\big )_\mathcal H\, a^-_1(e_k)Ra^-_2(e_l)\nonumber \\&\qquad +\big (K_2P_{\Delta _2} K_2 e_l,e_k\big )_\mathcal H\, a^+_2(e_k)Ra^-_2(e_l) +\big (K_1P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^-_1(e_k) Ra^+_1(e_l)\Big ], \end{aligned}$$

(56)

and similarly we write down the right-hand side of (55). (The appearing series converge strongly in $\mathcal L(\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G))$.) Through lengthy but rather straightforward calculations, one shows that both expression are equal. To give the reader a feeling how these calculations are carried out, we consider the following term appearing in (56):

$$\begin{aligned}&\sum _{i,j,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_2 e_j,e_i\big )_\mathcal H\big (K_2P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H a^-_1(e_i)a^-_2(e_j)a^+_2(e_k)Ra^+_1(e_l)\nonumber \\&\quad = \sum _{i,j,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_2 e_j,e_i\big )_\mathcal H\big (K_2P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^+_2(e_k)a^-_1(e_i)a^-_2(e_j)Ra^+_1(e_l)\nonumber \\&\qquad +\sum _{i,j,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_2 e_j,e_i\big )_\mathcal H\big (K_2P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, \delta _{j,k}\,a^-_1(e_i)R a^+_1(e_l), \end{aligned}$$

(57)

where we used the CAR. The second sum on the right-hand side of (57) is equal to

$$\begin{aligned}&\sum _{i,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_2 e_i,e_k\big )_\mathcal H\big (K_2P_{\Delta _2} K_1 e_l,e_i\big )_\mathcal H\, a^-_1(e_k)R\, a^+_1(e_l)\nonumber \\&\quad =\sum _{k,l=1}^{\infty }\big (K_2P_{\Delta _2} K_1 e_l,K_2P_{\Delta _1} K_1 e_k\big )_\mathcal H\, a^-_1(e_k)R\, a^+_1(e_l)\nonumber \\&\quad =\sum _{k,l=1}^{\infty }\big (K_1P_{\Delta _1}(\textbf{ 1}-K)P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^-_1(e_k)R\, a^+_1(e_l)\nonumber \\&\quad =\sum _{k,l=1}^{\infty }\big (K_1P_{\Delta _1\cap \Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^-_1(e_k)R\, a^+_1(e_l)\nonumber \\&\qquad -\sum _{k,l=1}^{\infty }\big (K_1P_{\Delta _1}KP_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^-_1(e_k)R\, a^+_1(e_l). \end{aligned}$$

(58)

On the other hand, another term appearing in (56) is:

$$\begin{aligned}&\sum _{i,j,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_1 e_j,e_i\big )_\mathcal H \big (K_1P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^-_1(e_i) a^+_1(e_j)a^-_1(e_k) R\,a^+_1(e_l) \nonumber \\&\quad = \sum _{i,j,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_1 e_j,e_i\big )_\mathcal H \big (K_1P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, a^-_1(e_k) a^-_1(e_i) a^+_1(e_j)R\,a^+_1(e_l) \nonumber \\&\qquad + \sum _{i,j,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_1 e_j,e_i\big )_\mathcal H \big (K_1P_{\Delta _2} K_1 e_l,e_k\big )_\mathcal H\, \delta _{j,k}\,a^-_1(e_i) R\,a^+_1(e_l). \end{aligned}$$

(59)

Similarly, the second sum on the right-hand side of (59) is equal to

$$\begin{aligned}&\sum _{i,k,l=1}^{\infty }\big (K_1P_{\Delta _1} K_1 e_i,e_k\big )_\mathcal H \big (K_1P_{\Delta _2} K_1 e_l,e_i\big )_\mathcal H a^-_1(e_k) R\,a^+_1(e_l) \nonumber \\&\quad = \sum _{k,l=1}^{\infty }\big (K_1P_{\Delta _2} K_1 e_l,K_1P_{\Delta _1} K_1 e_k\big )_\mathcal Ha^-_1(e_k) R\,a^+_1(e_l) \nonumber \\&\quad =\sum _{k,l=1}^{\infty }\big (K_1P_{\Delta _1} K P_{\Delta _2} K_1 e_l, e_k\big )_\mathcal H a^-_1(e_k) R\,a^+_1(e_l). \end{aligned}$$

(60)

Thus, we see that, in formula (57), the ‘wrong’ term given by (60) cancels out, the first sum on the right-hand side of (57) and the first sum on the right-hand side of (59) come from $W(\Delta _2,\rho (\Delta _1) R)$, and the first sum on the right-hand side of (58) comes from $W(\Delta _1\cap \Delta _2, R)$.

We leave the rest of calculations to the interested reader. $\square$

Proof of Proposition 4.2

For $\Delta _1,\Delta _2\in \mathcal B_0(X)$, we have by Lemma 4.4,

$$\begin{aligned} \rho (\Delta _1)\rho (\Delta _2)=\rho (\Delta _1)W(\Delta _2,\textbf{1})=W(\Delta _2,\rho (\Delta _1))+\rho (\Delta _1\cap \Delta _2). \end{aligned}$$

Hence, by (17),

$$\begin{aligned} {:}\rho (\Delta _1)\rho (\Delta _2){:}=W(\Delta _2,\rho (\Delta _1)), \end{aligned}$$

i.e., formula (53) holds for $n=2$. Assume formula (53) holds for n and let us prove it for $n+1$. By Lemma 4.4 and (17),

$$\begin{aligned}&\rho (\Delta _1){:}\rho (\Delta _2)\cdots \rho (\Delta _{n+1}){:}=\rho (\Delta _1) W\big (\Delta _{n+1},{:}\rho (\Delta _2)\cdots \rho (\Delta _{n}){:}\big )\\&\quad = W\big (\Delta _{n+1}, \rho (\Delta _1){:}\rho (\Delta _2)\cdots \rho (\Delta _{n}){:}\big )+W\big (\Delta _1\cap \Delta _{n+1},{:}\rho (\Delta _2)\cdots \rho (\Delta _{n}){:}\big )\\&\quad = W\big (\Delta _{n+1}, {:}\rho (\Delta _1)\rho (\Delta _2)\cdots \rho (\Delta _{n}){:}\big )\\&\quad \quad +\sum _{i=2}^n W\big (\Delta _{n+1},{:}\rho (\Delta _2)\cdots \rho (\Delta _1\cap \Delta _i)\cdots \rho (\Delta _n){:}\big )+{:}\rho (\Delta _1\cap \Delta _{n+1})\rho (\Delta _2)\cdots \rho (\Delta _n){:}\\&\quad = W\big (\Delta _{n+1}, {:}\rho (\Delta _1)\rho (\Delta _2)\cdots \rho (\Delta _{n}){:}\big )+\sum _{i=2}^{n+1} {:}\rho (\Delta _2)\cdots \rho (\Delta _1\cap \Delta _i)\cdots \rho (\Delta _{n+1}){:}, \end{aligned}$$

which, by (17), implies that formula (53) holds for $n+1$.

5 The Joint Spectral Measure of the Particle Density

Our aim now is to apply the results of Sect. 2.3 by setting $\mathcal F=\mathcal{A}\mathcal{F}(\mathcal G)$, $\mathcal D=\mathcal{A}\mathcal{F}_{\textrm{fin}}(\mathcal G)$, $\mathcal A$ to be the commutative $*$-algebra generated by the particle density$(\rho (\Delta ))_{\Delta \in \mathcal B_0(X)}$, and $\tau$ to be the vacuum state on $\mathcal A$, i.e., $\tau (a)=(a\Omega ,\Omega )_{\mathcal{A}\mathcal{F}(\mathcal G)}$ ($a\in \mathcal A$), where $\Omega$ is the vacuum in $\mathcal{A}\mathcal{F}(\mathcal G)$.

Recall the construction of the integral kernel $\mathbb K(x,y)$ of the operator $\mathbb K$ in Sect. 2.2.

Theorem 5.1

The operators $(\rho (\Delta ))_{\Delta \in \mathcal B_0(X)}$ have correlation measures $(\theta ^{(n)})_{n=1}^\infty$ respective the vacuum state state $\tau$. Furthermore, the corresponding correlation functions are given by

$$\begin{aligned} k^{(n)}(x_1,\dots ,x_n)=\det \big [\mathbb K(x_i,x_j)\big ]_{i,j=1,\dots ,n}.\end{aligned}$$

(61)

We will prove Theorem 5.1 below, in Sect. 6, but let us now formulate and prove the main theorem of the paper.

Theorem 5.2

The operators $(\rho (\Delta ))_{\Delta \in \mathcal B_0(X)}$, together with their correlation measures $(\theta ^{(n)})_{n=1}^\infty$, satisfy the assumptions of Theorem 2.5. Thus, the following statements hold:

• (i) Let $\mathfrak D:=\{a\Omega \mid a\in \mathcal A\}$ and let $\mathfrak F$ denote the closure of $\mathfrak D$ in $\mathcal{A}\mathcal{F}(\mathcal G)$. Each operator $(\rho (\Delta ),\mathfrak D)$ is essentially self-adjoint in $\mathfrak F$, i.e., the closure of $\rho (\Delta )$, denoted by $\widetilde{\rho }(\Delta )$, is a self-adjoint operator in $\mathfrak F$.

• (ii) For any $\Delta _1,\Delta _2\in \mathcal B_0(X)$, the projection-valued measures (resolutions of the identity) of the operators $\widetilde{\rho }(\Delta _1)$ and $\widetilde{\rho }(\Delta _2)$ commute.

• (iii) There exist a unique point process $\mu$ in X and a unique unitary operator$U:\mathfrak F\rightarrow L^2(\Gamma _X,\mu )$ satisfying $U\Omega =1$ and (20). In particular, (21) holds.

• (iv) The correlations functions of the point process $\mu$ are given by (61).

Proof

We first prove the following

Lemma 5.3

For any $\Delta \in \mathcal B_0(X)$,

$$\begin{aligned} \int _\Delta \mathbb K(x,x)\sigma (dx)={\text {Tr}}(\mathbb K_{\Delta \cap X_1})+{\text {Tr}}(\mathbb K_{\Delta \cap X_2}), \end{aligned}$$

(62)

and for any $\Delta _1,\dots ,\Delta _n\in \mathcal B_0(X)$ ($n\ge 2$), we have

$$\begin{aligned} \int _{\Delta _1\times \dots \times \Delta _{n}}\det \big [\mathbb K(x_i,x_j)]_{i,j=1,\dots ,n} \,\sigma (dx_1)\cdots \sigma (dx_{n})=\sum _{\xi \in S_{n}}(-1)^{{\text {sgn}}(\xi )}\prod _{\psi \in {\text {Cycles}}(\xi )} \mathbb {T}_{\psi },\end{aligned}$$

(63)

where, for a cycle $\psi =(l_1 l_2 \cdots l_k)$ in a permutation $\xi \in S_{n}$, we have

$$\begin{aligned} \mathbb T_\psi&= \int _{\Delta _{l_1}}\mathbb K(x,x)\sigma (dx)={\text {Tr}}(\mathbb K_{\Delta _{l_1}\cap X_1})+{\text {Tr}}(\mathbb K_{\Delta _{l_1}\cap X_2}),\quad \text {if }k=1, \end{aligned}$$

(64)

$$\begin{aligned} \mathbb T_\psi&={\text {Tr}}\big (P_{\Delta _{l_1}}\mathbb K P_{\Delta _{l_2}} \mathbb K P_{\Delta _{l_3}}\mathbb K \cdots \mathbb K P_{\Delta _{l_k}}\mathbb K P_{\Delta _{l_1}}\big ),\quad \text {if }k\ge 2. \end{aligned}$$

(65)

Proof

Recall that that, if $S, T\in \mathcal S_2(\mathcal H)$ with integral kernels S(x, y) and T(x, y), respectively, then

$$\begin{aligned} {\text {Tr}}(ST)=\int _X (ST)(x,x)\sigma (dx)=\int _{X^2}S(x,y)T(y,x)\sigma (dx)\,\sigma (dy). \end{aligned}$$

Hence, formula (62) holds by statement (ii) of Lemma 2.1 and the construction of the integral kernel $\mathbb K(x,y)$.

Next, we note that, for any $\Delta _1,\Delta _2\in \mathcal B_0(X)$, we have $P_{\Delta _1}\mathbb K P_{\Delta _2}\in \mathcal S_2(\mathcal H)$, hence the operator appearing in (65) is indeed of trace class. Formula (63) obviously holds with $\mathbb T_\psi$ given, for $k=1$ by (64), and for $k\ge 2$,

$$\begin{aligned} \mathbb T_\psi&= \int _{\Delta _{l_1}\times \Delta _{l_2}\times \dots \times \Delta _{l_{k}}} \mathbb K(x_{l_1},x_{l_2}) \mathbb K(x_{l_2},x_{l_3})\\&\quad \times \cdots \times \mathbb K(x_{l_{k-1}},x_{l_k})\mathbb K(x_{l_k},x_{l_1})\sigma (dx_{l_1})\cdots \sigma (dx_{l_k})\\&=\int _{X^k} \big (P_{\Delta _{l_1}}\mathbb K P_{\Delta _{l_2}}\big )(x_1,x_2) \big (P_{\Delta _{l_2}}\mathbb K P_{\Delta _{l_3}}\big )(x_2,x_3)\\&\quad \times \cdots \times \big (P_{\Delta _{l_{k-1}}}\mathbb K P_{\Delta _{l_k}}\big )(x_{k-1},x_k)\big (P_{\Delta _{l_k}}\mathbb K P_{\Delta _{l_1}}\big )(x_k,x_1)\sigma (dx_{1})\cdots \sigma (dx_{k})\\&=\int _X \big (P_{\Delta _{l_1}}\mathbb K P_{\Delta _{l_2}} \mathbb K P_{\Delta _{l_3}}\mathbb K \cdots \mathbb K P_{\Delta _{l_k}}\mathbb K P_{\Delta _{l_1}}\big )(x,x)\sigma (dx)\\&={\text {Tr}}\big (P_{\Delta _{l_1}}\mathbb K P_{\Delta _{l_2}} \mathbb K P_{\Delta _{l_3}}\mathbb K \cdots \mathbb K P_{\Delta _{l_k}}\mathbb K P_{\Delta _{l_1}}\big ). \end{aligned}$$

$\square$

Let us prove that condition (LB1) of Theorem 2.5 is satisfied. For $l\ge 2$, we have

$$\begin{aligned} \big |{\text {Tr}}(\mathbb K^l_\Delta )\big |\le \Vert \mathbb K^l_\Delta \Vert _1{} & {} \le \Vert \mathbb K_\Delta \Vert _2\,\Vert \mathbb K^{l-1}_\Delta \Vert _2\le \Vert \mathbb K_\Delta \Vert _2^2 \,\Vert \mathbb K^{l-2}_\Delta \Vert \nonumber \\{} & {} \le \max \big \{\Vert \mathbb K_\Delta \Vert _2,\,\Vert \mathbb K_\Delta \Vert \big \}^l, \end{aligned}$$

(66)

where $\Vert \cdot \Vert _1$ denotes the norm in $\mathcal S_1(\mathcal H)$. Theorem 5.1, Lemma 5.3 and formula (66) imply that condition (LB1) is satisfied with

$$\begin{aligned} C_\Delta = \max \big \{\Vert \mathbb K_{\Delta \cap X_1}\Vert _1+\Vert \mathbb K_{\Delta \cap X_2}\Vert _1,\Vert \mathbb K_\Delta \Vert _2,\,\Vert \mathbb K_\Delta \Vert \big \}. \end{aligned}$$

In the case where $\Delta$ is a subset of either $X_1$ or $X_2$, we can find a finer estimate of $\theta ^{(n)}(\Delta ^n)$. Indeed, assume, for example that $\Delta \subset X_1$. Then, for any $x,y\in \Delta$, we have, by (16),

$$\begin{aligned} \mathbb K(x,y)=(K_1(x,\cdot ),K_1(y,\cdot ))_{\mathcal H}, \end{aligned}$$

and so, for any $x_1,\dots ,x_n\in \Delta$,

$$\begin{aligned} \det \big [\mathbb K(x_i,x_j)]_{i,j=1,\dots ,n}=\big (K_1(x_1,\cdot )\wedge \dots \wedge K_1(x_n,\cdot ), K_1(x_1,\cdot )\wedge \dots \wedge K_1(x_n,\cdot )\big )_{\mathcal H^{\wedge n}}\, n!, \end{aligned}$$

which implies

$$\begin{aligned} \left| \det \big [\mathbb K(x_i,x_j)]_{i,j=1,\dots ,n}\right| \le \Vert K_1(x_1,\cdot )\Vert ^2_{\mathcal H}\cdots \Vert K_1(x_n,\cdot )\Vert ^2_{\mathcal H}\,n!. \end{aligned}$$

Hence, by Theorem 5.1, condition (LB1) is satisfied with

$$\begin{aligned} C_\Delta =\int _{\Delta \times X}|K_1(x,y)|^2\sigma (dx)\sigma (dy).\end{aligned}$$

(67)

Similarly, for $\Delta \in \mathcal B_0(X_2)$, (LB1) is satisfied with

$$\begin{aligned} C_\Delta =\int _{\Delta \times X}|K_2(x,y)|^2\sigma (dx)\sigma (dy).\end{aligned}$$

(68)

It follows from the proof of Theorem 2.5 in [20] that, when checking condition (LB2), it is sufficient to assume that all sets in the sequence $\{\Delta _{l}\}_{l\in \mathbb {N}}$ are subsets of either $X_1$ or $X_2$. But then (LB2) is an immediate consequence of formulas (67) and (68). $\square$

6 Proof of Theorem 5.1

We will now prove Theorem 5.1. Our strategy here is to prove the existence of correlation measures and that of correlation functions at the same time.

We first state, for any $x_1,\dots ,x_n\in X$,

$$\begin{aligned} \det \big [\mathbb K(x_i,x_j)\big ]_{i,j=1,\dots ,n}\ge 0. \end{aligned}$$

(69)

Indeed, if all points $x_1,\dots ,x_n$ belong to the same part, $X_1$ or $X_2$, then the matrix $\big [\mathbb K(x_i,x_j)\big ]_{i,j=1,\dots ,n}$ is Hermitian, hence its determinant is $\ge 0$. Otherwise, without loss of generality, we may assume that for some m with $1<m<n$, we have $x_1,\dots ,x_m\in X_1$ and $x_{m+1},\dots ,x_n\in X_2$. But then formula (69) follows from [22, Proposition 1.4].

Next, it is easy to see that, if among points $x_1,\dots ,x_n$, at least two points coincide, then $\det \big [\mathbb K(x_i,x_j)\big ]_{i,j=1,\dots ,n}=0$. Therefore, the measure

$$\begin{aligned} \det \big [\mathbb K(x_i,x_j)\big ]_{i,j=1,\dots ,n}\,\frac{1}{n!}\,\sigma (dx_1)\cdots \sigma (dx_n) \end{aligned}$$

is concentrated on $X^{(n)}$.

Hence, to prove Theorem 5.1, it suffices to show that, for any $\Delta _1,\dots ,\Delta _m\in \mathcal {B}_0(X_1)$, $\Delta _{m+1},\dots ,\Delta _{m+n}\in \mathcal {B}_0(X_2)$, $m,n\in \mathbb N_0$, $m+n\ge 1$, we have

$$\begin{aligned}&\tau \big ({:}\rho (\Delta _1)\cdots \rho (\Delta _{m+n}){:}\big )\nonumber \\&\quad =\int _{\Delta _1\times \cdots \times \Delta _{m+n}}\det \big [\mathbb K(x_i,x_j)\big ]_{i,j=1,\cdots ,m+n}\, \sigma (dx_1)\cdots \sigma (dx_{m+n}). \end{aligned}$$

(70)

We divide the proof of this formula into several steps.

Step 1. To shorten our notations, we denote, for $i,j\in \mathbb N$ and $\Delta \in \mathcal B_0(X_1)$,

$$\begin{aligned}&c_{ij}^{++}(\Delta )=\big (K_2P_\Delta K_1 e_j,e_i\big )_\mathcal H,\quad c_{ij}^{--}(\Delta )=\big (K_1P_\Delta K_2 e_j,e_i\big )_\mathcal H, \\&c_{ij}^{+-}(\Delta ) =\big (K_2P_\Delta K_2 e_j,e_i\big )_\mathcal H,\quad c_{ij}^{-+}(\Delta )=\big (K_1P_\Delta K_1 e_j,e_i\big )_\mathcal H , \end{aligned}$$

for $\Delta \in \mathcal {B}_{0}(X_2)$,

$$\begin{aligned}&c_{ij}^{++}(\Delta )=\big (K_2P_\Delta K_1 e_i,e_j\big )_\mathcal H,\quad c_{ij}^{--}(\Delta )=\big (K_1P_\Delta K_2 e_i,e_j\big )_\mathcal H ,\\&c_{ij}^{+-}(\Delta )= \big (K_1P_\Delta K_1 e_i,e_j\big )_\mathcal H ,\quad c_{ij}^{-+}(\Delta )=\big (K_2P_\Delta K_2 e_i,e_j\big )_\mathcal H, \end{aligned}$$

and

$$\begin{aligned} A^+_i:=a^+_2(e_i),\quad A^-_i:=a^-_1(e_i),\quad B_i^+:=a_1^+(e_i),\quad B_i^-:=a_2^-(e_i). \end{aligned}$$

Then, by Proposition 3.3, for $\Delta \in \mathcal {B}_0(X_1)$,

$$\begin{aligned}&\rho (\Delta )=\sum _{i,j=1}^\infty \quad \sum _{{\Diamond _1,\Diamond _2}\in \{+,-\}} c_{ij}^{ \Diamond _1\Diamond _2}(\Delta ) A^{\Diamond _1}_i B^{\Diamond _2}_j , \end{aligned}$$

and for $\Delta \in \mathcal {B}_0(X_2)$,

$$\begin{aligned}&\rho (\Delta )=\sum _{i,j=1}^\infty \quad \sum _{{\Diamond _1,\Diamond _2}\in \{+,-\}} c_{ij}^{ \Diamond _1\Diamond _2}(\Delta ) B^{\Diamond _1}_i A^{\Diamond _2}_j .\end{aligned}$$

We define an ordered set

$$\begin{aligned} \mathfrak {E}:=\{1,\dots ,m,m+1,\dots ,m+n,(m+n)',\dots ,(m+1)',m',(m-1)',\dots ,1'\}\end{aligned}$$

(71)

(the elements of $\mathfrak E$ being listed in (71) in the increasing order). By Proposition 4.2,

$$\begin{aligned}&\tau \big ({:}\rho (\Delta _1)\cdots \rho (\Delta _{m+n}){:}\big ) \nonumber \\&\quad =\sum _{i_1,\dots ,i_{m+n},i_{(m+n)'},\dots ,i_{1'}\in \mathbb N}\ \sum _{\Diamond _1,\dots ,\Diamond _{m+n},\Diamond _{(m+n)'},\dots ,\Diamond _{1'}\in \{+,-\}} c_{i_1i_{1'}}^{\Diamond _1\Diamond _{1'}}(\Delta _1)\cdots c_{i_{m+n}i_{(m+n)'}}^{\Diamond _{m+n}\Diamond _{(m+n)'}}(\Delta _{m+n})\nonumber \\&\qquad \times \tau \big (A_{i_1}^{\Diamond _1}\cdots A_{i_m}^{\Diamond _m}B_{i_{m+1}}^{\Diamond _{m+1}}\cdots B_{i_{m+n}}^{\Diamond _{m+n} }A_{i_{(m+n)'}}^{\Diamond _{(m+n)'}}\cdots A_{i_{(m+1)'}}^{\Diamond (m+1)'}B_{i_{m'}}^{\Diamond _{m'}}\cdots B_{i_{1'}}^{\Diamond _{1'}}\big )\nonumber \\&\quad =\sum _{i_1,\dots ,i_{m+n},i_{(m+n)'},\dots ,i_{1'}\in \mathbb N}\quad c_{i_1i_1'}^{- +}(\Delta _1)\cdots c_{i_{m}i_m'}^{- +}(\Delta _{m})\nonumber \\&\qquad \times \sum _{\Diamond _{m+1},\dots ,\Diamond _{m+n},\Diamond _{(m+n)'},\dots ,\Diamond _{(m+1)'}\in \{+,-\}} c_{i_{m+1}i_{(m+1)'}}^{\Diamond _{m+1}\Diamond _{(m+1)'}}(\Delta _{m+1})\cdots c_{i_{m+n}i_{(m+n)'}}^{\Diamond _{m+n}\Diamond _{(m+n)'}}(\Delta _{m+n})\nonumber \\ {}&\qquad \times \tau \big (A_{i_1}^{-}\cdots A_{i_m}^{-}B_{i_{m+1}}^{\Diamond _{m+1}}\cdots B_{i_{m+n}}^{\Diamond _{m+n} }A_{i_{(m+n)'}}^{\Diamond _{(m+n)'}}\cdots A_{i_{(m+1)'}}^{\Diamond (m+1)'}B_{i_{m'}}^{+}\cdots B_{i_{1'}}^{+}\big ). \end{aligned}$$

(72)

Step 2. We remind the reader that $\tau$ is a quasi-free state. As easily seen, the following lemma holds.

Lemma 6.1

Let $n\in \mathbb {N}$ and let $g_1,\dots ,g_{2n}\in \mathcal G$. Let $\Diamond _1,\dots ,\Diamond _{2n}\in \{+,-\}$ and assume that the number of plusses among $\Diamond _1,\dots ,\Diamond _{2n}$ is the same as the number of minuses. Then,

$$\begin{aligned} \big (a^{\Diamond _1}(g_1)\cdots a^{\Diamond _n}(g_{2n})\Omega ,\Omega \big )_{\mathcal{A}\mathcal{F}(\mathcal G)} =\sum (-1)^{{\text {Cross}}(\nu )} \bigg (\prod _{\begin{array}{c} \{i,j\}\in \nu \\ i< j \end{array}}(g_j,g_i)_\mathcal G\bigg ), \end{aligned}$$

where the summation is over all partitions $\nu =\big \{\{i_1,j_1\},\dots ,\{i_{n},j_{n}\}\big \}$ of $\{1,\dots ,2n\}$ with $i_k<j_k$ and such that $\Diamond _{i_k}=-$, $\Diamond _{j_k}=+$ ($k=1,\dots ,n$).

We define four (ordered) subsets of $\mathfrak E$ as follows:

$$\begin{aligned} \mathfrak A:=\{1,\dots ,m\},\quad \mathfrak B:=\{m+1,\dots ,m+n\}, \nonumber \\ \mathfrak C:=\{(m+n)',\dots ,(m+1)'\},\ \mathfrak D:=\{m',\dots ,1'\}. \end{aligned}$$

(73)

Denote by $\mathfrak R$ the collection of all partitions $\nu$ of $\mathfrak E$ into n two-point sets such that, if $\{i,j\}\in \nu$ with $i<j$, then one of the following four statements holds: (i) $i\in \mathfrak A$ and $j\in \mathfrak B$; (ii) $i\in \mathfrak B$ and $j\in \mathfrak C$; (iii) $i\in \mathfrak C$ and $j\in \mathfrak D$; (iv) $i\in \mathfrak A$ and $j\in \mathfrak D$.

By Lemma 6.1 and in view of the definition of $A^{\Diamond }_i$ and $B^{\Diamond }_i$ $(\Diamond \in \{+, -\})$, we continue (72) as follows:

$$\begin{aligned}&=\sum _{i_1,\dots ,i_{m+n},i_{(m+n)'},\dots ,i_{1'}\in \mathbb N} c_{i_1 i_{1'}}^{-+}(\Delta _1)\cdots c_{i_m i_{m'}}^{-+}(\Delta _n) \sum _{\nu \in \mathfrak R} (-1)^{{\text {Cross}}(\nu )} \quad c_{i_{m+1} i_{(m+1)'}}^{\Diamond _{m+1}(\nu ) \Diamond _{(m+1)'}(\nu )}(\Delta _{m+1}) \nonumber \\&\quad \times \dots \times c_{i_{m+n} i_{(m+)'}}^{\Diamond _{m+n}(\nu ) \Diamond _{(m+n)'}(\nu )}(\Delta _{m+n}) \prod _{\begin{array}{c} \{u,v\}\in \nu \end{array}}\delta _{i_u,i_v}. \end{aligned}$$

(74)

Here, for $u\in \mathfrak E$, we denote $\Diamond _u(\nu ):=-$ if $\{u,v\}\in \nu$ and $u< v$, and $\Diamond _u(\nu ):=+$ if $\{v,u\}\in \nu$ and $u<v$.

Step 3. Let $I:\mathfrak E\rightarrow \mathfrak E$ be the bijective map that is defined as follows: I acts as the identity on $\mathfrak A$ and $\mathfrak D$ and swaps the elements of $\mathfrak B$ and $\mathfrak C$, i.e., $I(i)=i'$ for all $i\in \mathfrak B$ and $I(i')=i$ for all $i'\in \mathfrak C$.

Denote by $\mathfrak S$ the collection of all partitions $\nu$ of $\mathfrak E$ into n two-point sets such that, for each $\{i,j\}\in \nu$, we have $i\in \mathfrak A\cup \mathfrak B$ and $j\in \mathfrak C\cup \mathfrak D$. As easily seen, the map I induces a bijection (still denoted by I) of $\mathfrak R$ onto $\mathfrak S$.

Lemma 6.2

(i) For $\nu \in \mathfrak R$, denote by $k_1(\nu )$ the number of all $\{i,j\}\in \nu$ such that $i\in \mathfrak A$, $j\in \mathfrak B$, and denote $k_2(\nu )$ the number of all $\{i,j\}\in \nu$ such that $i\in \mathfrak C$, $j\in \mathfrak D$. Then $k_1(\nu )=k_2(\nu )$ and we denote $\mathfrak l(\nu ):=k_1(\nu )=k_2(\nu )$.

(ii) We have, for each $\nu \in \mathfrak R$

$$\begin{aligned} (-1)^{{\text {Cross}}(\nu )}=(-1)^{{\text {Cross}}(I(\nu ))}(-1)^{\mathfrak l(\nu )}.\end{aligned}$$

(75)

Proof of Proposition 4.2

Part (i) is obvious, so we only prove part (ii). For $\nu \in \mathfrak R$, under the map I, the change in the number of crossings happens in the following three cases:

• (a) Let $\{i_1,j_1\}, \{i_2,j_2\}\in \nu$, $i_1,i_2\in \mathfrak A$, $j_1, j_2\in \mathfrak B$. Then $\{I(i_1),I(j_1)\}$, $\{I(i_2),I(j_2)\}$ have a crossing if and only if $\{i_1, j_1\}$, $\{i_2,j_2\}$ do not have a crossing;

• (b) Let $\{i_1,j_1\}, \{i_2,j_2\}\in \nu$, $i_1,i_2\in \mathfrak C$, $j_1, j_2\in \mathfrak D$. Then $\{I(i_1),I(j_1)\}$, $\{I(i_2),I(j_2)\}$ have a crossing if and only if $\{i_1, j_1\}$, $\{i_2,j_2\}$ do not have a crossing;

• (c) Let $\{i_1,j_1\}, \{i_2,j_2\}\in \nu$, $i_1\in \mathfrak A$, $j_1\in \mathfrak B$, $i_2\in \mathfrak C$, $j_2\in \mathfrak D$. Then $\{i_1,j_1\}$, $\{i_2,j_2\}$ do not have a crossing while $\{I(i_1),I(j_1)\}$ and $\{I(i_2),I(j_2)\}$ have a crossing.

Let $N_a(\nu )$, $N_b(\nu )$, and $N_c(\nu )$ denote the number of $\big \{\{i_1,j_1\}, \{i_2,j_2\}\big \}\subset \nu$ as in (a), (b), and (c), respectively. Since for any $i,j\in \mathbb N_0$, $(-1)^{i-j}=(-1)^{i+j}$, we therefore have:

$$\begin{aligned} (-1)^{{\text {Cross}}(I(\nu ))}=(-1)^{{\text {Cross}}(\nu ) +N_{a}(\nu )+N_{b}(\nu )+N_c(\nu )}.\end{aligned}$$

(76)

By part (i), $N_a(\nu )=N_b(\nu )$, while $N_c(\nu )=\mathfrak l(\nu )^2$. Hence, (76) implies (75).$\square$

Step 4. We will now identify the set of partitions $\mathfrak R$ with the symmetric group $S_{m+n}$. We define $\mathfrak m:\mathfrak C\cup \mathfrak D\rightarrow \mathfrak A\cup \mathfrak B$ by $\mathfrak { m}i'=i$ for $i'\in \mathfrak C\cup \mathfrak D$.

We define $\mathfrak I: \mathfrak R\rightarrow S_{m+n}$ as follows: for $\nu \in \mathfrak R$ let $\xi =\mathfrak I(\nu )$ be given by:

• for each $\{u,v\} \in \nu$ such that $u\in \mathfrak {A}$ and $v\in \mathfrak D$, $\xi (u)=\mathfrak { m} v$,

• for each $\{u,v\}\in \nu$ such that $u\in \mathfrak {A}$ and $v\in \mathfrak B$, $\xi (u)= v$,

• for each $\{u,v\}\in \nu$ such that $u\in \mathfrak {B}$ and $v\in \mathfrak C$, $\xi (\mathfrak { m}v)= u$,

• for each $\{u,v\}\in \nu$ such that $u\in \mathfrak {C}$ and $v\in \mathfrak D$, $\xi (\mathfrak { m}u)=\mathfrak {m} v$.

As easily seen, for each $\nu \in \mathfrak R$,

$$\begin{aligned} {\text {Cross}}(I(\nu ))={\text {sgn}}(\mathfrak I(\nu )). \end{aligned}$$

Then formula (75) implies

$$\begin{aligned} (-1)^{{\text {Cross}}(\mathfrak I^{-1}(\xi ))}=(-1)^{{\text {sgn}}(\xi )}(-1)^{\mathfrak l(\mathfrak I^{-1}(\xi ))},\quad \xi \in S_{m+n}. \end{aligned}$$

Hence, we continue (74) as follows:

$$\begin{aligned}&=\sum _{\xi \in S_{m+n}}(-1)^{{\text {sgn}}(\xi )}(-1)^{\mathfrak l(\mathfrak I^{-1}(\xi ))} \sum _{i_1,\dots ,i_{m+n},i_{(m+n)'},\dots ,i_{1'}\in \mathbb N} c_{i_1 i_{1'}}^{-+}(\Delta _1)\cdots c_{i_m i_{m'}}^{-+}(\Delta _n)\nonumber \\&\quad \times c_{i_{m+1} i_{(m+1)'}}^{\Diamond _{m+1}(\mathfrak I^{-1}(\xi )) \Diamond _{(m+1)'}(\mathfrak I^{-1}(\xi ))}(\Delta _{m+1})\cdots c_{i_{m+n} i_{(m+n)'}}^{\Diamond _{m+n}(\mathfrak I^{-1}(\xi )) \Diamond _{(m+n)'}(\mathfrak I^{-1}(\xi ))}(\Delta _{m+n})\nonumber \\&\quad \times \prod _{\begin{array}{c} \{u,v\}\in \mathfrak I^{-1}(\xi ) \end{array}}\delta _{i_u,i_v}. \end{aligned}$$

(77)

Step 5. We define mappings $\mathfrak r_1:\mathfrak A\cup \mathfrak B\rightarrow \mathfrak A\cup \mathfrak C$ and $\mathfrak r_2:\mathfrak A\cup \mathfrak B\rightarrow \mathfrak B\cup \mathfrak D$ as follows:

• for $u\in \mathfrak A$, $\mathfrak r_1(u):=u$ and $\mathfrak r_2(u):=u'$;

• for $u\in \mathfrak B$, $\mathfrak r_1(u):=u'$ and $\mathfrak r_2(u):=u$.

Then, for each $\xi \in S_{m+n}$, we have $\{u,v\}\in \mathfrak I^{-1}(\xi )$ if and only if, for some $i\in \{1,\dots ,m+n\}$, we have $\{u,v\}=\{\mathfrak r_1(u), \mathfrak r_2(\xi (u))\}$. Hence, by (72), (74), and (77),

$$\begin{aligned}&\tau \big ({:}\rho (\Delta _1)\cdots \rho (\Delta _{m+n}){:}\big ) \nonumber \\&\quad =\sum _{\xi \in S_{m+n}}(-1)^{{\text {sgn}}(\xi )}(-1)^{\mathfrak l(\mathfrak I^{-1}(\xi ))} \sum _{i_1,\dots ,i_{m+n},i_{(m+n)'},\dots ,i_{1'}\in \mathbb N} c_{i_1 i_{1'}}^{-+}(\Delta _1)\cdots c_{i_m i_{m'}}^{-+}(\Delta _m)\nonumber \\&\qquad \times c_{i_{m+1} i_{(m+1)'}}^{\Diamond _{m+1}(\mathfrak I^{-1}(\xi )) \Diamond _{(m+1)'}(\mathfrak I^{-1}(\xi ))}(\Delta _{m+1})\cdots c_{i_{m+m} i_{(m+n)'}}^{\Diamond _{m+n}(\mathfrak I^{-1}(\xi )) \Diamond _{(m+n)'}(\mathfrak I^{-1}(\xi ))}(\Delta _{m+n})\nonumber \\&\qquad \times \prod _{i=1}^{m+n}\delta _{i_{\mathfrak r_1(u)},i_{\mathfrak r_2(\xi (u))}}. \end{aligned}$$

(78)

Step 6. Next, we prove

Lemma 6.3

Let $\xi \in S_{m+n}$ and let $(l_1\,l_2\cdots l_k)$ be a cycle in $\xi$. Then

$$\begin{aligned}&\sum _{i_{l_1},i_{l_2},\dots ,i_{l_k},i_{l_1'},i_{l_2'},\dots ,i_{l_k'} \in \mathbb N}\, c_{i_{l_1}i_{l_1'}}^{\Diamond _{l_1}(\mathfrak I^{-1}(\xi )) \Diamond _{l_1'}(\mathfrak I^{-1}(\xi ))} (\Delta _{l_1}) c_{i_{l_2}i_{l_2'}}^{\Diamond _{l_2}(\mathfrak I^{-1}(\xi )) \Diamond _{l_2'}(\mathfrak I^{-1}(\xi ))} (\Delta _{l_2})\nonumber \\&\quad \times \cdots \times c_{i_{l_k}i_{l_k'}}^{\Diamond _{l_k}(\mathfrak I^{-1}(\xi )) \Diamond _{l_k'}(\mathfrak I^{-1}(\xi ))} (\Delta _{l_k})\delta _{i_{\mathfrak r_1(l_1)},i_{\mathfrak r_2(l_2)}}\delta _{i_{\mathfrak r_1(l_2)},i_{\mathfrak r_2(l_3)}}\cdots \delta _{i_{\mathfrak r_1(l_k)},i_{\mathfrak r_2(l_1)}}\nonumber \\&\quad ={\text {Tr}}\big (P_{\Delta _{l_k}} R(l_k,l_{k-1})P_{\Delta _{l_{k-1}}} R(l_{k-1},l_{k-2})P_{\Delta _{l_{k-2}}}\cdots P_{\Delta _{l_1}}R(l_1,l_k)P_{\Delta _{l_k}} \big ). \end{aligned}$$

(79)

Here, for $u,v\in \{1,2,\dots ,m+n\}$,

$$\begin{aligned} R(u,v):={\left\{ \begin{array}{ll}K,&{} \quad \text {if } \min \{u,v\}\le m,\\ \textbf{1}-K, &{} \quad \text {if } \min \{u,v\}\ge m+1. \end{array}\right. }\end{aligned}$$

(80)

Proof

Let us first consider the case where $l_1,l_2,\dots ,l_k\in \{1,\dots ,m\}$. Then the left-hand side of (79) becomes

$$\begin{aligned}&\sum _{i_{l_1},i_{l_2},\dots ,i_{l_k},i_{l_1'},i_{l_2'},\dots ,i_{l_k'}\in \mathbb N}\, c_{i_{l_1}i_{l_1'}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_2'}}^{- +}(\Delta _{l_2})\cdots c_{i_{l_k}i_{l_k'}}^{-+}(\Delta _{l_k}) \delta _{i_{ l_1},i_{l_2'}}\delta _{i_{ l_2},i_{l_3'}}\dots \delta _{i_{ l_k},i_{l_1'}}\\&\quad =\sum _{i_{l_1},i_{l_2},\dots ,i_{l_k}\in \mathbb N}\, c_{i_{l_1}i_{l_k}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_1}}^{- +}(\Delta _{l_2})\cdots c_{i_{l_k}i_{l_{k-1}}}^{-+}(\Delta _{l_k}). \end{aligned}$$

We have

$$\begin{aligned}&\sum _{ i_{l_1}\in \mathbb N}\, c_{i_{l_1}i_{l_k}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_1}}^{- +}(\Delta _{l_2}) = \sum _{ i_{l_1}\in \mathbb N}\, \big (K_1P_{\Delta _{l_1}} K_1 e_{i_{l_k}},e_{i_{l_1}}\big )_\mathcal H \big (K_1P_{\Delta _{l_2}} K_1 e_{i_{l_1}},e_{i_{l_2}}\big )_\mathcal H \\&\quad = \big (K_1P_{\Delta _{l_2}}KP_{\Delta _{l_1}} K_1 e_{i_{l_k}},e_{i_{l_2}}\big )_\mathcal H . \end{aligned}$$

Next,

$$\begin{aligned}&\sum _{ i_{l_2}\in \mathbb N}\, \big (K_1P_{\Delta _{l_2}}KP_{\Delta _{l_1}} K_1 e_{i_{l_k}},e_{i_{l_2}}\big )_\mathcal H\, c_{i_{l_3}i_{l_2}}^{-+}(\Delta _{l_3}) \\&\quad =\sum _{ i_{l_2}\in \mathbb N}\, \big (K_1P_{\Delta _{l_2}}KP_{\Delta _{l_1}} K_1 e_{i_{l_k}},e_{i_{l_2}}\big )_\mathcal H \big (K_1P_{\Delta _{l_3}} K_1 e_{i_{l_2}},e_{i_{l_3}}\big )_\mathcal H\\&\quad = \big (K_1P_{\Delta _{l_3}} K P_{\Delta _{l_2}}KP_{\Delta _{l_1}} K_1 e_{i_{l_k}},e_{i_{l_3}}\big )_\mathcal H . \end{aligned}$$

Continuing by analogy, we conclude:

$$\begin{aligned}&\sum _{i_{l_1},i_{l_2},\dots ,i_{l_{k-1}}\in \mathbb N}\, c_{i_{l_1}i_{l_k}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_1}}^{-+}(\Delta _{l_2})\dots c_{i_{l_k}i_{l_{k-1}}}^{-+}(\Delta _{l_k}) \\&\qquad = \big (K_1P_{\Delta _{l_k}} K P_{\Delta _{l_{k-1}}}K\cdots K P_{\Delta _{l_1}} K_1 e_{i_{l_k}},e_{i_{l_k}}\big )_\mathcal H , \end{aligned}$$

which implies

$$\begin{aligned}&\sum _{i_{l_1},i_{l_2},\dots ,i_{l_k}\in \mathbb N}\, c_{i_{l_1}i_{l_k}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_1}}^{-+}(\Delta _{l_2})\cdots c_{i_{l_k}i_{l_{k-1}}}^{-+}(\Delta _{l_k}) \\&\qquad = \sum _{ i_{l_k}\in \mathbb N}\, \big (K_1P_{\Delta _{l_k}} K P_{\Delta _{l_{k-1}}}K\cdots K P_{\Delta _{l_1}} K_1 e_{i_{l_k}},e_{i_{l_k}}\big )_\mathcal H \\&\qquad ={\text {Tr}}\big (K_1P_{\Delta _{l_k}} K P_{\Delta _{l_{k-1}}}K\cdots K P_{\Delta _{l_1}} K_1\big )\\&\qquad ={\text {Tr}}\big (P_{\Delta _{l_k}} K P_{\Delta _{l_{k-1}}}K\cdots K P_{\Delta _{l_1}} K\big )\\&\qquad ={\text {Tr}}\big (P_{\Delta _{l_k}} K P_{\Delta _{l_{k-1}}}K\cdots K P_{\Delta _{l_1}} K P_{\Delta _{l_k}}\big ). \end{aligned}$$

We similarly treat the case where $l_1,l_2,\dots ,l_k\in \{m+1,\dots ,m+n\}$.

Finally we consider the case where

$$\begin{aligned} \{l_1,l_2,\dots ,l_k\}\cap \{1,\dots ,m\}\ne \varnothing ,\quad \{l_1,l_2,\dots ,l_k\}\cap \{m+1,\dots ,m+n\}\ne \varnothing . \end{aligned}$$

Without loss of generality, we may assume that $l_1\in \{1,\dots ,m\}$ and $l_k\in \{m+1,\dots ,m+n\}.$ To simplify the notation, we will additionally assume that, for some $\alpha \in \{1,\dots ,k-1\}$,

$$\begin{aligned} l_1,\dots ,l_\alpha \in \{1,\dots ,m\},\quad l_{\alpha +1},\dots ,l_k\in \{m+1,\dots ,m+n\}. \end{aligned}$$

The interested reader can easily extend our arguments to the more general case.

We consider separately three cases:

Case 1: $\alpha =k-1$. Then the left-hand side of (79) becomes

$$\begin{aligned} \sum _{i_{l_1},\dots ,i_{l_{k-1}},i_{l_k'}\in \mathbb N}\, c_{i_{l_1}i_{l_k'}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_1}}^{- +}(\Delta _{l_2})\cdots c_{i_{l_{k-1}}i_{l_{k-2}}}^{-+}(\Delta _{l_{k-1}})c_{i_{l_{k-1}}i_{l_{k'}}}^{+-}(\Delta _{l_{k}}). \end{aligned}$$

Analogously to the above calculations, we get

$$\begin{aligned}&\sum _{i_{l_1},\dots ,i_{l_{k-2}}\in \mathbb N}\, c_{i_{l_1}i_{l_k'}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_1}}^{- +}(\Delta _{l_2})\cdots c_{i_{l_{k-1}}i_{l_{k-2}}}^{-+}(\Delta _{l_{k-1}}) \\&\qquad =\big (K_1P_{\Delta _{l_{k-1}}}KP_{\Delta _{l_{k-2}}}K\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_{k-1}}}\big )_\mathcal H. \end{aligned}$$

Then

$$\begin{aligned}&\sum _{i_{l_{k-1}}\in \mathbb N}\,\big (K_1P_{\Delta _{l_{k-1}}}KP_{\Delta _{l_{k-2}}}K\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_{k-1}}}\big )_\mathcal H\, c_{i_{l_{k-1}}i_{l_{k'}}}^{+-}(\Delta _{l_{k}}) \\&\quad = \sum _{i_{l_{k-1}}\in \mathbb N}\,\big (K_1P_{\Delta _{l_{k-1}}}KP_{\Delta _{l_{k-2}}}K\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_{k-1}}}\big )_\mathcal H \big (K_1P_{\Delta _{l_k}}K_1 e_{i_{l_{k-1}}},e_{i_{l'_{k}}}\big )_\mathcal H\\&\quad =\big (K_1P_{\Delta _{l_k}}KP_{\Delta _{l_{k-1}}}KP_{\Delta _{l_{k-2}}}K\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_k'}}\big )_\mathcal H, \end{aligned}$$

and

$$\begin{aligned}&\sum _{i_{l_{k}'}\in \mathbb N}\,\big (K_1P_{\Delta _{l_k}}KP_{\Delta _{l_{k-1}}}KP_{\Delta _{l_{k-2}}}K\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_k'}}\big )_\mathcal H \\&\quad ={\text {Tr}}\big (P_{\Delta _{l_k}}KP_{\Delta _{l_{k-1}}}KP_{\Delta _{l_{k-2}}}K\cdots KP_{\Delta _{l_1}}KP_{\Delta _{l_k}}\big ). \end{aligned}$$

Case 2: $\alpha =k-2$. Then the left-hand side of (79) becomes

$$\begin{aligned}&\sum _{i_{l_1},\dots ,i_{l_{k-2}},i_{l_{k-1}'},i_{l_k'}\in \mathbb N}\, c_{i_{l_1}i_{l_k'}}^{-+}(\Delta _{l_1}) c_{i_{l_2}i_{l_1'}}^{- +}(\Delta _{l_2})\cdots c_{i_{l_{k-2}}i_{l_{k-3}}}^{-+}(\Delta _{l_{k-2}})\\&\quad \qquad \times c_{i_{l_{k-2}}i_{l_{k-1}'}}^{++}(\Delta _{l_{k-1}})c_{i_{l_{k-1}'}i_{l_{k}'}}^{--}(\Delta _{l_{k}})\\&\qquad =\sum _{i_{l_{k-2}},i_{l_{k-1}'},i_{l_k'}\in \mathbb N}\,\big (K_1P_{\Delta _{l_{k-2}}}KP_{\Delta _{l_{k-3}}}K\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_{k-2}}}\big )_\mathcal H\\&\quad \qquad \times \big (K_2P_{\Delta _{l_{k-1}}}K_1 e_{i_{l_{k-2}}},e_{i_{l_{k-1}'}}\big )_\mathcal H \big (K_1P_{\Delta _{l_{k}}}K_2 e_{i_{l_{k-1}'}},e_{i_{l_k'}}\big )_\mathcal H\\&\qquad =\sum _{i_{l_{k-1}'},i_{l_k'}\in \mathbb N}\, \big (K_2P_{\Delta _{l_{k-1}}}P_{\Delta _{l_{k-2}}}KP_{\Delta _{l_{k-3}}}K\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_{k-1}'}}\big )_\mathcal H \\&\quad \qquad \times \big (K_1P_{\Delta _{l_{k}}}K_2 e_{i_{l_{k-1}'}},e_{i_{l_k'}}\big )_\mathcal H\\&\qquad =\sum _{i_{l_k'}\in \mathbb N}\,\big ( K_1 P_{\Delta _{l_{k}}} (\textbf{ 1}-K) P_{\Delta _{l_{k-1}}} K P_{\Delta _{l_{k-2}}}\cdots KP_{\Delta _{l_1}}K_1e_{i_{l_k'}},e_{i_{l_k'}}\big )_\mathcal H\\&\qquad ={\text {Tr}}\big (P_{\Delta _{l_{k}}} (\textbf{ 1}-K) P_{\Delta _{l_{k-1}}} K P_{\Delta _{l_{k-2}}}K\cdots KP_{\Delta _{l_1}}KP_{\Delta _{l_{k}}} \big ). \end{aligned}$$

Case 3: $\alpha \leqslant k-3$. Then the left-hand side of (79) becomes

$$\begin{aligned}&=\sum _{i_{l_1},\dots ,i_{l_\alpha },i_{l_{\alpha +1}'},i_{l_{\alpha +2}'},\dots ,i_{l_k'}\in \mathbb N}\,c_{i_{l_1}i_{l_k'}}^{-+}(\Delta _{l_1})c_{i_{l_2}i_{l_1}}^{-+}(\Delta _{l_2}) \cdots c_{i_{l_\alpha }i_{l_{\alpha -1}}}^{-+}(\Delta _{l_\alpha }) \\&\quad \times c_{i_{l_\alpha }i_{l_{\alpha +1}'}}^{++}(\Delta _{l_{\alpha +1}}) c_{i_{l_{\alpha +1}'}i_{l_{\alpha +2}'}}^{-+}(\Delta _{l_{\alpha +2}})\cdots c_{i_{l_{k-2}'}i_{l_{k-1}'}}^{-+}(\Delta _{l_{k-1}})c_{i_{l_{k-1}'}i_{l_{k}'}}^{--} (\Delta _{l_{k}}). \end{aligned}$$

Similarly to Case 2, we obtain:

$$\begin{aligned}&\sum _{i_{l_1},\dots ,i_{l_\alpha }\in \mathbb N}\, c_{i_{l_1}i_{l_k'}}^{-+}(\Delta _{l_1})c_{i_{l_2}i_{l_1}}^{-+}(\Delta _{l_2}) \cdots c_{i_{l_\alpha }i_{l_{\alpha -1}}}^{-+}(\Delta _{l_\alpha }) c_{i_{l_\alpha }i_{l_{\alpha +1}'}}^{++}(\Delta _{l_{\alpha +1}}) \\&\quad =\big (K_2P_{\Delta _{l_{\alpha +1}}} KP_{\Delta _{l_{\alpha }}}K\cdots KP_{\Delta _{l_{1}}}K_1 e_{i_{l_k'}},e_{i_{l_{\alpha +1}'}}\big )_\mathcal H. \end{aligned}$$

Then

$$\begin{aligned}&\sum _{i_{l_{\alpha +1}'},\dots ,i_{l_{k-2}'}\in \mathbb N}\, \big (K_2P_{\Delta _{l_{\alpha +1}}} KP_{\Delta _{l_{\alpha }}}K\cdots KP_{\Delta _{l_{1}}}K_1 e_{i_{l_k'}},e_{i_{l_{\alpha +1}'}}\big )_\mathcal H \\&\qquad \times c_{i_{l_{\alpha +1}'}i_{l_{\alpha +2}'}}^{-+}(\Delta _{l_{\alpha +2}})\cdots c_{i_{l_{k-2}'}i_{l_{k-1}'}}^{-+}(\Delta _{l_{k-1}})\\&\quad = \sum _{i_{l_{\alpha +1}'},\dots ,i_{l_{k-2}'}\in \mathbb N}\, \big (K_2P_{\Delta _{l_{\alpha +1}}} KP_{\Delta _{l_{\alpha }}}K\cdots KP_{\Delta _{l_{1}}}K_1 e_{i_{l_k'}},e_{i_{l_{\alpha +1}'}}\big )_\mathcal H \\&\qquad \times \big (K_2 P_{\Delta _{l_{\alpha +2}}}K_2 e_{i_{l_{\alpha +1}'}},e_{i_{l_{\alpha +2}'}}\big )_\mathcal H\cdots \big (K_2 P_{\Delta _{l_{k-1}}}K_2 e_{i_{l_{k-2}'}},e_{i_{l_{k-1}'}}\big )_\mathcal H\\&\quad = \big (K_2 P_{\Delta _{l_{k-1}}}(\textbf{ 1}-K) P_{\Delta _{l_{k-2}}}(\textbf{ 1}-K) \cdots (\textbf{ 1}-K) P_{\Delta _{l_{\alpha +1}}}KP_{\Delta _{l_{\alpha }}}K\\&\qquad \cdots KP_{\Delta _{l_{1}}}K_1 e_{i_{l_{k}'}},e_{i_{l_{k-1}'}}\big )_\mathcal H. \end{aligned}$$

Finally,

$$\begin{aligned}&\sum _{i_{l_{k-1}'},i_{l_k'}\in \mathbb N}\, \big (K_2 P_{\Delta _{l_{k-1}}}(\textbf{ 1}-K) P_{\Delta _{l_{k-2}}}(\textbf{ 1}-K) \cdots (\textbf{ 1}-K) P_{\Delta _{l_{\alpha +1}}}KP_{\Delta _{l_{\alpha }}}K\\&\quad \cdots KP_{\Delta _{l_{1}}}K_1 e_{i_{l_{k}'}},e_{i_{l_{k-1}'}}\big )_\mathcal H\, c_{i_{l_{k-1}'}i_{l_{k}'}}^{--} (\Delta _{l_{k}}) \\&= \sum _{i_{l_{k-1}'},i_{l_k'}\in \mathbb N}\, \big (K_2 P_{\Delta _{l_{k-1}}}(\textbf{ 1}-K) P_{\Delta _{l_{k-2}}}(\textbf{ 1}-K) \cdots (\textbf{ 1}-K) P_{\Delta _{l_{\alpha +1}}}KP_{\Delta _{l_{\alpha }}}K\\&\quad \cdots KP_{\Delta _{l_{1}}}K_1 e_{i_{l_{k}'}},e_{i_{l_{k-1}'}}\big )_\mathcal H \big (K_1 P_{\Delta _{l_{k}}}K_2 e_{i_{l_{k-1}'}},e_{i_{l_{k}'}}\big )_\mathcal H\\&={\text {Tr}}\big ( P_{\Delta _{l_{k}}}(\textbf{ 1}-K)P_{\Delta _{l_{k-1}}}(\textbf{ 1}-K)\cdots (\textbf{ 1}-K) P_{\Delta _{l_{\alpha +1}}}K P_{\Delta _{l_\alpha }} K \cdots K P_{\Delta _{l_1}}K P_{\Delta _{l_{k}}}\big )_\mathcal H.\quad \end{aligned}$$

Step 7. For a given cycle $\theta = (l_1 l_2\cdots l_k)$ in a permutation $\xi \in S_{m+n}$, we denote by $\widetilde{\mathbb T}_\theta$ the value given by (the right hand-side of) formula (79). Denote by $\mathfrak {t}(\theta )$ the number of $i\in \{1,\dots ,k\}$ such that $l_i\in \{m+1,\dots ,m+n\}$ but $l_{i+1}\in \{1,\dots ,m\}$, where $l_{k+1}:=l_1$. Then, by (64), (65), and (80),

$$\begin{aligned} (-1)^{\mathfrak {t}(\theta )}\,\widetilde{\mathbb T}_\theta ={\text {Tr}}\big (P_{\Delta _{l_{k}}} \mathbb K P_{\Delta _{l_{k-1}}} \mathbb K \cdots \mathbb K P_{\Delta _{l_1}}\mathbb K P_{\Delta _{l_{k}}}\big ) =\mathbb {T}_{\theta ^{-1}}\,.\end{aligned}$$

(81)

By Lemma 6.2 (i),

$$\begin{aligned} \sum _{\theta \in {\text {Cycles}}(\xi )}\mathfrak {t}(\theta )= \mathfrak {l}(\mathfrak {I}^{-1}(\xi )).\end{aligned}$$

(82)

Thus, by (78), (81), (82) and Lemma 6.3,

$$\begin{aligned} \tau \big ({:}\rho (\Delta _1)\cdots \rho (\Delta _{m+n}){:}\big )&=\sum _{\xi \in S_{m+n}}(-1)^{{\text {sgn}}(\xi )}\prod _{\theta \in {\text {Cycles}}(\xi )} \mathbb {T}_{\theta ^{-1}}\nonumber \\&=\sum _{\xi \in S_{m+n}}(-1)^{{\text {sgn}}(\xi )}\prod _{\theta \in {\text {Cycles}}(\xi )} \mathbb {T}_{\theta }. \end{aligned}$$

(83)

Formulas (63) and (83) imply(70).