Remarks on BEC on graphs

TL;DR

This paper investigates Bose--Einstein condensation on graphs with transient adjacency matrices, revealing a non-factor quasi-free state exhibiting BEC that decomposes into generalized coherent states, and reviews conditions for their properties.

Contribution

It introduces a non-factor quasi-free state exhibiting BEC on graphs and analyzes the properties of generalized coherent states in this context.

Findings

The BEC state is non-factor and decomposes into generalized coherent states.

Conditions for generalized coherent states to be faithful, factor, and pure are reviewed.

Generalized coherent states are shown to be quasi-equivalent under certain conditions.

Abstract

We consider Bose--Einstein condensation (BEC) on graphs with transient adjacency matrix and obtain a quasi-free state exhibiting BEC is non-factor and decompose into generalized coherent states. We review necessary and sufficient conditions that a generalized coherent state is faithful, factor, and pure and generalized coherent states are quasi-equivalent as well.

Equations

$$\phi (W(f))=\exp \{-{\Vert f\Vert }_{\mathfrak{h}}^2/4+i\mathrm{Re}\lambda (f)\}$$

$$\phi (W(f))=\exp \{-S(f,f)/4+i\lambda (f)\},f\in \mathfrak{h}.$$

$$\alpha (P)(B(f_1)B(f_2)\cdots B(f_n))$$

$$\alpha (P)B(f)$$

$$W\phi (f)W_{\phi }(g)=\exp (-{\gamma }_{\tilde{K}}(f,g)/2)W_{\phi }(f+g),f,g\in \mathrm{Re}\tilde{K}.$$

$$\phi (B(f_1)\cdots B(f_{2n-1}))=0,$$

$$\phi (B(f_1)\cdots B(f_{2n}))=\sum \prod _{j=1}^n\phi (B(f_{s(j)})B(f_{s(j+n)})),$$

$$S_{\tilde{K}}(f,g)=\phi (B(f{)}^{\ast }B(g)),f,g\in \tilde{K}$$

$${\gamma }_{\tilde{K}}(f,g)=S_{\tilde{K}}(f,g)-S_{\tilde{K}}(\Gamma g,\Gamma f),f,g\in \tilde{K}.$$

$$(f,g{)}_S:=S_{\tilde{K}}(f,g)+S_{\tilde{K}}({\Gamma }_{\tilde{K}}g,{\Gamma }_{\tilde{K}}f),f,g\in \tilde{K}.$$

$$(\xi ,S_K\eta {)}_S$$

$$(\xi ,{\gamma }_K\eta {)}_S$$

$${\gamma }_L({\xi }_1\oplus {\xi }_2,{\eta }_1\oplus {\eta }_2)$$

$$\widetilde{{\Gamma }_L}$$

$$({\xi }_1\oplus {\xi }_2,{\eta }_1\oplus {\eta }_2{)}_L$$

$$(\xi ,{\gamma }_L\eta {)}_L$$

$${\Pi }_L$$

$$\tau (W_{S_1}(f))=W_{S_2}(f),f\in \mathrm{Re}\tilde{K},$$

$$\sigma (f,g)=-\sigma (g,f),f,g\in V.$$

$$\sigma (Jf,g)=-\sigma (f,Jg),J^2=-1,$$

$$(c_1+i{c}_2)f=c_{1}f+c_{2}Jf,c_1,c_2\in \mathbb{R},f\in V.$$

$${\Gamma }_{\tilde{K}}f$$

$${\gamma }_{\tilde{K}}(f,g)$$

$$a_{\phi }(f):=\{{\pi }_{\phi }(B(f))+i{\pi }_{\phi }(B(if))\}/\sqrt{2},a^{†}(f):=\{{\pi }_{\phi }(B(f))-i{\pi }_{\phi }(B(if))\}/\sqrt{2},$$

$${\tau }_{\lambda }(W(f)):=e^{i\lambda (f)}W(f),f\in V.$$

$${\phi }_{S,\lambda }(W(f)):={\phi }_S\circ {\tau }_{\lambda }(W(f))=e^{i\lambda (f)}{\phi }_S(W(f)),f\in V.$$

$$R_S=R_{S,\lambda },$$

$$\{{\pi }_S(W(f))\}{f\in V}^{\prime \prime }=\{e^{i\lambda (f)}{\pi }_S(W(f))\}{f\in V}^{\prime \prime }$$

$$\tau ({\pi }_{S_1,{\lambda }_1}(A))={\pi }_{S_2,{\lambda }_2}(A),A\in \mathcal{W}(V,\sigma ).$$

$${\pi }_{S_1,{\lambda }_1}(W(h))=e^{i{\lambda }_1(h)}\text{mathbbm}1$$

$${\pi }_{S_2,{\lambda }_2}(W(h))=e^{i{\lambda }_2(h)}{\pi }_2(W(h))=\tau ({\pi }_{S_1,{\lambda }_1}(W(h)))=e^{i{\lambda }_1(h)}\text{mathbbm}1.$$

$$\tau ({\pi }_{S_1,\lambda }(A))={\pi }_{S_2}(A),A\in \mathcal{W}(V,\sigma ).$$

$$\exp (i\lambda (f)-S_1(f,f)/2)=⟨{\xi }_{S_1},{\tau }^{-1}({\pi }_{S_2}(W(f))){\xi }_{S_1}⟩$$

$$e^{i\lambda (f)}=\tau ({\pi }_{S_1,\lambda }(W(f)))={\pi }_{S_2}(W(f))=\text{mathbbm}1$$

Full text

Remarks on BEC on Graphs

Tomohiro Kanda

Graduate School of Mathematics, Kyushu University,

744 Motoka, Nishi-ku, Fukuoka 819-0395, JAPAN

t-kanda@math.kyushu-u.ac.jp

Abstract: We consider Bose–Einstein condensation (BEC) on graphs with transient adjacency matrix. We prove the equivalence of BEC and non-factoriality of a quasi-free state. Moreover, quasi-free states exhibiting BEC decompose into generalized coherent states. We review necessary and sufficient conditions that a quasi-free state is faithful, factor, and pure and quasi-free states are quasi-equivalent, including the paper of H. Araki and M. Shiraishi (1971/72), H. Araki (1971/72), and H. Araki and S. Yamagami (1982). Using their formats and results, we prove necessary and sufficient conditions that a generalized coherent state is faithful, factor, and pure and generalized coherent states are quasi-equivalent as well.

Keywords: CCR algebra, generalized coherent state, quasi-equivalence, Bose–Einstein condensation.

AMS subject classification: 82B10

1 Introduction

In [14], T. Matsui studied the condition for Bose–Einstein condensation (BEC for short.) in terms of the random walk on a graph. In [6], F. Fidaleo, D. Guido, and T. Isola, in [7] and [8], F. Fidaleo studied some spectral properties of the adjacency matrix of graphs and BEC. They obtained the criterion for BEC on graphs. In [11], J. T. Lewis and J. V. Pulè obtained the non-factoriality of a quasi-free state exhibiting BEC in $L^{2}({\mathbb{R}}^{3})$ case. However, in case of graphs, BEC implies non-factoriality of a quasi-free states exhibiting BEC is not clear, thus, we study a quasi-free state exhibiting BEC and prove the equivalence of the occurrence of BEC and non-factoriality of a quasi-free state. Moreover, we give factor decomposition of quasi-free states exhibiting BEC into generalized coherent states which are factor and mutually disjoint (Theorem 4.9.). Generalized coherent states are generalization of coherent states in the following sense. Let ${\mathfrak{h}}$ be a Hilbert space and $\sigma$ be the symplectic form defined by $\sigma(f,g):={\rm Im}{\left\langle f,g\right\rangle}_{{\mathfrak{h}}}$. In mathematics, a coherent state $\varphi$ on the Weyl CCR algebra ${\cal W}({\mathfrak{h}},\sigma)$ is given by

[TABLE]

for each $f\in{\mathfrak{h}}$, where $W(f)$, $f\in{\mathfrak{h}}$, are unitaries which generate ${\cal W}({\mathfrak{h}},\sigma)$ and $\lambda$ is a ${\mathbb{C}}$-linear functional on ${\mathfrak{h}}$. (See [10, Theorem 3.1.].) A state $\varphi$ on ${\cal W}({\mathfrak{h}},\sigma)$ is a generalized coherent state, if there exists a positive semi-definite sesquilinear form $S$ on ${\mathfrak{h}}\times{\mathfrak{h}}$ and an ${\mathbb{R}}$-linear functional $\lambda:{\mathfrak{h}}\to{\mathbb{R}}$ such that

[TABLE]

In Section 2, we review works of H. Araki and M. Shiraishi [1], H. Araki [2], and H. Araki and S. Yamagami [3]. In [1], H. Araki and M. Shiraishi and in [2], H. Araki considered quasi-free states on the CCR algebra and obtained a condition that a quasi-free state is faithful, factor, and pure. In [3], H. Araki and S. Yamagami got necessary and sufficient conditions that quasi-free states are quasi-equivalent. In [12], J. Manuceau and A. Verbeure and in [13], J. Manuceau, F. Rocca, and D. Testard obtained a condition that a quasi-free state on the Weyl CCR algebra is pure and factor. In [18], A. van Daele obtained conditions of quasi-equivalence of quasi-free states on the Weyl CCR algebra as well. To consider conditions of factoriality, purity and faithfulness of a generalized coherent state and conditions of quasi-equivalence of generalized coherent states in a unified framework, we use formats in [1], [2], and [3].

In Section 3, we consider generalized coherent states on the Weyl CCR algebra. We prove necessary and sufficient conditions that a generalized coherent state is faithful, factor, and pure and necessary and sufficient conditions that generalized coherent states are quasi-equivalent. Moreover, we give an explicit form of factor decomposition of non-factor generalized coherent state. In [9], R. Honegger considered the decomposition of gauge-invariant quasi-free states. In the present paper, we only assume that a state on the Weyl CCR algebra is quasi-free or generalized coherent.

In Section 4, we review works of F. Fidaleo [8] and consider the non-factoriality of quasi-free states with BEC. We show that a quasi-free state exhibiting BEC is non-factor and such state decomposes into generalized coherent states which are mutually disjoint. In [15], J. V. Pulé, A. F. Verbeure, and V. A. Zagrebnov considered inhomogeneous BEC on $L^{2}({\mathbb{R}}^{\nu})$, $\nu\geq 1$, and obtained that the occurrence of BEC implies spontaneous symmetry breaking and an equilibrium state exhibiting BEC decompose into periodic states. In [14], T. Matsui obtained that the occurrence of BEC implies spontaneous symmetry breaking in case of graphs with some assumptions (See [14, Assumption 1.1.].) as well. In the present paper, generalized coherent states appeared in factor decomposition of a quasi-free state are not periodic. Thus, we give another decomposition of a quasi-free state exhibiting BEC.

2 Preliminaries

In this section, we review works of H. Araki and M. Shiraishi [1], H. Araki [2], and H. Araki and S. Yamagami [3]. In [1], H. Araki and M. Shiraishi and in [2], H. Araki considered quasi-free states on the CCR algebra and obtained necessary and sufficient conditions that a quasi-free state is factor, pure, and faithful. In [3], H. Araki and S. Yamagami obtained necessary and sufficient conditions that quasi-free states are quasi-equivalent. We use facts presented in this section to consider necessary and sufficient conditions that a generalized coherent state is factor, pure, and faithful and generalized coherent states are quasi-equivalent and to prove non-factoriality of quasi-free states exhibiting BEC.

2.1 Some Properties of a Quasi-free state

Let $\tilde{K}$ be a ${\mathbb{C}}$-linear space and $\gamma_{\tilde{K}}:\tilde{K}\times\tilde{K}\to{\mathbb{C}}$ be a sesquilinear form. Let $\Gamma_{\tilde{K}}$ be an anti-linear involution ($\Gamma_{\tilde{K}}^{2}=\mathbbm{1}$) satisfying $\gamma_{\tilde{K}}(\Gamma_{\tilde{K}}f,\Gamma_{\tilde{K}}g)=-\gamma_{\tilde{K}}(g,f)$. A CCR algebra ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ over $(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ is the quotient of the complex $*$-algebra generated by $B(f)$, $f\in\tilde{K}$, its adjoint $B(f)^{*}$, $f\in\tilde{K}$ and an identity over the following relations:

$B(f)$ is complex linear in $f$, 2. 2.

$B(f)^{*}B(g)-B(g)B(f)^{*}=\gamma_{\tilde{K}}(f,g)\mathbbm{1}$, 3. 3.

$B(\Gamma_{\tilde{K}}f)^{*}=B(f)$.

Any linear operator $P$ on $\tilde{K}$ satisfying

$P^{2}=P$, 2. 2.

$\gamma_{\tilde{K}}(Pf,g)>0$, if $Pf\neq 0$, 3. 3.

$\gamma_{\tilde{K}}(Pf,g)=\gamma_{\tilde{K}}(f,Pg)$, 4. 4.

$\Gamma_{\tilde{K}}P\Gamma_{\tilde{K}}=1-P$,

is called a basis projection.

Let ${\mathfrak{h}}$ be a complex pre-Hilbert space. A CCR ($*$-)algebra ${\cal A}_{{\rm CCR}}({\mathfrak{h}})$ over ${\mathfrak{h}}$ is the quotient of the $*$-algebra generated by $a^{\dagger}(f)$ and $a(f)$, $f\in{\mathfrak{h}}$, and an identity by the following relations:

$a^{\dagger}(f)$ is complex linear in $f$, 2. 2.

$(a^{\dagger}(f))^{*}=a(f)$, 3. 3.

${\left[a(f),a^{\dagger}(g)\right]}=(f,g)_{{\mathfrak{h}}}\mathbbm{1}$ and ${\left[a^{\dagger}(f),a^{\dagger}(g)\right]}=0={\left[a(f),a(g)\right]}$.

Let $P$ be a basis projection. Then the mapping $\alpha(P)$ from ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ to ${\cal A}_{{\rm CCR}}(P\tilde{K})$ defined by

[TABLE]

is a $*$-isomorphism of ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ onto ${\cal A}_{{\rm CCR}}(P\tilde{K})$.

Let ${\cal A}$ be a $*$-algebra with identity. A linear functional $\varphi$ on ${\cal A}$ is said to be state, if $\varphi$ satisfies $\varphi(A^{*}A)\geq 0$, $A\in{\cal A}$, and $\varphi(\mathbbm{1})=1$. For a state $\varphi$ on ${\cal A}$, we have the GNS-representation space $({\mathfrak{H}}_{\varphi},\pi_{\varphi},\xi_{\varphi})$ associated with $\varphi$. We set ${\rm Re}\tilde{K}:=\set{f\in\tilde{K}}{\Gamma_{\tilde{K}}f=f}$. Then $f\in{\rm Re}\tilde{K}$ if and only if $B(f)^{*}=B(f)$.

On ${\rm Re}\tilde{K}$, the operators $B(f)$, $f\in{\rm Re}\tilde{K}$, correspond to field operators. Moreover, $a^{\dagger}(f)$ and $a(f)$ correspond to the creation operators and the annihilation operators. We give examples of $\tilde{K}$, $\gamma_{\tilde{K}}$, and $\Gamma_{\tilde{K}}$ in Section 3 and 4.

Let $\varphi$ be a state on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ such that $\pi_{\varphi}(B(f))$ is essentially self-adjoint for all $f\in{\rm Re}\tilde{K}$. Then we put $W_{\varphi}(f)=\exp(i\pi_{\varphi}(B(f)))$, $f\in{\rm Re}\tilde{K}$. Such state $\varphi$ is said to be regular if $W_{\varphi}(f)$ satisfies the Weyl–Segal relations:

[TABLE]

In general, the Weyl CCR algebra is the universal ${\rm C}^{*}$-algebra generated by unitaries $W(f)$, $f\in{\rm Re}\tilde{K}$, which satisfy (2.2) and we denote ${\cal W}({\rm Re}\tilde{K},\gamma_{\tilde{K}})$ the Weyl CCR algebra. (See also [5, Theorem 5.2.8.].)

A state $\varphi$ on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ is said to be quasi-free, if $\varphi$ satisfies the following equations:

[TABLE]

where $n\in{\mathbb{N}}$ and the sum is over all permutations $s$ satisfying $s(1)<s(2)<\cdots<s(n),s(j)<s(j+n)$, $j=1,2,\cdots,n$. For any quasi-free state $\varphi$ over ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$, the sesquilinear form $S_{\tilde{K}}:\tilde{K}\times\tilde{K}\to{\mathbb{C}}$ defined by

[TABLE]

is positive semi-definite and satisfies

[TABLE]

(See [1, Lemma 3.2.].) Any quasi-free state on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ determines the positive semi-definite sesquilinear form $S$, which satisfies the equation (2.5). Conversely, for any positive semi-definite sesquilinear form $S_{\tilde{K}}$ on $\tilde{K}\times\tilde{K}$ satisfying (2.5), there exists a unique quasi-free state $\varphi$ satisfying (2.4) and $\varphi$ is regular. (See [1, Lemma 3.5.].) Thus, there exists a one-to-one correspondence between a positive semi-definite sesquilinear form $S_{\tilde{K}}$ on $\tilde{K}\times\tilde{K}$ and a quasi-free state $\varphi$ on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$. We denote the quasi-free state on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ determined by a positive semi-definite sesquilinear form $S_{\tilde{K}}$ by $\varphi_{S}$ defined in (2.4). We define the positive semi-definite form $(\cdot,\cdot)_{S}$ on $\tilde{K}\times\tilde{K}$ by the following equation:

[TABLE]

We set $N_{S}:=\set{f\in\tilde{K}}{{\left\|f\right\|}_{S}=0}$, where ${\left\|f\right\|}_{S}=(f,f)_{S}^{1/2}$. We denote the completion of $\tilde{K}/N_{S}$ with respect to the norm ${\left\|\cdot\right\|}_{S}$ by $K$. Since $S_{\tilde{K}}(f,f)\leq{\left\|f\right\|}^{2}_{S}$, ${\left|\gamma_{\tilde{K}}(f,f)\right|}\leq{\left\|f\right\|}^{2}_{S}$, and ${\left\|\Gamma_{\tilde{K}}f\right\|}_{S}={\left\|f\right\|}_{S}$ for any $f\in\tilde{K}$, we can extend the sesquilinear form $S_{\tilde{K}}$ and $\gamma_{{\tilde{K}}}$ to the sesquilinear form on $K\times K$ and the operator $\Gamma_{\tilde{K}}$ to the operator on $K$. We denote the extensions of $S_{\tilde{K}}$, $\gamma_{\tilde{K}}$, and $\Gamma_{\tilde{K}}$ by $S_{K}$, $\gamma_{K}$, and $\Gamma_{K}$, respectively. We define the bounded operators $S_{K}$ and $\gamma_{K}$ on $K$ by the following equations:

[TABLE]

A quasi-free state $\varphi_{S}$ is said to be Fock type if $N_{S}=\{0\}$ and the spectrum of the operator $S_{K}$ defined in (2.7) is contained in $\{0,1/2,1\}$. For any positive semi-definite sesquilinear form $S_{\tilde{K}}$ on $\tilde{K}\times\tilde{K}$, we can construct a Fock type state as follows. Let $\tilde{L}=K\oplus K$. For $\xi_{1},\xi_{2},\eta_{1},\eta_{2}\in K$, we set

[TABLE]

Let $N_{L}=\set{\xi\in\tilde{L}}{(\xi,\xi)_{L}=0}$. Then we denote the completion of $\tilde{L}/N_{L}$ with respect to the norm ${\left\|\cdot\right\|}_{L}$ by $L$. We define the bounded operators $\gamma_{L}$ and $\Pi_{L}$ on $L$ satisfying

[TABLE]

Then the spectrum of $\Pi_{L}$ on $L$ is contained in $\{0,1/2,1\}$. (See [1, Lemma 5.8.] and [1, Lemma 6.1.].) Moreover the following three lemmas hold:

Lemma 2.1**.**

[1, Corollary 6.2.]* The map $f\in\tilde{K}\mapsto[f]\in L$, where $[f]:=(f\oplus 0)+N_{L}$, induces a $*$-homomorphism $\alpha_{\tilde{K}}$ of ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ into ${\cal A}(L,\gamma_{L},\Gamma_{L})$. The restriction of a Fock type state $\varphi_{\Pi_{L}}$ of ${\cal A}(L,\gamma_{L},\Gamma_{L})$ to $\alpha_{\tilde{K}}({\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}}))$ gives a quasi-free state $\varphi_{S}$ of ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ through $\varphi_{\Pi_{L}}(\alpha_{\tilde{K}}(A))=\varphi_{S}(A)$.*

Lemma 2.2**.**

[2, Lemma 2.3.]* Let $R_{S}$ be the von Neumann algebra generated by spectral projections of all $\pi_{\Pi_{L}}(B(f))$, $f\in{\rm Re}{\tilde{K}}$, on the GNS representation space $({\mathfrak{H}}_{\Pi_{L}},\pi_{\Pi_{L}},\xi_{\Pi_{L}})$ of ${\cal A}(L,\gamma_{L},\Gamma_{L})$ associated with $\varphi_{\Pi_{L}}$. Then the following conditions are equivalent:*

The GNS cyclic vector $\xi_{\Pi_{L}}$ is cyclic for $R_{S}$. 2. 2.

The GNS cyclic vector $\xi_{\Pi_{L}}$ is separating for $R_{S}$. 3. 3.

The operator $S_{K}$ on $K$ does not have an eigenvalue [math]. 4. 4.

The operator $S_{K}$ on $K$ does not have an eigenvalue $1$.

Lemma 2.3**.**

[2, Lemma 2.4.]* The center of $R_{S}$ is generated by $\exp(i\pi_{\Pi_{L}}(B(h)))$, $h\in{\rm Re}\overline{(E_{0}K\oplus 0)}^{L}$, where $E_{0}$ is the spectral projection of $S_{K}$ for $1/2$ and $\overline{(E_{0}K\oplus 0)}^{L}$ is the closure of $E_{0}K\oplus 0$ with respect to the norm ${\left\|\cdot\right\|}_{L}$. In particular, $R_{S}$ is factor if and only if $K_{0}=E_{0}K=\{0\}$.*

2.2 Quasi-equivalence of Quasi-free states

We recall the definitions of quasi-equivalence of representations and states.

Definition 2.4**.**

[2, Definition 6.1.]* Let $\pi_{S_{1}}$ and $\pi_{S_{2}}$ be representations associated with quasi-free states $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$, respectively. The representations $\pi_{S_{1}}$ and $\pi_{S_{2}}$ are said to be quasi-equivalent, if there exists an isomorphism $\tau$ from $R_{S_{1}}=\set{W_{S_{1}}(f)}{f\in{\rm Re}\tilde{K}}^{\prime\prime}$ onto $R_{S_{2}}=\set{W_{S_{2}}(f)}{f\in{\rm Re}\tilde{K}}^{\prime\prime}$ such that*

[TABLE]

where $W_{S_{1}}(f)=\exp(i\pi_{S_{1}}(B(f)))$ and $W_{S_{2}}(f)=\exp(i\pi_{S_{2}}(B(f)))$. Let $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ be quasi-free states on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$. The states $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ are said to be quasi-equivalent, if for each GNS-representations $({\mathfrak{H}}_{S_{i}},\pi_{S_{i}})$, $i=1,2$ associated with $\varphi_{S_{i}}$, respectively, are quasi-equivalent.

This definition is equivalent to the definition of quasi-equivalence of states on a ${\rm C}^{*}$-algebra. (See [4, Definition 2.4.25.] and [4, Theorem 2.4.26.].)

Let $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ be quasi-free states on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$. In [3], H. Araki and S. Yamagami showed the following theorem:

Theorem 2.5**.**

[3, Theorem]* Two quasi-free states $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ are quasi-equivalent if and only if the following conditions hold:*

The topologies induced by ${\left\|\cdot\right\|}_{S_{1}}$ and ${\left\|\cdot\right\|}_{S_{2}}$ are equal. 2. 2.

Let $K$ be the completion of $\tilde{K}$ with respect to the topology ${\left\|\cdot\right\|}_{S_{1}}$ or ${\left\|\cdot\right\|}_{S_{2}}$. Then $S_{1}^{1/2}-S_{2}^{1/2}$ is in the Hilbert–Schmidt class on $K$, where the $S_{1}$ and $S_{2}$ are operators on $K$ defined in (2.7).

3 Generalized Coherent states

In this section, we consider generalized coherent states on the Weyl CCR algebra. Using facts in the previous section, we give necessary and sufficient conditions that a generalized coherent state is factor, pure, and faithful and generalized coherent states are quasi-equivalent as well.

3.1 The Weyl CCR algebra

Let $V$ be an ${\mathbb{R}}$-linear space with a symplectic form $\sigma:V\times V\to{\mathbb{R}}$, i.e., $\sigma$ is a bilinear form on $V$ and satisfy the following relations:

[TABLE]

We assume that there exists an operator $J$ on $V$ with the properties

[TABLE]

then $V$ is a ${\mathbb{C}}$-linear space with scalar multiplication defined by

[TABLE]

Then we define the complexification $V^{\mathbb{C}}$ of $V$ by (3.3). We set $(f+ig)^{*}=f-ig$ for $f,g\in V$. We fix a symplectic space $(V,\sigma)$ with an operator $J$ satisfying (3.2). We puts $\tilde{K}=V^{\mathbb{C}}$,

[TABLE]

Then on the GNS-representation space $({\mathfrak{H}}_{\varphi},\pi_{\varphi})$ associated with a regular state $\varphi$ on ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$, ${\cal W}({\rm Re}\tilde{K},\gamma_{\tilde{K}})={\cal W}(V,\sigma)$. Moreover, $\pi_{\varphi}(B(f))$, $f\in{\rm Re}\tilde{K}$, correspond to filed operators. We define the annihilation operators $a(f)$ and the creation operators $a^{\dagger}(f)$ on ${\mathfrak{H}}_{\varphi}$ by the following equation:

[TABLE]

for any $f\in{\rm Re}\tilde{K}$.

In this section, we identify the Weyl CCR algebra ${\cal W}(V,\sigma)$ with a regular state $\varphi$ and ${\cal A}(\tilde{K},\gamma_{\tilde{K}},\Gamma_{\tilde{K}})$ with $\varphi$, where $\tilde{K}$, $\gamma_{\tilde{K}}$ and $\Gamma_{\tilde{K}}$ defined in (3.4).

3.2 Generalized coherent states

For an ${\mathbb{R}}$-linear functional $\lambda:V\to{\mathbb{R}}$, there exists a $*$-automorphism $\tau_{\lambda}$ on ${\cal W}(V,\sigma)$ defined by

[TABLE]

Let $\varphi_{S}$ be a quasi-free state on ${\cal W}(V,\sigma)$. Then we define the generalized coherent state $\varphi_{S,\lambda}$ by the following equation:

[TABLE]

We sets $N_{S}=\set{f\in V^{\mathbb{C}}}{{\left\|f\right\|}_{S}=0}$, where ${\left\|\cdot\right\|}_{S}=(\cdot,\cdot)^{1/2}_{S}$ is the semi-norm defined in (2.6) and $V^{\mathbb{C}}_{S}$ is the completion of $V^{\mathbb{C}}/N_{S}$ by the norm ${\left\|\cdot\right\|}_{S}$. We denote the GNS-representation space with respect to $\varphi_{S}$ and $\varphi_{S,\lambda}$ by $({\mathfrak{H}}_{S},\pi_{S},\xi_{S})$ and $({\mathfrak{H}}_{S,\lambda},\pi_{S,\lambda},\xi_{S,\lambda})$, respectively.

Lemma 3.1**.**

Let $\varphi_{S}$ and $\varphi_{S,\lambda}$ be a quasi-free state and a generalized coherent state on ${\cal W}(V,\sigma)$, respectively. Then

[TABLE]

where $R_{S}$ and $R_{S,\lambda}$ is the von Neumann algebra generated by $\set{\pi_{S}(W(f))}{f\in V}$ and $\set{\pi_{S,\lambda}(W(f))}{f\in V}$, respectively.

Proof. Since $\varphi_{S}$ is regular, there exist self-adjoint operators $\Psi_{S}(f)$, $f\in V$ such that $\pi_{S}(W(f))=\exp(i\Psi_{S}(f))$. By definition of generalized coherent states, we have $\pi_{S,\lambda}(W(f))=e^{i\lambda(f)}\pi_{S}(W(f))$ and $({\mathfrak{H}}_{S,\lambda},\pi_{S,\lambda},\xi_{S,\lambda})=({\mathfrak{H}}_{S},\pi_{S,\lambda},\xi_{S})$. On ${\mathfrak{H}}_{S}$, we have

[TABLE]

Thus, $R_{S}=R_{S,\lambda}$ by the double commutant theorem. $\blacksquare$

Theorem 3.2**.**

Let $\varphi_{S,\lambda}$ be a generalized coherent state on ${\cal W}(V,\sigma)$. Then $\varphi_{S,\lambda}$ is faithful if and only if $S$ does not have an eigenvalue [math] on $V^{\mathbb{C}}_{S}$.

Proof. Note that $\varphi_{S}$ and $\varphi_{S,\lambda}$ has the same GNS cyclic vector space $\xi_{\Pi_{L}}$. By Lemma 2.2, $\varphi_{S,\lambda}$ is faithful if and only if $S$ does not have an eigenvalue [math] on $V^{\mathbb{C}}_{S}$. $\blacksquare$

Theorem 3.3**.**

Let $\varphi_{S,\lambda}$ be a generalized coherent state on ${\cal W}(V,\sigma)$. Then $\varphi_{S,\lambda}$ is factor if and only if $S$ does not have an eigenvalue $1/2$ on $V^{\mathbb{C}}_{S}$.

Proof. By Lemma 2.3 and Lemma 3.1, we have the statement. $\blacksquare$

Theorem 3.4**.**

Let $(V,\sigma)$ be a non-degenerate symplectic space and $\varphi_{S,\lambda}$ be a generalized coherent state on ${\cal W}(V,\sigma)$. Then $\varphi_{S,\lambda}$ is pure if and only if $S$ is a basis projection.

Proof. If $S$ is a basis projection, then by Lemma 3.1 and [1, Lemma 5.5.] $\varphi_{S}$ is pure.

We use the notation in Section 2. Thus, $\tilde{K}=V^{\mathbb{C}}$, $K=V^{\mathbb{C}}_{S}$, and $L$ is the completion of $V^{\mathbb{C}}_{S}\oplus V^{\mathbb{C}}_{S}/N_{L}$ with respect to the norm ${\left\|\cdot\right\|}_{L}$ defined in (2.11). If $\varphi_{S,\lambda}$ is pure, then by Theorem 3.3, $S$ does not have an eigenvalue $1/2$. Then $\Pi_{L}$ defined in (2.13) does not have an eigenvalue $1/2$ because the eigenspace of $\Pi_{L}$ with $1/2$ is the completion of the set $\set{f\oplus f}{f\in E_{0}K}$ with respect to the norm ${\left\|\cdot\right\|}_{L}$, where $E_{0}$ is the spectral projection of $S$ onto $\ker(S-1/2)$. (See also the proof of (4) of [1, Lemma 6.1.].) Thus, $\Pi_{L}$ is a basis projection. Using the notation of [1, Lemma 5.5.], we have $R_{S}=R_{\Pi_{L}}(H_{1})$, with $H_{1}=[{\rm Re}\tilde{K}]\oplus 0\subset L$ and $\overline{H_{1}}=\overline{{\rm Re}K\oplus 0}^{L}\oplus 0$. If $\Pi_{L}\neq S$, then $K\neq L$. Thus, we have $R_{\Pi_{L}}(H_{1})^{\prime}=R_{\Pi_{L}}(H_{1}^{\perp})$ by [1, Lemma 5.5.] and $H_{1}^{\perp}\neq\{0\}$, where $H_{1}^{\perp}$ is the orthogonal complement with respect to the inner product $(\cdot,\cdot)_{L}$ defined in (2.11). It leads $R_{S}^{\prime}\neq{\mathbb{C}}\mathbbm{1}$. It contradict to the purity of $\varphi_{S}$. Thus, $S$ is a basis projection. $\blacksquare$

We have necessary and sufficient conditions that a generalized coherent state is faithful, factor, and pure. Next, we consider the quasi-equivalence of generalized coherent states.

Lemma 3.5**.**

Let $\varphi_{S,\lambda}$ be a generalized coherent state on ${\cal W}(V,\sigma)$. Then $f\in N_{S}$ if and only if $\pi_{S,\lambda}(W(f))=e^{i\lambda(f)}\mathbbm{1}$.

Proof. If $f\in N_{S}$, then $\varphi_{S}(W(tf))=1$ for any $t\in{\mathbb{R}}$. Thus, by regularity of $\varphi_{S}$, $\pi_{S}(W(f))=\mathbbm{1}$. By definition of generalized coherent state, $\pi_{S,\lambda}(W(f))=e^{i\lambda(f)}\mathbbm{1}$.

If $\pi_{S,\lambda}(W(f))=e^{i\lambda(f)}\mathbbm{1}$, $f\in V$, then $\pi_{S}(W(f))=\mathbbm{1}$. Since $g^{*}=g$ for any $g\in V$, we have that $(f,f)_{S}=0$. $\blacksquare$

Lemma 3.6**.**

Let $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ be generalized coherent states on ${\cal W}(V,\sigma)$. If $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ are quasi-equivalent, then $N_{S_{1}}=N_{S_{2}}$.

Proof. Since $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ are quasi-equivalent, then there exists $\tau:\pi_{S_{1},\lambda_{1}}({\cal W}(V,\sigma))^{\prime\prime}\to\pi_{S_{2},\lambda_{2}}({\cal W}(V,\sigma))^{\prime\prime}$ such that

[TABLE]

If $N_{S_{1}}\neq N_{S_{2}}$, then there exists $f\in V^{\mathbb{C}}$ such that $f\in N_{S_{1}}$ and $f\not\in N_{S_{2}}$. Put $h=f+f^{*}$. Then $h\in V={\rm Re}V^{\mathbb{C}}$ and $h\in N_{S_{1}}$ and $h\not\in N_{S_{2}}$. For such $h$, we have

[TABLE]

by Lemma 3.5. However, we have

[TABLE]

It contradict to Lemma 3.5. $\blacksquare$

Theorem 3.7**.**

Let $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ be generalized coherent states on ${\cal W}(V,\sigma)$. Then $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ are quasi-equivalent if and only if the following conditions hold:

${\left\|\cdot\right\|}_{S_{1}}$* and ${\left\|\cdot\right\|}_{S_{2}}$ induce the same topology,* 2. 2.

$S_{1}^{1/2}-S_{2}^{1/2}$* is a Hilbert–Schmidt class operator,* 3. 3.

$\lambda_{1}=\lambda_{2}$* on $N_{S_{1}}=N_{S_{2}}$,* 4. 4.

$\lambda_{1}-\lambda_{2}$* is continuous with respect to the norm ${\left\|\cdot\right\|}_{S_{1}}$ or ${\left\|\cdot\right\|}_{S_{2}}$.*

Proof. Assume that the topologies induced by ${\left\|\cdot\right\|}_{S_{1}}$ and ${\left\|\cdot\right\|}_{S_{2}}$ are equivalent, $S_{1}^{1/2}-S_{2}^{1/2}$ is Hilbert-Schmidt class, $\lambda_{1}-\lambda_{2}$ is continuous with respect to ${\left\|\cdot\right\|}_{S_{1}}$, and $\lambda_{1}=\lambda_{2}$ on $N_{S_{1}}=N_{S_{2}}$. Then $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ are quasi-equivalent by [3, Theorem] and $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ are quasi-equivalent by continuity of $\lambda_{1}-\lambda_{2}$ and $\lambda_{1}=\lambda_{2}$ on $N_{S_{1}}=N_{S_{2}}$.

Next, we assume that $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ are quasi-equivalent. The quasi-equivalence of $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ induces the quasi-equivalence of $\varphi_{S_{1},\lambda_{1}-\lambda_{2}}$ and $\varphi_{S_{2}}$. Put $\lambda:=\lambda_{1}-\lambda_{2}$. Then there exists a $*$-isomorphism $\tau$ from $\pi_{S_{1},\lambda}({\cal W}(V,\sigma))^{\prime\prime}$ onto $\pi_{S_{2}}({\cal W}(V,\sigma))^{\prime\prime}$ such that

[TABLE]

For any $f\in V$,

[TABLE]

is ${\left\|\cdot\right\|}_{S_{2}}$-continuous in $f\in V$. Thus, $\lambda$ and $S_{1}$ are ${\left\|\cdot\right\|}_{S_{2}}$-continuous. By symmetry, $\lambda$ and $S_{2}$ are ${\left\|\cdot\right\|}_{S_{1}}$-continuous as well. By Lemma 3.5, $N_{S}:=N_{S_{1}}=N_{S_{2}}$. If $\lambda\neq 0$ on $N_{S}$, then there exists $f\in N_{S}\backslash\{0\}$ such that $\lambda(f)\neq 0$. If $\lambda(f)=2n\pi$ for some $n\in{\mathbb{Z}}$, then we replace $f$ by $f/\pi$. For such $f$, we have

[TABLE]

by Lemma 3.5. It contradicts to the quasi-equivalence of $\varphi_{S_{1},\lambda}$ and $\varphi_{S_{2}}$. Thus, $\lambda=0$ on $N_{S}$. Let $\tau^{\prime}$ be the map from $\pi_{S_{1},\lambda}({\cal W}(V,\sigma))$ to $\pi_{S_{1}}({\cal W}(V,\sigma))$ defined by

[TABLE]

Since $\lambda$ is continuous with respect to the norm ${\left\|\cdot\right\|}_{S_{1}}$ and $\lambda=0$ on $N_{S}$, then we can extend $\tau^{\prime}$ to a map from $\pi_{S_{1},\lambda}({\cal W}(V,\sigma))^{\prime\prime}$ onto $\pi_{S_{1}}({\cal W}(V,\sigma))^{\prime\prime}$. Then $\tau^{\prime}$ induce the quasi-equivalence of $\varphi_{S_{1},\lambda}$ and $\varphi_{S_{1}}$. Thus, $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ are quasi-equivalent and by Theorem 2.5, we have the statement. $\blacksquare$

Remark 3.8**.**

In [20], S. Yamagami obtained quasi-equivalence conditions of (generalized) coherent states in terms of the transition amplitude. For applications to concrete models Hilbert-Schmidt conditions in Theorem 3.7 are easier to handle. Let $\varphi_{S_{1},\lambda_{1}}$ and $\varphi_{S_{2},\lambda_{2}}$ be generalized coherent states on the Weyl CCR algebra ${\cal W}(V,\sigma)$. Assume that $\varphi_{S_{1}}$ and $\varphi_{S_{2}}$ are quasi-equivalent. If $\lambda_{1}-\lambda_{2}$ is not continuous in ${\left\|\cdot\right\|}_{S_{1}}$ or ${\left\|\cdot\right\|}_{S_{2}}$ or $\lambda_{1}\neq\lambda_{2}$, then the transition amplitude $(\varphi_{S_{1},\lambda_{1}}^{1/2},\varphi_{S_{2},\lambda_{2}}^{1/2})=0$, where $\varphi_{S_{1}}^{1/2}$ and $\varphi_{S_{2},\lambda_{2}}^{1/2}$ is GNS-vector in the universal representation space $L^{2}({\cal W}(V,\sigma)^{**})$. (See [20, Theorem 5.3.].)

Factor decompositions of quasi-free states are given in [9], [16] and [19], e.t.c.. For the convenience of the reader, we give an explicit form of factor decomposition of a non-factor generalized coherent state. We recall the definition of the disjointness of states. (See also [4, Definition 4.1.20.] and [4, Lemma 4.2.8.].)

Definition 3.9**.**

Let $\varphi_{1}$ and $\varphi_{2}$ be positive linear functionals on a ${\rm C}^{*}$-algebra ${\cal A}$. The positive linear functionals $\varphi_{1}$ and $\varphi_{2}$ are said to be disjoint, if for $\omega=\varphi_{1}+\varphi_{2}$, there is a projection $P\in\pi_{\omega}({\cal A})^{\prime\prime}\cap\pi_{\omega}({\cal A})^{\prime}$ such that

[TABLE]

where $\pi_{\omega}$ is the GNS-representation and $\xi_{\omega}$ is the GNS-cyclic vector associated with $\omega$.

Note that factor representations are either quasi-equivalent or disjoint. (See e.g. [4, Proposition 2.4.22.], [4, Theorem 2.4.26. (1)], and [4, Proposition 2.4.27.].)

Theorem 3.10**.**

Let $\varphi_{S,\lambda}$ be a generalized coherent state on ${\cal W}(V,\sigma)$. If $\varphi_{S,\lambda}$ is non-factor, then there exists a probability measure $\mu$ on ${\mathbb{R}}^{2I}$ and $\varphi_{S,\lambda}$ has factor decomposition of the form

[TABLE]

where $\varphi_{SE_{0}^{\perp},x\cdot\rho+\lambda}(W(f))=\exp(-S(E_{0}^{\perp}f,E_{0}^{\perp}f)/4+ix\cdot\rho(f)+i\lambda(f))$ and $\rho(f)=({\rm Re}(e_{k},f)_{S},{\rm Im}(e_{k},f)_{S})_{k\in I}\in{\mathbb{R}}^{2I}$. Moreover, $\varphi_{SE_{0}^{\perp},x\cdot\rho+\lambda}$ and $\varphi_{SE_{0}^{\perp},y\cdot\rho+\lambda}$ are disjoint unless $x\neq y$, $x,y\in{\mathbb{R}}^{2I}$.

Proof. If a generalized coherent state $\varphi_{S,\lambda}$ on ${\cal W}(V,\sigma)$ is non-factor, then on $V^{\mathbb{C}}_{S}$, $S$ has the spectral decomposition

[TABLE]

where $E_{0}$ is the spectral projection of $S$ with an eigenvalue $1/2$, $I$ is an index set such that ${\left|I\right|}={\rm dim}\ker(S-1/2)$, and $\{e_{k}\}_{k\in I}$ is an orthonormal basis for $\ker(S-1/2)$. Thus, for any $W(f)$, $f\in V$, we have

[TABLE]

By a theorem of Bochner–Minlos (See e.g. [17, Theorem 2.2.]), there exists a probability measure $\mu$ on ${\mathbb{R}}^{2I}$ such that

[TABLE]

where $\rho(f)=({\rm Re}(e_{k},f)_{S},{\rm Im}(e_{k},f)_{S})_{k\in I}\in{\mathbb{R}}^{2I}$. For $\varphi_{SE_{0}^{\perp},x\cdot\rho+\lambda}$, we have $N_{SE_{o}^{\perp}}=E_{0}V^{\mathbb{C}}\neq\{0\}$. Since $E_{0}V^{\mathbb{C}}\neq\{0\}$, there exists a $f\in V$ such that ${\rm Re}(e_{k},f)_{S}\neq 0$ or ${\rm Im}(e_{k},f)_{S}\neq 0$. We put $f_{n}:=E_{0}f+1/nE_{0}^{\perp}f$. Then ${\left\|f_{n}\right\|}_{SE_{0}^{\perp}}\to 0$ and ${\rm Re}(e_{k},f_{n})_{S}\not\to 0$ or ${\rm Im}(e_{k},f_{n})_{S}\not\to 0$ as $n\to\infty$. Thus, the generalized coherent states $\varphi_{SE_{0}^{\perp},x\cdot\rho+\lambda}$ and $\varphi_{SE_{0}^{\perp},y\cdot\rho+\lambda}$, $x,y\in{\mathbb{R}}^{2I}$ are not quasi-equivalent unless $x=y$ by Theorem 3.7. Since ${\left\|\cdot\right\|}_{S}$ and ${\left\|\cdot\right\|}_{SE_{0}^{\perp}}$ induce the same topology on $V^{\mathbb{C}}$ and $SE_{0}^{\perp}$ on $V^{\mathbb{C}}_{SE_{0}^{\perp}}$ does not have an eigenvalue $1/2$, $\varphi_{SE_{0}^{\perp},x\cdot\rho+\lambda}$ is factor and $\varphi_{SE_{0}^{\perp},x\cdot\rho+\lambda}$ and $\varphi_{SE_{0}^{\perp},y\cdot\rho+\lambda}$ are disjoint unless $x\neq y$, $x,y\in{\mathbb{R}}^{2I}$. $\blacksquare$

4 BEC and Non-factor states

In this section, we consider quasi-free states on ${\cal W}({\mathfrak{h}},\sigma)$, where ${\mathfrak{h}}$ is a pre-Hilbert space over ${\mathbb{C}}$ with an inner product ${\left\langle\cdot,\cdot\right\rangle}_{{\mathfrak{h}}}$ and $\sigma(f,g)={\rm Im}{\left\langle f,g\right\rangle}_{{\mathfrak{h}}}$, $f,g\in{\mathfrak{h}}$. We give the decomposition of quasi-free states on ${\cal W}({\mathfrak{h}},\sigma)$ into generalized coherent states which are mutually disjoint.

4.1 General properties

In this subsection, we use the following notations. Let ${\mathfrak{h}}$ be a subspace of a Hilbert space over ${\mathbb{C}}$. We assume that ${\mathfrak{h}}$ is equipped with positive definite inner products ${\left\langle\cdot,\cdot\right\rangle}_{{\mathfrak{h}}}$ and ${\left\langle\cdot,\cdot\right\rangle}_{0}$. Let $q$ be a linear functional on ${\mathfrak{h}}$. We consider the quasi-free state $\varphi_{q,D}$, $D\geq 0$, on ${\cal W}({\mathfrak{h}},\sigma)$ defined by

[TABLE]

where $a(f)$ and $a^{\dagger}(f)$, $f\in{\mathfrak{h}}$, are the annihilation operators and the creation operators on the GNS representation space ${\mathfrak{H}}_{\varphi_{q,D}}$, respectively. Note that the annihilation operators $a(f)$, $f\in{\mathfrak{h}}$, and the creation operators $a^{\dagger}(f)$, $f\in{\mathfrak{h}}$ satisfy the following equation:

[TABLE]

Our aim is to show that $\varphi_{q,D}$ is non-factor if $q$ is not continuous with respect to the norm ${\left\|\cdot\right\|}_{q,D}$ defined in (4.8) and $D>0$, and to get factor decomposition of $\varphi_{q,D}$, in this subsection. Let $\{e_{n}\}_{n\in{\mathbb{N}}}$ be an orthonormal basis on a Hilbert space which is contained in ${\mathfrak{h}}$. Fix $\{e_{n}\}_{n\in{\mathbb{N}}}$. We set

[TABLE]

for $f=\sum_{n\in{\mathbb{N}}}f_{n}e_{n}\in{\mathfrak{h}}$, where $f_{n}\in{\mathbb{C}}$, $n\in{\mathbb{N}}$ and $\overline{f_{n}}$ is the complex conjugate of $f_{n}$. For a linear functional $q$ and $D\geq 0$, we put $\tilde{K}_{q,D}={\mathfrak{h}}\oplus{\mathfrak{h}}$. For $f_{1},f_{2},g_{1},g_{2}\in{\mathfrak{h}}$, we sets

[TABLE]

We define the inner product on $\tilde{K}_{q,D}$ by

[TABLE]

Let $N_{K_{q,D}}=\set{f\in\tilde{K}_{q,D}}{{\left\|f\right\|}_{q,D}=0}$. Then we denote the completion of $\tilde{K}_{q,D}/N_{K_{q,D}}$ with respect to the norm ${\left\|\cdot\right\|}_{q,D}$ by $K_{q,D}$. In this case, ${\left\|f_{1}\oplus f_{2}\right\|}_{q,D}=0$ leads $f_{1}=0$ and $f_{2}=0$. Thus, $N_{K_{q,D}}=\{0\}$.

We put

[TABLE]

and ${\left\|\cdot\right\|}_{{\mathfrak{K}}}={\left\langle\cdot,\cdot\right\rangle}_{{\mathfrak{K}}}^{1/2}$. We define the Hilbert space ${\mathfrak{K}}$ by the completion of ${\mathfrak{h}}$ with respect to the norm ${\left\|\cdot\right\|}_{{\mathfrak{K}}}$.

Lemma 4.1**.**

The space $K_{q,D}$ has the following form:

• $1.$

If $D>0$ and $q$ is not continuous with respect to the norm ${\left\|\cdot\right\|}_{{\mathfrak{K}}}$, then we have

[TABLE]

• $2.$

If $D=0$ or $q$ is continuous with respect to the norm ${\left\|\cdot\right\|}_{{\mathfrak{K}}}$, then we have

[TABLE]

Proof. We consider the case of $D>0$ and $q$ is not continuous with respect to the norm ${\left\|\cdot\right\|}_{{\mathfrak{K}}}$. It suffices to show that ${\mathbb{C}}\oplus{\mathfrak{K}}=\overline{{\mathfrak{h}}}$, where $\overline{{\mathfrak{h}}}$ is the completion of ${\mathfrak{h}}$ with respect to the norm ${\left\|\cdot\right\|}^{\prime}$ defined by

[TABLE]

We define $\pi:{\mathfrak{h}}\to{\mathbb{C}}\oplus{\mathfrak{K}}$ by

[TABLE]

Since $q$ is not continuous, for any $f\in{\mathfrak{h}}$, there exists a sequence $f_{n}$ in ${\mathfrak{h}}$ such that $\lim_{n\to\infty}{\left\|f_{n}-f\right\|}_{{\mathfrak{K}}}=0$ and $\lim_{n\to\infty}q(f_{n})=0$. For such $f_{n}$ and $f$, we have

[TABLE]

The case of $D=0$ is clear. We assume that $q$ is continuous with respect to the norm ${\left\|\cdot\right\|}_{{\mathfrak{K}}}$. By continuity of $q$, the norm ${\left\|\cdot\right\|}^{\prime}$, defined in (4.12), and ${\left\|\cdot\right\|}_{{\mathfrak{K}}}$ induce the same topology. $\blacksquare$

Theorem 4.2**.**

For a linear space ${\mathfrak{h}}$ with positive definite inner products ${\left\langle\cdot,\cdot\right\rangle}_{{\mathfrak{h}}}$ and ${\left\langle\cdot,\cdot\right\rangle}_{0}$, if $D>0$ and $q$ is not continuous with respect to the norm ${\left\|\cdot\right\|}_{\mathfrak{K}}$, then the two-point function $\varphi_{q,D}$ defined in (4.1) is a non-factor state on ${\cal W}({\mathfrak{h}},\sigma)$.

Proof. First, we consider the case of $D>0$. By Lemma 2.1 and Lemma 2.2, it suffices to show that $1/2\in\sigma_{P}(S_{q,D})$. By Lemma 4.1, an element of $K_{q,D}$ has the form $(a_{1},f_{1},a_{2},f_{2})$, $a_{1},a_{2}\in{\mathbb{C}}$, $f_{1},f_{2}\in{\mathfrak{K}}$. For any $(a_{1}\oplus f_{1}\oplus a_{2}\oplus f_{2}),(b\oplus 0\oplus 0\oplus 0)\in K_{q,D},b\in{\mathbb{C}}$, the operator $S_{q,D}$ satisfies

[TABLE]

Thus, we have $S_{q,D}(b\oplus 0\oplus 0\oplus 0)=1/2(b\oplus 0\oplus 0\oplus 0)$ for any $b\in{\mathbb{C}}$ and $1/2\in\sigma_{P}(S_{q,D})$. $\blacksquare$

Proposition 4.3**.**

For a linear space ${\mathfrak{h}}$ with positive definite inner products ${\left\langle\cdot,\cdot\right\rangle}_{{\mathfrak{h}}}$ and ${\left\langle\cdot,\cdot\right\rangle}_{0}$, if $D=0$ or $q$ is continuous with respect to the norm ${\left\|\cdot\right\|}_{\mathfrak{K}}$, the two-point function $\varphi_{q,D}$ defined in (4.1) is a factor state on ${\cal W}({\mathfrak{h}},\sigma)$.

Proof. If $q$ is continuous with respect to the norm ${\left\|\cdot\right\|}_{\mathfrak{K}}$, then $\varphi_{q,D}$ is quasi-equivalent to $\varphi_{0,0}$ by Theorem 3.7. Thus, it suffice to show the case of $D=0$. There exists the positive contraction operator $A$ on ${\mathfrak{K}}$ such that ${\left\langle\xi,A\eta\right\rangle}_{{\mathfrak{K}}}={\left\langle\xi,\eta\right\rangle}_{{\mathfrak{h}}}/2$ and ${\left\langle\xi,(\mathbbm{1}-A)\eta\right\rangle}_{{\mathfrak{K}}}={\left\langle\xi,\eta\right\rangle}_{0}$, $\xi,\eta\in{\mathfrak{K}}$. Then $S_{0,0}$ has the following form:

[TABLE]

for $\eta_{1},\eta_{2}\in{\mathfrak{K}}$. If $1/2\in\sigma_{P}(S_{0,0})$, then $(\mathbbm{1}+A)\eta_{1}=\eta_{1}$ and $(\mathbbm{1}-A)\eta_{2}=\eta_{2}$. Thus, $\eta_{1},\eta_{2}\in\ker A$. Since the positive definiteness of ${\left\langle\cdot,\cdot\right\rangle}_{\mathfrak{h}}$ and ${\left\langle\cdot,\cdot\right\rangle}_{0}$ on ${\mathfrak{h}}$, ${\mathfrak{h}}\cap\ker A=\{0\}$. Thus, $\ker A=\{0\}$ and $\varphi_{0,0}$ is factor. $\blacksquare$

Next, we consider factor decomposition of $\varphi_{q,D}$, if $q$ is not continuous in ${\left\|\cdot\right\|}_{\mathfrak{K}}$. Let $({\mathfrak{H}}_{0},\pi_{0},\xi_{0})$ be the GNS-representation space with respect to $\varphi_{0}:=\varphi_{q,0}=\varphi_{0,D}$. Since $\varphi_{0}$ is regular state on ${\cal W}({\mathfrak{h}},\sigma)$, there exist self-adjoint operators $\Psi_{0}(f)$, $f\in{\mathfrak{h}}$, such that

[TABLE]

Now we define the field operators $\Psi_{s_{1},s_{2}}(f)$, $s_{1},s_{2}\in{\mathbb{R}}$, $f\in{\mathfrak{h}}$, on ${\mathfrak{H}}_{0}$ by

[TABLE]

Let $\pi_{s_{1},s_{2}}$ be the representation of ${\cal W}({\mathfrak{h}},\sigma)$ on ${\mathfrak{H}}_{0}$ defined by

[TABLE]

Using the $\pi_{s_{1},s_{2}}$, we define the state $\varphi_{s_{1},s_{2}}$ on ${\cal W}({\mathfrak{h}},\sigma)$ by

[TABLE]

Then we have the following theorem.

Theorem 4.4**.**

If $q$ is not continuous in ${\left\|\cdot\right\|}_{\mathfrak{K}}$, then for each $s_{1},s_{2},t_{1},t_{2}\in{\mathbb{R}}$, $\varphi_{s_{1},s_{2}}$ and $\varphi_{t_{1},t_{2}}$ are factor and disjoint unless $t_{1}=s_{1}$ and $t_{2}=s_{2}$.

Proof. By Lemma 3.1 and Proposition 4.3, $\varphi_{s_{1},s_{2}}$ and $\varphi_{t_{1},t_{2}}$ are factor. Since $q$ is not continuous with respect to the norm, $\varphi_{s_{1},s_{2}}$ and $\varphi_{t_{1},t_{2}}$ are disjoint unless $t_{1}=s_{1}$ and $t_{2}=s_{2}$ by Theorem 3.7. $\blacksquare$

Finally, we obtain factor decomposition of $\varphi_{q,D}$.

Theorem 4.5**.**

If $q$ is not continuous in ${\left\|\cdot\right\|}_{\mathfrak{K}}$, then for any $D>0$, factor decomposition of $\varphi_{q,D}$ defined in (4.1) is given by

[TABLE]

Proof. By Theorem 3.10, we are done. $\blacksquare$

4.2 On graphs

In this subsection, let $X=(VX,EX)$ be an undirected graph, where $VX$ is the set of all vertices in $X$ and $EX$ is the set of all edges in $X$. Two vertices $x,y\in VX$ are said to be adjacent if there exists an edge $(x,y)\in EX$ joining $x$ and $y$, and we write $x\sim y$. We denote the set of all the edges connecting $x$ with $y$ by $E_{x,y}$. Since the graph is undirected, $E_{x,y}=E_{y,x}$. Let $\ell^{2}(VX)$ be the set of all square summable sequence labeled by the vertices in $VX$. Let $A_{X}$ be the adjacency operator of $X$ defined by

[TABLE]

In addition, for any $x\in VX$, we set the degree of $x$ by ${\rm deg}(x)$ and

[TABLE]

We assume that $X$ is connected, countable and ${\rm deg}<\infty$. Then, the adjacency operator $A_{X}$ acting on $\ell^{2}(VX)$ is bounded. If for any $\delta_{x}$, $x\in VX$, $A_{X}$ satisfies the condition

[TABLE]

then $A_{X}$ is said to be transient. Let $H$ be the Hamiltonian on $\ell^{2}(VX)$ defined by $H:={\left\|A_{X}\right\|}\mathbbm{1}-A_{X}$.

A bounded operator $B$ on $\ell^{2}(VX)$ is called positivity preserving if $B_{x,y}:={\left\langle\delta_{x},B\delta_{y}\right\rangle}\geq 0$ for any $x,y\in VX$. A sequence $\set{v(x)}{x\in VX}$ is called a Perron–Frobenius weight for $B$ if it has positive entries and

[TABLE]

for any $x\in VX$.

In [8], F. Fidaleo considered BEC on graphs and showed the following two results.

Proposition 4.6**.**

[8, Proposition 4.1.]* Let $A_{X}$ be the adjacency operator of $X$ on $\ell^{2}(VX)$ and $H$ be the Hamiltonian defined by $H={\left\|A_{X}\right\|}\mathbbm{1}-A_{X}$. Let ${\mathfrak{h}}$ be a subspace of $\ell^{2}(VX)$ satisfying the following three conditions: For each $\beta>0$,*

1. $1.$

$e^{itH}{\mathfrak{h}}={\mathfrak{h}}$, $t\in{\mathbb{R}}$; 2. $2.$

For each entire function $f$, $f(H){\mathfrak{h}}\subset{\cal D}((e^{\beta H}-1)^{-1/2})$; 3. $3.$

$\sum_{x\in VX}{\left|(f(H)u)(x)\right|}v(x)<\infty$, and ${\left\langle f(H)u,v\right\rangle}=\overline{f(0)}{\left\langle u,v\right\rangle}$, where $v$ is a Perron–Frobenius weight for $A_{X}$.

Then for $D\geq 0$, the two–point function

[TABLE]

satisfies the KMS condition at inverse temperature $\beta>0$ on the Weyl CCR algebra ${\cal W}({\mathfrak{h}},\sigma)$ with respect to the dynamics generated by the Bogoliubov transformations

[TABLE]

By the above proposition and [14, Proposition 1.1.], we are said to be BEC occur if the case of $D>0$ and BEC does not occur if the case of $D=0$.

Theorem 4.7**.**

[8, Theorem 4.5.]* Suppose that $A_{X}$ is transient. Let ${\mathfrak{h}}_{1}$ be the subspace of $\ell^{2}(VX)$ defined by*

[TABLE]

Then ${\mathfrak{h}}_{1}$ satisfies the conditions $1$, $2$, and $3$ in Proposition 4.6. Thus, for ${\mathfrak{h}}_{1}$ and any $D\geq 0$, the two-point function given in (4.26) defines KMS state on the Weyl CCR algebra ${\cal W}({\mathfrak{h}}_{1},\sigma)$.

We give another example of ${\mathfrak{h}}$. Let ${\cal P}({\mathbb{C}})$ be the set of all polynomial functions on ${\mathbb{C}}$. Let ${\mathfrak{h}}_{2}$ be the subspace defined by

[TABLE]

Lemma 4.8**.**

The space ${\mathfrak{h}}_{2}$ satisfies the following conditions;

1. $1^{\prime}$.

$e^{itH}{\mathfrak{h}}_{2}={\mathfrak{h}}_{2}$, $t\in{\mathbb{R}}$; 2. $2^{\prime}$.

$e^{\beta H}{\mathfrak{h}}_{2}\subset{\cal D}((e^{\beta H}-1)^{-1/2})$; 3. $3^{\prime}$.

$\sum_{x\in VG}{\left|(e^{\beta H}u)(x)\right|}<\infty$, and ${\left\langle e^{\beta H}u,v\right\rangle}={\left\langle u,v\right\rangle}$, $u\in{\mathfrak{h}}_{2}$.

Proof. The condition $1^{\prime}$, $e^{itH}{\mathfrak{h}}_{2}\subset{\mathfrak{h}}_{2}$ is clear. Now we prove the condition $2^{\prime}$, $e^{\beta H}{\mathfrak{h}}_{2}\subset{\cal D}((e^{\beta H}-\mathbbm{1})^{-1/2})$. Note that $(e^{\beta x}-1)^{-1}-(\beta x)^{-1}$ is continuous on $[0,\infty)$. Thus, it enough to show that $e^{\beta H}{\mathfrak{h}}_{2}\subset{\cal D}(H^{-1/2})$. Since $A_{X}$ is transient and $p(t)e^{-(t-a)^{2}/b}$ is a rapidly decreasing function on ${\mathbb{R}}$, for a generator of ${\mathfrak{h}}_{2}$, $\int_{\mathbb{R}}p(t)e^{-\frac{(t-a)^{2}}{b}}e^{itH}\delta_{x}dt$, we have

[TABLE]

where $C_{1}$ is a positive constant satisfying

[TABLE]

Next, we show that $\sup_{n\in{\mathbb{N}}}\sum_{x\in V\Lambda}{\left|(e^{\beta H}u)(x)\right|}v(x)<\infty$, $u\in{\mathfrak{h}}_{2}$, where $\Lambda_{n}$ is a finite subgraph of $X$ such that $\Lambda_{n}\nearrow X$. Let $C_{R}$ be a circle centered at the origin with radius $R>{\left\|A_{X}\right\|}$. We have

[TABLE]

for any $x,y\in VX$, where $C_{2}$ is a positive constant satisfying

[TABLE]

By the above inequality (LABEL:eq:ineq_of_generator), we get

[TABLE]

Finally, we show the latter part of the condition $3^{\prime}$. For any $f\in{\mathfrak{h}}_{2}$, by definition of $v$,

[TABLE]

Thus, we are done. $\blacksquare$

Theorem 4.9**.**

Suppose that the adjacency operator $A_{X}$ of a graph $X$ is transient. For $D>0$, the two-point function $\varphi_{D}$ defined in (4.26) is a non-factor KMS state on ${\cal W}({\mathfrak{h}}_{1},\sigma)$ or ${\cal W}({\mathfrak{h}}_{2},\sigma)$. Moreover, we have factor decomposition of $\varphi_{D}$ into extremal KMS states

[TABLE]

Proof. Since ${\left\langle\cdot,(e^{\beta H}+\mathbbm{1})(e^{\beta H}-\mathbbm{1})^{-1}\cdot\right\rangle}$ is positive definite inner product on ${\mathfrak{h}}_{1}$ and ${\mathfrak{h}}_{2}$, it suffice to show that ${\left\langle v,f\right\rangle}$, $f\in{\mathfrak{h}}_{1}$ or $f\in{\mathfrak{h}}_{2}$ is not continuous with respect to the norm ${\left\langle\cdot,(e^{\beta H}+\mathbbm{1})(e^{\beta H}-\mathbbm{1})^{-1}\cdot\right\rangle}$ by Theorem 4.4 and 4.5. Let $p_{n}$ be the polynomial defined by

[TABLE]

For any $f\in{\mathfrak{h}}_{1}$, $(p_{n}(H)-\mathbbm{1})f\in{\mathfrak{h}}_{1}$. Put $f_{n}=(p_{n}(H)-\mathbbm{1})f$. Then

[TABLE]

and

[TABLE]

for any $n\in{\mathbb{N}}$. Thus, we have that ${\left\langle v,\cdot\right\rangle}$ is not continuous.

For any $f\in{\mathfrak{h}}_{2}$, we put $f_{n}=p_{n}(H)f$. We can show $f_{n}\in{\mathfrak{h}}_{2}$. Similarly the case of ${\mathfrak{h}}_{1}$, we can prove the statement. $\blacksquare$

Acknowledgments

The author would like to thank Professor Taku Matsui for discussions and comments for this work.