Variable Planck's constant and scaling properties of states on Weyl algebra

TL;DR

This paper investigates how variations in Planck's constant affect states on Weyl algebra, revealing invariance conditions, state transformations, and the impact on quantum dynamics and algebraic structures.

Contribution

It analyzes the effects of changing Planck's constant on CCR-algebra states, especially quasi-free and KMS-states, and explores conditions for invariance and algebraic restrictions.

Findings

Universal invariant states are convex combinations of Fock states with different Planck's constants.

Rescaling Planck's constant nontrivially alters the dynamics of KMS-states.

Restrictions on the algebra are necessary to accommodate larger variations in Planck's constant.

Abstract

We consider the possible quantum effect for infinite systems produced by variations of the Planck's constant. Using the algebraic formulation of quantum theory we study behaviour of states $\omega$ defined as positive, normalized functionals on the canonical commutation relations algebra (CCR-algebra) under the changes of the defining relations of the CCR. These defining relations of the multiplication in the CCR-algebra depend explicitly on the value of the Planck's constant. We analyse to what extend changes of the $\hbar$ preserve the original state space (this gives restrictions on the admissible changes of the Plank's constant) and what properties have original quantum states $\omega$ as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of Planck's constant. The second important class of states we study are the KMS-states, here the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes of the $\hbar$, but then one has to restrict the CCR-algebra to a subalgebra.

Equations

$$[\Phi (f),\Phi (g)]=i\sigma (f,g)$$

$$W_f=e^{i\Phi (f)}$$

$$W_f{W}_g=e^{-\frac{i}{2}\sigma (f,g)}{W}_{f+g}$$

$${\sigma }_h(f,g)=h\sigma (f,g),f,g\in \mathbb{H}$$

$$W_f(g)=\{\begin{array}{ll}1,& f=g\\ 0,& f\ne g\end{array}$$

$$W_f{W}_g=e^{-\frac{i}{2}{\sigma }_h(f,g)}{W}_{f+g}$$

$$W_f^{\ast }=W_{-f}$$

$$||\sum _{j=1}^n{a}_j{\delta }_{f_j}|{|}_1=\sum _{j=1}^n|a_j|$$

$$||A||=\underset{\pi }{\sup }||\pi (A)||,A\in {\overline{{\mathfrak{W}}_0(H,{\sigma }_h)}}^{||\cdot |{|}_1}$$

$${\gamma }_h(W_f^h)=W_{\sqrt{h}f}^{h=1},f\in \mathbb{H}$$

$$W_f^{h\ast }=W_{-f}^h$$

$${\gamma }_h(W_f^h{)}^{\ast }=W_{\sqrt{h}f}^{\ast }={\gamma }_h(W_f^{h\ast })$$

$${\gamma }_h(A{)}^{\ast }={\gamma }_h(A^{\ast }),A\in {\mathfrak{W}}_0(\mathbb{H},{\sigma }_h)$$

$${\gamma }_h(W_f^h{W}_g^h)=e^{-\frac{i}{2}h\sigma (f,g)}{\gamma }_h(W_{f+g}^h)=e^{-\frac{i}{2}\sigma (\sqrt{h}f,\sqrt{h}g)}{W}_{\sqrt{h}f+\sqrt{h}g}={\gamma }_h(W_f^h){\gamma }_h(W_g^h)$$

$${\gamma }_h(AB)={\gamma }_h(A){\gamma }_h(B)$$

$$\sum _{j,k=1}^n{a}_j{\overline{a}}_k{e}^{-\frac{i}{2}{\sigma }_h(f_j,f_k)}\phi (f_j-f_k)\ge 0$$

$${\phi }_h(f)=\phi (\frac{1}{\sqrt{h}}f)$$

$$\phi \in {\mathcal{F}}_h\iff {\phi }_h\in \mathcal{F}$$

$$\sum _{j,k=1}^n{a}_j{\overline{a}}_k{e}^{-\frac{i}{2}\sigma (\sqrt{h}{f}_j,\sqrt{h}{f}_k)}\phi (f_j-f_k)=\sum _{j,k=1}^n{a}_j{\overline{a}}_k{e}^{-\frac{i}{2}\sigma ({\tilde{f}}_j,{\tilde{f}}_k)}{\phi }_h({\tilde{f}}_j-{\tilde{f}}_k)$$

$${\omega }_h=\omega \circ {\tau }_h,{\tau }_h(W_f)=W_{(1/\sqrt{h})f}$$

$$S(f,g)=⟨f,Ag⟩$$

$${\phi }_S(f)=e^{-\frac{1}{4}S(f,f)}$$

$$|\sigma (f,g){|}^2\le S(f,f)S(g,g)$$

$$A\ge \text{openone}$$

$$\omega (W_f)={\phi }_S(f)$$

$$T_1=\frac{1}{\sqrt{2}}(A+\text{openone}{)}^{1/2},T_2=\frac{1}{\sqrt{2}}(A-\text{openone}{)}^{1/2}$$

$${\pi }_{\omega }(W_f)={\pi }_0(W_{T_{1}f})\otimes {\pi }_0(W_{T_{2}f})$$

$${\tilde{\pi }}_{\omega }(W_f)={\pi }_0(W_{T_{2}f})\otimes {\pi }_0(W_{T_{1}f})$$

$$D(a_{\omega }(f))=D(a_{\omega }^{\ast }(f))=D({\Phi }_{\omega }(f))\cap D({\Phi }_{\omega }(if))$$

$$a_{\omega }(f)=\frac{1}{\sqrt{2}}\left({\Phi }_{\omega }(f)+i{\Phi }_{\omega }(if)\right),a_{\omega }^{\ast }=\frac{1}{\sqrt{2}}\left({\Phi }_{\omega }(f)-i{\Phi }_{\omega }(if)\right)$$

Full text

Weyl algebra, quasi - free states, variable Planck constant

I Introduction

One can observe recently mounting interest in the fundamental question of the possible variability of the Planck’s constant $\hbar$ Dirac ; Uzan1 ; Uzan2 ; Webb ; Webb-sp ; Mangano ; Hutchin . As it sounds paradoxical, questioning of the absolutness of the $\hbar$ as a fundamental constant entity and considering it as ”material” coefficient Hutchin characterizing locally the quantumness of the fabric of space-time, is of principal interest. Dirac’s suggestion that the fundamental constants can depend on the epoch of the Universe Dirac inspired many authors. As already existing works indicate, it touches not only the questions concerning models of the Universe including with dark energy and matter, its beginings, but also experiments performed locally at human scales. The topic is around for a long time, formerly scattered in isolated research, it has grown into a more definite flow of works as theoretical as experimental. It is worth noting that some experiments verifying variability of the Planck’s constant are already run for decades Hutchin . As spatial as temporal Mangano variations of $\hbar$ are taken into consideration. Possible influences of such variations on macroscopic world are being considered Yang . On the other hand, a formal variability of the $\hbar$ has always been accepted in the delicate context of the so called classical limit for given quantum system, or commutative limit of canonical commutation relations within the canonical quantization scheme. An example of strict formalism, where the Planck’s constant is related to a running parameter controlling the non-commutativity, is the formalism of the (strict) deformation quantization (see e.g. Rieffel ; Bordemann ; Landsman-jmp ; Landsman-book ), where classical limit is literarily performed as the vanishing $\hbar$ limit.

Recently, M.A. de Gosson Gosson considered quantum mechanical consequences of possible changes of Planck’s constant. Using the Wigner formalism this author shows that the purity of the state is extremely sensitive to such changes. In our article we study the similar problems, but in the case of quantum systems with infinite number of degrees of freedom, such as quantum fields or quantum statistical systems. In the algebraic setting, possible changes of Planck’s constant which influence commutation relations of basic objects, modify the structure of the algebra of canonical commutation relations (CCR). In particular, in the Weyl form of CCR, the product of elements of the algebra is defined in terms of symplectic form $\sigma$ on the one - particle space $\mathbb{H}$, which manifestly depends on the Planck’s constant. In the algebraic formulation of quantum theory, the states are identified with positive and normalized functionals $\omega$ on the corresponding. algebra of CCR, so the problem of sensitivity of quantum states to the changes of Planck’s constant can be formulated as follows. Suppose that $\omega$ is the state on the CCR algebra given by the Planck’s constant fixed at its ”standard” value. Consider now the same functional $\omega$ but as a state on the algebra with changed Planck’s constant i.e. with changed multiplication law. In this context, the natural questions arise:

(i) under which condition this is possible?

(ii) what are the properties of $\omega$ considered as a state on the new algebra?

In the present paper we consider these questions in the case of quasi - free states. To simplify the analysis we fix standard Planck’s constant $\hbar=1$ and possible changes of Planck’s constant are measured by dimensionless scaling parameter $h$. The starting point is the CCR algebra defined by symplectic form $\sigma$ with $h=1$, denoted by $\mathfrak{W}(\mathbb{H},\sigma)$. The algebras corresponding to changed Planck’s constant are defined in terms of rescaled symplectic forms $\sigma_{h}$ and are denoted by $\mathfrak{W}(\mathbb{H},\sigma_{h})$. We show that given state $\omega$ on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$ can be also considered as a state on $\mathfrak{W}(\mathbb{H},\sigma_{h})$ if a properly defined rescaled functional $\omega_{h}$ is a state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$. It turns out that this condition gives a bound on the admissible values of the scaling parameter $h$. On the other hand, the rescaled state $\omega_{h}$ on $\mathfrak{W}(\mathbb{H},\sigma)$ can be treated as isomorphic image of the state $\omega$ on the algebra $\mathfrak{W}(\mathbb{H},\sigma_{h})$, under the natural $\ast$ - isomorphism between the algebras $\mathfrak{W}(\mathbb{H},\sigma)$ and $\mathfrak{W}(\mathbb{H},\sigma_{h})$.

The study of the properties of these rescaled states is the main purpose of the present work. The first result shows that for any quasi-free state $\omega$ defined by some operator $A\geq\openone$, the rescaled functional $\omega_{h}$ is a quasi - free state on $\mathfrak{W}(\mathbb{H},\sigma)$, provided the scaling parameter $h$ is not larger that the bottom of the spectrum of $A$. In the case of Fock state $\omega_{0}$ where $A=\openone$, $0<h<1$ and all rescaled states $\omega_{0,h}$ are non - Fock states with non - zero expectation value of the one - particle number operator. On the other hand, the total number operator does not exists, since the states $\omega_{0,h}$ are not quasi - equivalent to the Fock state $\omega_{0}$. The states $\omega_{0,h}$ have another interesting property. These states are extreme points of the convex set of all universally invariant states of the algebra $\mathfrak{W}(\mathbb{H},\sigma)$ (see Sect. IIIC). So any universally invariant state can be interpreted as a convex combination of Fock states with different values of Planck’s constant. Next we study the properties of equilibrium states. We adopt the general definition in terms of Kubo - Martin - Schwinger condition. Again the changes of Planck’s constant alter the properties of states, but as we show the state $\omega$ being KMS - state at the inverse temperature $\beta$, after rescaling remains $\beta$ - KMS state but with respect to the dynamics altered in a quite complicated way. The new hamiltonian is affiliated to the algebra generated by the hamiltonian defining the starting dynamics of the system. When the scaling parameter $h$ is beyond the admissible set of values, in particular $h$ is larger then the bottom of the spectrum of $A$, the quasi - free states $\omega_{h}$ have remarkable properties. It turns out that $\omega_{h}$ are positive - definite only after restriction to some subalgebra of the Weyl algebra $\mathfrak{W}(\mathbb{H},\sigma)$. On the other hand, such restricted states can be extended to some non - regular states $\widetilde{\omega}_{h}$ on the whole Weyl algebra. It is worth to stress that for some physically interesting systems (especially in connection with quantum gauge theories) the lack of regularity of states is not a mathematical pathology, but has interesting theoretical consequences (for detailed discussion see Strocchi ). When the scaling parameter achieves its maximal value $h_{\ast}=||A||_{\mathrm{op}}$, the corresponding non - regular states are all equal to the unique trace - state $\omega_{\infty}$ i.e. the state satisfying $\omega_{\infty}(AB)=\omega_{\infty}(BA)$ for all $A,B\in\mathfrak{W}(\mathbb{H},\sigma)$. Such a state, formally can be interpreted as the equilibrium state at infinite ($\beta=0$) temperature.

II Algebra of commutation relation and states

II.1 Weyl algebra of canonical commutation relation

A Weyl algebra of canonical commutation relations usually is constructed in terms of exponentials of canonical field operators $\Phi(f)$, with $f$ belonging to real vector space $\mathbb{H}$ endowed with symplectic form $\sigma(f,g)$, which encodes CCR relations (we put $\hbar=1$)

[TABLE]

If the operators $\Phi(f)$ are self - adjoint, we define

[TABLE]

and obtain the relations

[TABLE]

Generally, we define an abstract CCR algebra in the Weyl form MSTV as follows. Let $\mathbb{H}$ be a real linear space and $\sigma$ a symplectic form on $\mathbb{H}$. We consider also symplectic forms $\sigma_{h},\;h\in\mathbb{R},\;h>0$ as

[TABLE]

Let $W_{f}$ be the function on $\mathbb{H}$ given by

[TABLE]

Let $\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})$ be the $\ast$ - algebra generated by $\{W_{f}\,:\,f\in\mathbb{H}\}$. The elements $W_{f}$ satisfy

[TABLE]

and

[TABLE]

On the algebra $\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})$ we define the norm

[TABLE]

and let $\overline{\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})}^{||\cdot||_{1}}$ be the completion of $\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})$ with respect to this norm. So defined algebra is $\ast$ - Banach algebra. To obtain C*∗* - algebra of CCR one introduces minimal regular norm

[TABLE]

where $\pi$ are $\ast$ - representations of $\overline{\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})}^{||\cdot||_{1}}$. The completion of $\overline{\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})}^{||\cdot||_{1}}$ with respect to this norm is C*∗* - algebra of CCR denoted by $\mathfrak{W}(\mathbb{H},\sigma_{h})$.

Theorem II.1

For any $h>0$ there exists a unique $\ast$ - isomorphisms $\gamma_{h}$ from $\mathfrak{W}(\mathbb{H},\sigma_{h})$ onto $\mathfrak{W}(\mathbb{H},\sigma)$.

Proof: We follow here the arguments from HR . Let us denote the Weyl elements generating the algebra $\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})$ by $W_{f}^{h}$. Define

[TABLE]

Linear extension of Eq. (II.8) gives a bijection $\gamma_{h}$ from $\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})$ onto $\mathfrak{W}_{0}(\mathbb{H},\sigma)$. Since

[TABLE]

we have

[TABLE]

which by linear extension leads to the relation

[TABLE]

Since

[TABLE]

it follows that for all $A,B\in\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})$

[TABLE]

and $\gamma_{h}$ is $\ast$ - isomorphisms from $\mathfrak{W}_{0}(\mathbb{H},\sigma_{h})$ onto $\mathfrak{W}_{0}(\mathbb{H},\sigma)$, which can be extended by continuity.

II.2 States on $\mathfrak{W}(\mathbb{H},\sigma_{h})$

A state on $\mathfrak{W}(\mathbb{H},\sigma_{h})$ is normalized and positive linear functional.

Definition II.1

The function $\varphi\,:\,\mathbb{H}\to\mathbb{C}$ is $\sigma_{h}$ - positive if for all $n\in\mathbb{N}$ and $a_{k}\in\mathbb{C},\;f_{k}\in\mathbb{H};\>k=1,\ldots,n$

[TABLE]

We say that $\varphi$ is $\sigma$ - positive if it is $\sigma_{h}$ - positive for $h=1$. The set of all $\sigma_{h}$ - positive functions on $\mathbb{H}$ will be denoted by ${\mathcal{F}}_{h}$. We write ${\mathcal{F}}_{1}={\mathcal{F}}$. Define

[TABLE]

Theorem II.2

[TABLE]

Proof: If $\varphi\in{\mathcal{F}}_{h}$, then condition (II.9) is satisfied. Now the lefthand side of (II.9) can be written in the following way

[TABLE]

where $\tilde{f}_{j}=\sqrt{h}f_{j}$. It means that the function $\varphi_{h}\in{\mathcal{F}}$. Similarly we show that if $\varphi_{h}$ is $\sigma$ - positive, then $\varphi$ is $\sigma_{h}$ - positive. The following result is standard:

Theorem II.3

The functional $\omega$ is a state on $\mathfrak{W}(\mathbb{H},\sigma_{h})$ if and only if the generating function of $\omega$ i.e the function $\varphi_{\omega}(f)=\omega(W_{f}),\;f\in\mathbb{H}$, belongs to ${\mathcal{F}}_{h}$.

Fix some value of the parameter $h$ and consider the state $\omega$ on CCR algebra $\mathfrak{W}(\mathbb{H},\sigma_{h})$. $\omega$ is a linear functional on $\ast$ - algebra generated by elements $W_{f},\;f\in\mathbb{H}$. By definition this functional is positive - definite, when the product of elements of the algebra is defined in terms of symplectic form $\sigma_{h}$, so $\omega$ can be extended to positive - definite functional on C*∗* - algebra $\mathfrak{W}(\mathbb{H},\sigma_{h})$. The corresponding generating function $\varphi_{\omega}$ is $\sigma_{h}$ - positive. Notice that if $\tilde{h}\neq h$, the same functional can be considered on the algebra $\mathfrak{W}_{0}(\mathbb{H},\sigma_{\tilde{h}})$ where the multiplication is differently defined. But it can happen that $\omega$ is still positive - definite, so in this case the state $\omega$ can be extended to C*∗* - algebra $\mathfrak{W}(\mathbb{H},\sigma_{\tilde{h}})$. In particular we have

Theorem II.4

Let $\omega$ be a state on CCR algebra $\mathfrak{W}(\mathbb{H},\sigma)$. The functional $\omega$ can also be considered as a state on the algebra $\mathfrak{W}(\mathbb{H},\sigma_{h})$ if the rescaled functional $\omega_{h}$ given by

[TABLE]

is a state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$.

Remark II.1

Notice that $\tau_{h}=\gamma_{h}^{-1}$ and rescaled state $\omega_{h}$ is the isomorphic image of the functional $\omega$ considered as a state on the algebra $\mathfrak{W}(\mathbb{H},\sigma_{h})$.

III Quasi - free states

III.1 General definition

In the following we consider the case when $\mathbb{H}$ is a complex, infinite - dimensional Hilbert space with a scalar product $\langle\,{\cdot},\,{\cdot}\,\rangle$ and $\sigma(f,g)=\mathrm{Im}\,\langle\,{f},\,{g}\,\rangle$. Let $S(f,g)$ be a positive sesqulinear form on $\mathbb{H}$. Assume also that $S(f,g)$ is bounded. Then there exists a bounded linear operator $A$, such that

[TABLE]

The function $\varphi_{S}$ defined by

[TABLE]

is a generating function of some state $\omega$ on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$, if it is $\sigma$ - positive. One can show (see e.g. Petz ) that $\sigma$ - positivity of (III.2) is equivalent to the following condition

[TABLE]

In the case when the form $S(f,g)$ is defined by bounded operator $A$, the condition (III.3) is satisfied if and only if

[TABLE]

So if (III.4) is satisfied, the functional

[TABLE]

extended by linearity and continuity to the whole $\mathfrak{W}(\mathbb{H})$ is so called gauge - invariant quasi - free state on CCR algebra $\mathfrak{W}(\mathbb{H},\sigma)$.

The explicit construction of GNS representation $(\pi_{\omega},{\mathcal{H}}_{\omega},\Omega_{\omega})$ of the Weyl algebra $\mathfrak{W}(\mathbb{H},\sigma)$ defined by quasi - free state (III.5), can be given as follows MRT : Let ${\mathcal{H}}=\Gamma(\mathbb{H})$ be the bosonic Fock space over $\mathbb{H}$. Define

[TABLE]

GNS representation induced by $\omega$ is unitary equivalent to the representation defined on ${\mathcal{H}}_{\omega}={\mathcal{H}}\otimes{\mathcal{H}}$ and

[TABLE]

where $\pi_{0}$ is the Fock representation on ${\mathcal{H}}$. One can check that $\Omega_{\omega}=\Omega_{0}\otimes\Omega_{0}$ (where $\Omega_{0}$ is the vacuum vector in ${\mathcal{H}}$) is a cyclic vector for $\pi_{\omega}$. The representation $\pi_{\omega}$ is reducible, in fact $\tilde{\pi}_{\omega}$ given by

[TABLE]

commutes with $\pi_{\omega}$ and has $\Omega_{\omega}$ as a cyclic vector, too. So $\Omega_{\omega}$ is cyclic and separating for the von Neumann algebra $\mathfrak{M}_{\omega}$ generated by operators $\pi_{\omega}(W_{f})$.

The quasi - free state $\omega$ is regular, or even analytic i.e. the mapping $t\to\omega(W_{tf})$ for all $f\in\mathbb{H}$ is analytic in an open neighborhood of the origin. For each $f\in\mathbb{H}$, denote by $\Phi_{\omega}(f)$ the infinitesimal generator of the unitary group $t\to\pi_{\omega}(W_{tf})$. Define also the annihilation and creation operators, by

[TABLE]

and

[TABLE]

The operators $a_{\omega}(f),\;a_{\omega}^{\ast}(f)$ are densely defined, closed and $a_{\omega}(f)^{\ast}=a_{\omega}^{\ast}(f)$. Let us introduce the self - adjoint operator

[TABLE]

We take $N_{f}$ as a number operator for the one particle state $f\in\mathbb{H}$. One can check that

[TABLE]

The definition of total number operator is much more involved (see e.g. BR ). One can proceed as follows. The finite - dimensional subspaces $F\subseteq\mathbb{H}$ form a directed set when ordered by inclusion. If $\{f_{j}\}$ is an orthonormal basis for $F$, define

[TABLE]

and

[TABLE]

The set $\{n_{\omega,F}\}$ forms monotonically increasing net of positive, closed quadratic forms. Their limit will be also positive, closed quadratic form on ${\mathcal{H}}_{\omega}$. If this form is densely defined, it determines a unique self - adjoint total number operator $N_{\omega}$ on the Hilbert space ${\mathcal{H}}_{\omega}$ and one can show CMR that $N_{\omega}$ exists when the state is quasi - equivalent to the Fock state $\omega_{0}$ i.e. quasi - free state with $A=\openone$. More precisely, it means that the GNS representation $\pi_{\omega}$ is unitary equivalent to direct sum of copies of the Fock representation.

III.2 Rescaled quasi - free state $\omega_{h}$

Now we would like to study the properties of $\omega$ considered as functional defined on CCR algebra $\mathfrak{W}(\mathbb{H},\sigma_{h})$. To this end we consider the properties of rescaled state $\omega_{h}$. In this case the state $\omega_{h}$ is defined by the function

[TABLE]

which belongs to ${\mathcal{F}}$ if

[TABLE]

Obviously this condition is satisfied if $h\in(0,\;1)$. More generally,

[TABLE]

Theorem III.1

The quasi - free state $\omega$ defined by the operator $A\geq\openone$ can be considered as a quasi - free state on all algebras $\mathfrak{W}(\mathbb{H},\sigma_{h})$, for $h\in(0,\,h_{\mathrm{max}}]$, where $h_{\mathrm{max}}=\inf\,\mathrm{spec}(A)$. In particular, for all such values of the parameter $h$, the rescaled state $\omega_{h}$ is a quasi - free state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$.

III.3 Rescaled Fock state

The simplest quasi - free state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$ is the Fock state $\omega_{0}$ given by $A=\openone$. This state is pure and it means that GNS representation corresponding to $\omega_{0}$ is irreducible (see e.e. Petz ). Now we consider the rescaled functional $\omega_{0,h}$. This functional will be positive - definite for all $h\in(0,1)$ and can be written as

[TABLE]

so

[TABLE]

and we have

Theorem III.2

The rescaled Fock state $\omega_{0,h}$ on the CCR algebra $\mathfrak{W}(\mathbb{H},\sigma)$ is the quasi - free (non - Fock) state for all $0<h<1$.

Notice that in the state $\omega_{0,h}$, the number operator $N_{f}$ has non - zero expectation value

[TABLE]

but the total number operator does not exists, since the following theorem is true:

Theorem III.3

The state $\omega_{0,h}$ is not quasi - equivalent to the Fock state $\omega_{0}$.

Proof: The proof is based on the standard result: quasi - free state defined by operator $A\geq\openone$ is quasi - equivalent to the Fock state if and only if $A-\openone$ has finite trace (see e.g. C ). In the case of the state $\omega_{0,h}$ we have

[TABLE]

which in the case of infinite dimensional space $\mathbb{H}$ is not trace class.

It is worth to notice that re scaled Fock states $\omega_{0,h}$ are in fact the extreme points of the convex set of all universally invariant states Segal . To define this notion, let

[TABLE]

for any unitary operator on the Hilbert space $\mathbb{H}$. The state $\omega$ on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$ is universally invariant if for all unitary $U$

[TABLE]

It can be shown that any regular state $\omega$ on the Weyl algebra $\mathfrak{W}(\mathbb{H},\sigma)$ which is universally invariant has the form

[TABLE]

where $m$ is a probability measure on $[0,1)$ and the state $\omega_{c}$ is given by

[TABLE]

so

[TABLE]

IV Equilibrium states

IV.1 KMS state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$

We start with the general definition of the equilibrium state by the Kubo - Martin - Schwinger (KMS) condition BR . Let $\mathfrak{A}$ be the C*∗* - algebra of observables and $\{\alpha_{t}\}_{t\in\mathbb{R}}$ be the one - parameter group of $\ast$ - atomorphisms of $\mathfrak{A}$ describing time evolution. The state $\omega$ is $(\alpha_{t},\,\beta)$ - KMS state at the inverse temperature $\beta$ if for any pair $A,\,B\in\mathfrak{A}$, there exists a complex function $f_{A,B}(z)$ which is analytic on the strip

[TABLE]

and bounded and continuous on $\overline{{\mathcal{T}}_{\beta}}$, such that

[TABLE]

for all $t\in\mathbb{R}$.

Now we apply this definition to he case of quasi - free state on the CCR algebra $\mathfrak{W}(\mathbb{H},\,\sigma)$ defined by the operator $A\geq\openone$. Let $T_{t}$ be a group of unitary operators on $\mathbb{H}$ with self - adjoint generator $\mathbf{h}$. Assume that the form $S(f,g)=\langle\,{f},\,{Ag}\,\rangle$ is $T_{t}$ - invariant and define

[TABLE]

For quasi - free states, two - point correlation functions

[TABLE]

have the form

[TABLE]

where

[TABLE]

Now $\omega$ is $(\alpha_{t},\,\beta)$ -KMS with state if for every pair $f,g\in\mathbb{H}$ there exists a function $\Phi(f,g\,;\,z)$, analytic on the strip $\mathcal{T}_{\beta}$ and continuous on the boundary, such that

[TABLE]

It can be proved GJO that above condition implies that $\mathbf{h}\geq\epsilon\openone$ for $\epsilon>0$ and

[TABLE]

Observe that

[TABLE]

where

[TABLE]

Notice also that

[TABLE]

Let us define positive operator $\Delta=e^{\mathbf{h}}$ on the canonical domain $D$. Then

[TABLE]

and

[TABLE]

Observe that

[TABLE]

so the operators

[TABLE]

are bounded. Obviously operators $(A+\openone)\Delta^{iz},\;z\in\overline{\mathcal{T}_{\beta}}$ are bounded, so for all pairs of vectors $f,\,g\in\mathbb{H}$ we obtain the explicit formula for the analytic function $\Phi(f,g\,;\,z)$:

[TABLE]

IV.2 Rescaled KMS state $\omega_{h}$

In this subsection we study the properties of rescaled KMS state $\omega_{h}$. It is natural to assume that one - particle hamiltonian $\mathbf{h}$ is unbounded, so

[TABLE]

and scaling parameter $h\in(0,1)$. Now the state $\omega_{h}$ is defined by the operator

[TABLE]

which still satisfies $A_{h}\geq\openone$. In the analogy to the case $h=1$, we define

[TABLE]

or more precisely

[TABLE]

on the canonical domain $D^{h}$, where

[TABLE]

and $dP(\lambda)$ is the spectral measure of the operator $\Delta$. The function (IV.10) satisfies

[TABLE]

We define also the unitary group

[TABLE]

By $\mathbf{h}_{h}$ we denote the generator of this group. The operator $\mathbf{h}_{h}$ can be recovered from $\Delta_{h}$ by the formula

[TABLE]

Notice that the bottom of the spectrum of the operator $\mathbf{h}_{h}$ is given by the constant $\delta$ in the formula (IV.11). Define the function

[TABLE]

Using the same arguments as before, we show that the functions

[TABLE]

are analytic on the strip $\mathcal{T}_{\beta}$ and continuous on $\overline{\mathcal{T}_{\beta}}$. Moreover

[TABLE]

and

[TABLE]

Existence of the analytic functions (IV.15) satisfying boundary conditions (IV.16) and (IV.17) means the state $\omega_{h}$ is $(\alpha_{t}^{h},\,\beta)$ - KMS state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$, where

[TABLE]

So we arrive at the result

Theorem IV.1

Let $h\in(0,1)$ and $\omega$ is $(\alpha_{t},\,\beta)$ - KMS state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$. The rescaled state $\omega_{h}$ is $(\alpha_{t}^{h},\,\beta)$ - KMS state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$.

V Beyond the admissible values of the scaling parameter

As we have shown, the admissible values of the scaling parameter $h$ in the case of quasi - free state belong to the interval $(0,h_{\mathrm{max}})$, where $h_{\mathrm{max}}=\inf\,\mathrm{spec}(A)$. In this Section we study the properties of rescaled quasi - free states for the values of $h$ beyond the admissible interval. First we consider the state $\omega_{\infty}$ obtained as the limit

[TABLE]

Obviously

[TABLE]

This state is a unique trace - state on the algebra $\mathfrak{W}(\mathbb{H},\sigma)$ i.e. $\omega_{\infty}(AB)=\omega_{\infty}(BA)$ for all $A,B\in\mathfrak{W}(\mathbb{H},\sigma)$. However, the state $\omega_{\infty}$ is non - regular.

Consider now rescaled quasi free state for the scaling parameter

[TABLE]

It turns out that the quasi - free state $\omega$ on the CCR algebra constructed using symplectic form $\sigma$ can be positive functional with respect to the multiplication given by $\sigma_{h}$ only after restriction to some subalgebra of CCR algebra $\mathfrak{W}(\mathbb{H},\,\sigma)$. The restriction is constructed as follows. Let $E$ be the spectral measure of the operator $A$ which defines the state $\omega$. Let

[TABLE]

Take $\mathbb{H}_{h}=E_{h}\mathbb{H}$ and consider the restriction of the operator $A$ to this subspace

[TABLE]

Then

[TABLE]

so the quasi - free state $\omega^{(h)}$ defined by this operator, equals to the restriction of $\omega$ to the sub - algebra $\mathfrak{W}(\mathbb{H}_{h},\sigma)$. Moreover, it is the quasi - free state on the algebra $\mathfrak{W}(\mathbb{H}_{h},\,\sigma_{h})$. In the case of KMS state, $\inf\,\mathrm{spec}(A)=1$ and

[TABLE]

Let $P$ be the spectral measure of the operator $\Delta$ (see formula (IV.3)). Since

[TABLE]

where

[TABLE]

The restriction $\Delta^{(h)}$ of the operator $\Delta$ to the Hilbert space $\mathbb{H}_{h}$ is given by

[TABLE]

Since

[TABLE]

the operator $\Delta^{(h)}$ is bounded. Now we have

Theorem V.1

Let $h\in(1,h_{\ast})$. The restriction $\omega^{(h)}$ of KMS state $\omega$ to the subalgebra $\mathfrak{W}(\mathbb{H}_{h},\sigma)$ is KMS state with respect to the time evolution defined by unitary group $(\Delta^{(h)})^{it}$ with bounded generator.

Similarly, the restriction of the operator $\Delta_{h}$ defined by the formula (IV.8), can be written as

[TABLE]

Define also the unitary operators giving the time evolution

[TABLE]

and

[TABLE]

Using the similar method as in subsection IV. B, we can show the following result:

Theorem V.2

Let $h\in(1,\,h_{\ast})$ and $\omega^{(h)}$ is KMS state on the subalgebra $\mathfrak{W}(\mathbb{H}_{h},\,\sigma)$. Then the rescaled state $\omega^{(h)}_{h}$ is $(\widetilde{\alpha}_{t}^{(h)},\,\beta)$ - KMS state on the algebra $\mathfrak{W}(\mathbb{H}_{h},\,\sigma)$.

This result can be formulated in the following way. Let ${\mathcal{A}}^{(h)}=\mathfrak{W}(\mathbb{H}_{h},\sigma)$. Notice that for $h^{\prime}>h>1$

[TABLE]

Instead of state $\omega_{h}^{(h)}$ of subalgebras ${\mathcal{A}}^{(h)}$, consider its extension $\widetilde{\omega}_{h}$ to the whole algebra $\mathfrak{W}(\mathbb{H},\sigma)$, defined as

[TABLE]

The functional (V.9) is $\sigma$ - positive, so it defines a state on $\mathfrak{W}(\mathbb{H},\sigma)$, but this state is non - regular. In this way we obtain the family of non - regular states of Weyl algebra $\mathfrak{W}(\mathbb{H},\sigma)$, which are regular on subalgebras ${\mathcal{A}}^{(h)}$. Notice also that

[TABLE]

where $\omega_{\infty}$ is a trace - state defined by (V.1).

VI Conclusions

In the present work we have studied outcomes of the variability of the Planck’s constant in the quantum theory of infinite systems. The strict formulation of the problem within the algebraic quantum theory allowed to obtain new precise bounds on the changes of the $\hbar$ and interesting conclusions concerning the rescaled states. At first we have given the description of isomorphisms of $C^{*}$-algebras $\mathfrak{W}(\mathbb{H},\sigma_{h})$ for differing Planck’s constants and characterization of states in terms of the rescaling, what provided the tool to study the influence of rescaling on states. Then, the quasi-free states were discussed. It turns out that such a state $\omega$ defined by the operator $A\geq\openone$ remains quasi-free state on all algebras $\mathfrak{W}(\mathbb{H},\sigma_{h})$, for $h\in(0,\,h_{\mathrm{max}})$, with $h_{\mathrm{max}}=\inf\,\mathrm{spec}(A)$. The Fock-states after the rescaling $\omega_{0,h}$ do not become the new Fock-states, and even more, are not quasi-equivalent to the original Fock-state $\omega_{0}$. However, the rescaled Fock-states $\omega_{0,h}$ form the set of extreme points of the convex set of universally invariant states. The interesting situation we have for the rescaled KMS-states, namely, they remain of the KMS-class for the same value of temperature ($\beta$-parameter), but for nontrivially modified unitary evolution $\alpha^{h}_{t}$. In the Sec. V. we analyse what happens when states are reparametrized with values of parameter from the outside the allowed set $(0,h_{max})$, $h_{max}=\inf\,\mathrm{spec}(A)$. Such quasi-free states $\omega^{(h)}$ are non-regular, however can be interpreted as still quasi-free states on the appropriate restriction of the CCR-algebra. Similar conclusion for such ’unproper’ reparametrization is valid for the reparametrized KMS-states, provided the relevant evolution is appropriately adapted.

In conclusion, the algebraic description of infinite quantum systems yields some restrictions for the variability of the Planck’s constant and allows the comparison of states for different physical realities defined by its actual value.