Deformation quantization with minimal length

TL;DR

This paper develops a comprehensive non-formal deformation quantization framework incorporating a minimal length scale, defining a star-product, and constructing associated algebraic and operator structures.

Contribution

It introduces a new integral formula for the star-product, extends it to Hilbert spaces and distributions, and constructs a C*-algebra of observables with explicit operator representations.

Findings

A well-defined star-product with minimal length uncertainty

Construction of a C*-algebra of observables

Examples of maximally localized states and position eigenvectors

Abstract

We develop a complete theory of non-formal deformation quantization exhibiting a nonzero minimal uncertainty in position. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. Basic properties of the star-product are proved and the extension of the star-product to a certain Hilbert space and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, examples of position eigenvectors and states of maximal localization are given.

Equations

$$\Delta q\Delta p\ge \frac{\hslash }{2}\left(1+\beta (\Delta p{)}^2+\beta ⟨\hat{p}{⟩}^2\right),$$

$$\Delta q_0=\hslash \sqrt{\beta }.$$

$$[\hat{q},\hat{p}]=i\hslash \left(\hat{1}+\beta {\hat{p}}^2\right)$$

$$[x_{\mu },x_{\nu }]=i{\theta }_{\mu \nu }$$

$$[x_{\mu },x_{\nu }]=i{C}_{\mu \nu }^{\beta }{x}_{\beta },$$

$$[{\hat{q}}_i,{\hat{q}}_j]=i{\theta }_{ij},[{\hat{q}}_i,{\hat{p}}_j]=i\hslash {\delta }_{ij},[{\hat{p}}_i,{\hat{p}}_j]=i{\eta }_{ij},$$

$$\{q,p\}=1.$$

$$[f,g{]}_{\star }=i\hslash \{f,g\}+\cdots .$$

$$[q,p{]}_{\star }=i\hslash .$$

$$f\star g=f\exp \left(i\hslash (1-\lambda )\overleftarrow{{\partial }_q}\overrightarrow{{\partial }_p}-i\hslash \lambda \overleftarrow{{\partial }_p}\overrightarrow{{\partial }_q}\right)g,$$

$$[q,p{]}_{\star }=i\hslash (1+\beta p\star p).$$

$$f\star g$$

$$=\sum _{k=0}^{\infty }\frac{1}{k!}(i\hslash {)}^k\sum _{l=0}^k(\genfrac{}{}{0ex}{}{k}{l})(1-\lambda {)}^l(-\lambda {)}^{k-l}({\partial }_q^l{D}_p^{k-l}f)({\partial }_q^{k-l}{D}_p^{l}g),$$

$$q\star q=q^2,p\star p=p^2,$$

$$\bar{q}$$

$$\bar{p}$$

$$f\star g=f\exp \left(i\hslash (1-\lambda )\overleftarrow{{\partial }_{\bar{q}}}\overrightarrow{{\partial }_{\bar{p}}}-i\hslash \lambda \overleftarrow{{\partial }_{\bar{p}}}\overrightarrow{{\partial }_{\bar{q}}}\right)g,$$

$$\{q,p\}=1+\beta p^2.$$

$$x\oplus y=\frac{x+y}{1-\beta xy}.$$

$$x\oplus (y\oplus z)=\frac{x+\frac{y+z}{1-\beta yz}}{1-\beta x\frac{y+z}{1-\beta yz}}=\frac{x+y+z-\beta xyz}{1-\beta (xy+yz+zx)}=(x\oplus y)\oplus z,$$

$$\tan ({\theta }_1+{\theta }_2)=\frac{\tan {\theta }_1+\tan {\theta }_2}{1-\tan {\theta }_1\tan {\theta }_2}.$$

$$x\ominus y=x\oplus (-y).$$

$$\lambda \circ x=\frac{1}{\sqrt{\beta }}\tan \left(\lambda \arctan (\sqrt{\beta }x)\right).$$

$$(\lambda \circ x)\oplus (\lambda \circ y)$$

$$\displaystyle\displaystyle=\frac{1}{\sqrt{\beta}}\tan\left(\lambda\arctan(\sqrt{\beta}x)+\lambda\arctan(\sqrt{\beta}y)\right)=\frac{1}{\sqrt{\beta}}\tan\left(\lambda\arctan\bigl{(}\sqrt{\beta}(x\oplus y)\bigr{)}\right)$$

$$=\lambda \circ (x\oplus y),$$

$$({\lambda }_1\circ x)\oplus ({\lambda }_2\circ x)$$

$$=\frac{1}{\sqrt{\beta }}\tan \left(({\lambda }_1+{\lambda }_2)\arctan (\sqrt{\beta }x)\right)=({\lambda }_1+{\lambda }_2)\circ x,$$

$${\lambda }_1\circ ({\lambda }_2\circ x)$$

$$=\frac{1}{\sqrt{\beta }}\tan \left({\lambda }_1{\lambda }_2\arctan (\sqrt{\beta }x)\right)=({\lambda }_1{\lambda }_2)\circ x.$$

$$\tan \frac{\theta }{2}=\frac{\sqrt{1+{\tan }^2\theta }-1}{\tan \theta },\theta \in (-\frac{\pi }{2},\frac{\pi }{2})$$

$$\frac{1}{2}\circ x=\frac{\sqrt{1+\beta x^2}-1}{\beta x}.$$

$$x\oplus \frac{1}{\beta x}$$

Full text

Keywords and phrases: quantum mechanics, deformation quantization, star-product, minimal length, generalized uncertainty principle

1 Introduction

The idea that there should exists a minimal length scale was born in 1930s with the advent of quantum field theory. It was believed that it could help to remove divergences occurring in quantum theory. The interest in a minimal length scale was later on renewed when it was suggested that the gravity will play a significant role at short distances and effectively making it impossible to measure distances to a precision better than Planck’s length [1, 2, 3, 4]. Recently a minimal length arose also in many theories of quantum gravity and in string theory [5, 6, 7, 8, 9, 10, 11, 12, 13]. For a broad review of this topic and discussion of different implementations of a minimal length scale in quantum mechanics and quantum field theory we refer to [14].

A way of achieving minimal length scale is through modification of uncertainty relations for position and momentum. Such modified uncertainty relations are referred in the literature as a Generalized Uncertainty Principle and can be reproduced from modified commutation relations for position and momentum operators. In [15] authors proposed the following modified uncertainty relation for position and momentum in a one-dimensional case and nonrelativistic regime

[TABLE]

where $\displaystyle\beta$ is some positive constant. This is the simplest form of the uncertainty relation which leads to a nonzero minimal uncertainty $\displaystyle\Delta q_{0}$ in position. More general uncertainty relations were considered in [16, 17, 18, 19] exhibiting also a minimal uncertainty in momentum. The nonzero value of $\displaystyle\Delta q_{0}$ is a manifestation of the quantization of space. There exists a lower bound to the possible resolution with which we can measure distances. From (1) we can infer that the absolutely smallest uncertainty in position has the value

[TABLE]

We see that the constant $\displaystyle\beta$ describes the quantization of space where for $\displaystyle\beta=0$ we recover the usual uncertainty relation and nonquantized space. The uncertainty relation (1) can be derived from the following commutation relations

[TABLE]

for the operators of position and momentum. In literature can be found several works in which the value of the constant $\displaystyle\beta$ is assessed by means of different approaches. We mention papers [20, 21] in which authors consider generalized uncertainty principle in quantum gravity and calculate bounds for the parameter $\displaystyle\beta$ from well-known astronomical measurements as well as compute its value by using quantum corrections to the Newtonian potential.

It should be noted that in literature can be found approaches to quantum mechanics and field theory with different modified commutation relations than the one considered in this work. For instance in [22, 23, 24] authors developed quantum field theory on a noncommutative space-time characterized by the commutation relations of space-time variables either of the canonical type

[TABLE]

or the Lie-algebra type

[TABLE]

where $\displaystyle\theta_{\mu\nu}$ and $\displaystyle C^{\beta}_{\mu\nu}$ are real constant tensors. However, such commutation relations lead to different physical and mathematical consequences.

The purpose of the paper is to develop a deformation quantization approach to quantum mechanics which will exhibit a nonzero minimal uncertainty in position. A deformation quantization is a method of quantizing a classical Hamiltonian system by means of a suitable deformation of a Poisson algebra of the system. The deformation is performed with respect to some parameter which in the context of quantization is taken as the Planck’s constant $\displaystyle\hbar$. For a detailed description of deformation quantization theory we refer to [25, 26, 27, 28].

To our knowledge in the literature nothing can be found on the subject of deformation quantization with minimal length. Worth noting are papers [22, 24] cited earlier where star-products on a noncommutative space-time were introduced and papers [29, 30, 31] where authors consider quantum mechanics from a deformation quantization perspective with the following modified commutation relations of position and momentum operators

[TABLE]

where $\displaystyle\theta_{ij}$ and $\displaystyle\eta_{ij}$ are anti-symmetric real constant matrices. However, no one investigated deformation quantization approach to quantum mechanics with commutation relations (3), which is the topic of this paper.

The paper is organized as follows. In Section 2 we introduce a formal star-product which incorporates a minimal length scale. This formal star-product serves as a motivation for introducing, in next sections, a non-formal deformation quantization with minimal length. The term non-formal means that the star-product of two complex-valued functions results in a complex-valued function and not in a function with values in a ring $\displaystyle\mathbb{C}[\![\hbar]\!]$ of formal power series in $\displaystyle\hbar$ with coefficients in $\displaystyle\mathbb{C}$. First, in Section 3 is developed a generalized arithmetics on $\displaystyle\mathbb{R}$ which plays a fundamental role in the presented theory. Next, in Section 4 we introduce integral formulas for the star-product and a suitable space of functions on $\displaystyle\mathbb{R}^{2}$ for which the introduced integral formulas are well defined. We also prove basic properties of the star-product. Section 5 presents an extension of the star-product to a certain Hilbert subspace of $\displaystyle L^{2}(\mathbb{R}^{2},\mathrm{d}{l})$, and in Section 6 we extend the star-product to a suitable space of distributions. In Section 7 we define a $\displaystyle C^{*}$-algebra of observables and a space of states. We also give a characterization of states in terms of quasi-probabilistic distribution functions. Section 8 presents a construction of an operator representation in momentum space of the developed formalism of quantum mechanics. In Sections 9 and 10 is derived a form of quasi-probabilistic distribution functions describing position eigenvectors and states of maximal localization. We end the paper with some final remarks and conclusions given in Section 11.

2 Motivation

The starting point of our considerations is a classical system defined on a phase space $\displaystyle\mathbb{R}^{2}$ with a canonical Poisson bracket

[TABLE]

We can perform quantization of the above classical system by means of a deformation quantization methods. For this we have to introduce a star-product on the phase space $\displaystyle\mathbb{R}^{2}$, which will be a deformation, with respect to $\displaystyle\hbar$, of the ordinary point-wise product in the Poisson algebra $\displaystyle C^{\infty}(\mathbb{R}^{2})$, such that the star-product will remain associative but will loose commutativity. The first order term in the expansion with respect to $\displaystyle\hbar$ of the star-commutator should be equal

[TABLE]

Moreover, the following canonical commutation relation should hold

[TABLE]

The most natural family of star-products on $\displaystyle\mathbb{R}^{2}$ satisfying the above conditions is of the form

[TABLE]

where $\displaystyle 0\leq\lambda\leq 1$ describes different orderings of position and momentum operators in a corresponding operator representation.

We can now incorporate a minimal length scale into the above picture by deformation of the $\displaystyle\star$-product (10) with respect to $\displaystyle\beta$ to a new product satisfying, on account of (3), the following relation

[TABLE]

One of such deformations which has particularly simple form is given by

[TABLE]

where $\displaystyle D_{p}=(1+\beta p^{2})\partial_{p}$. Note, that

[TABLE]

where $\displaystyle q^{2}$ and $\displaystyle p^{2}$ are the usual products of $\displaystyle q$ and $\displaystyle p$ with themselves. Moreover, after performing the following noncanonical transformation of coordinates

[TABLE]

the $\displaystyle\star$-product (12) takes the form as in (10):

[TABLE]

and the star-commutator of $\displaystyle\bar{q}$, $\displaystyle\bar{p}$ is equal $\displaystyle i\hbar$.

In the limit $\displaystyle\hbar\to 0$ the $\displaystyle\star$-product (12) reduces to the ordinary point-wise product and $\displaystyle\frac{1}{i\hbar}[\,\cdot\,,\,\cdot\,]_{\star}$ reduces to a Poisson bracket which acts on observables of position and momentum through the following relation

[TABLE]

Thus in the limit $\displaystyle\hbar\to 0$ we received a classical system with a noncanonical Poisson bracket. Investigations of this classical system could reveal something about the parameter $\displaystyle\beta$. For example in [32] author argued that in order to preserve the correspondence principle the parameter $\displaystyle\beta$ has to depend on the mass of a particle which dynamics is described by the system.

In what follows we will investigate properties of the star-product (12) and develop a complete theory of quantum mechanics on phase space exhibiting a nonzero minimal uncertainty in position. As we will see the presented quantization procedure is connected with a modified arithmetics on $\displaystyle\mathbb{R}$, which we will present in the next section.

3 Generalized arithmetics on $\displaystyle\mathbb{R}$

In what follows we will develop a modified arithmetic on $\displaystyle\mathbb{R}$. It will be a particular case of a general approach considered in [33].

Let $\displaystyle\beta>0$. For $\displaystyle x,y\in\mathbb{R}$ such that $\displaystyle\beta xy\neq 1$ we define the generalized addition by

[TABLE]

The set $\displaystyle\{(x,y)\in\mathbb{R}^{2}\mid\beta xy=1\}$ on which the operation $\displaystyle\oplus$ is not well defined is of Lebesgue measure zero. The generalized addition $\displaystyle\oplus$ has the following properties:

1. (i)

$\displaystyle x\oplus(y\oplus z)=(x\oplus y)\oplus z$ (associativity), 2. (ii)

$\displaystyle x\oplus y=y\oplus x$ (commutativity), 3. (iii)

$\displaystyle x\oplus 0=0\oplus x=x$ (0 is the neutral element), 4. (iv)

$\displaystyle x\oplus(-x)=(-x)\oplus x=0$ ($\displaystyle-x$ is the inverse element to $\displaystyle x$), 5. (v)

$\displaystyle x\oplus y=\frac{1}{\sqrt{\beta}}\tan\left(\arctan(\sqrt{\beta}x)+\arctan(\sqrt{\beta}y)\right)$.

Indeed, we calculate that

[TABLE]

which proves property (i). Properties (ii), (iii), and (iv) are an immediate consequence of the definition. Property (v) follows from the formula for the tangent of the sum of angles:

[TABLE]

For every $\displaystyle y\in\mathbb{R}$ such that $\displaystyle y\neq 0$ the map $\displaystyle x\mapsto x\oplus y$ is a bijection of the set $\displaystyle\{x\in\mathbb{R}\mid x\neq\tfrac{1}{\beta y}\}$ onto the set $\displaystyle\{z\in\mathbb{R}\mid z\neq-\tfrac{1}{\beta y}\}$. We define the generalized subtraction by

[TABLE]

For a scalar $\displaystyle\lambda\in\mathbb{R}$ such that $\displaystyle-1\leq\lambda\leq 1$ and a vector $\displaystyle x\in\mathbb{R}$ we define the generalized multiplication by scalar:

[TABLE]

Note, that for a fixed $\displaystyle\lambda\in[-1,1]$ the map $\displaystyle x\mapsto\lambda\circ x$ is a bijection of $\displaystyle\mathbb{R}$ into $\displaystyle\mathbb{R}$ and for a fixed $\displaystyle x\in\mathbb{R}$ the map $\displaystyle\lambda\mapsto\lambda\circ x$ is a bijection of $\displaystyle[-1,1]$ into $\displaystyle\mathbb{R}$. The operation $\displaystyle\circ$ has the following properties:

1. (i)

$\displaystyle\lambda\circ(x\oplus y)=(\lambda\circ x)\oplus(\lambda\circ y)$, 2. (ii)

$\displaystyle(\lambda_{1}\circ x)\oplus(\lambda_{2}\circ x)=(\lambda_{1}+\lambda_{2})\circ x$, 3. (iii)

$\displaystyle\lambda_{1}\circ(\lambda_{2}\circ x)=(\lambda_{1}\lambda_{2})\circ x$.

Indeed, we calculate that

[TABLE]

which proves (i). Property (ii) follows from

[TABLE]

and property (iii) is a consequence of

[TABLE]

From the half-angle formula for the tangent function

[TABLE]

we get that in particular

[TABLE]

The definition of the generalized addition $\displaystyle\oplus$ can be continuously extended to the projectively extended space of real numbers $\displaystyle\overline{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$, so that it will be well defined for every pair of points $\displaystyle x,y\in\overline{\mathbb{R}}$, if we put

[TABLE]

Then, the space $\displaystyle\overline{\mathbb{R}}$ together with the extended operation $\displaystyle\oplus$ becomes a compact abelian topological group isomorphic to the circle group $\displaystyle\mathbb{T}^{1}$. For example if we define $\displaystyle\mathbb{T}^{1}$ as the quotient group $\displaystyle\mathbb{R}/2\pi\mathbb{Z}$ equipped with the usual addition of numbers modulo $\displaystyle 2\pi$, then the map

[TABLE]

is an isomorphism of $\displaystyle\mathbb{T}^{1}$ onto $\displaystyle\overline{\mathbb{R}}$.

For a function $\displaystyle f\colon\mathbb{R}\to\mathbb{C}$ we may define its generalized derivative by

[TABLE]

There holds

[TABLE]

Indeed, introducing $\displaystyle L_{p}(\eta)=p\oplus\eta=\frac{p+\eta}{1-\beta p\eta}$ we calculate that

[TABLE]

and

[TABLE]

The following identities hold

[TABLE]

Indeed, introducing the notation $\displaystyle h_{\lambda}(\eta)=\lambda\circ\eta$ we have that $\displaystyle h^{\prime}_{\lambda}(\eta)=\lambda\frac{1+\beta h_{\lambda}^{2}(\eta)}{1+\beta\eta^{2}}$ and

[TABLE]

For $\displaystyle q,p\in\mathbb{R}$ we will use the following notation

[TABLE]

We easily check that

[TABLE]

The operation $\displaystyle(q,p)$ can be treated as a generalized scalar product of $\displaystyle q$ and $\displaystyle p$.

On $\displaystyle\mathbb{R}$ we can introduce a measure which will be invariant with respect to generalized translations and which will reduce to the Lebesgue measure for $\displaystyle\beta=0$. The proper measure is equal

[TABLE]

One easily checks that indeed for $\displaystyle f\in L^{1}(\mathbb{R})$ and $\displaystyle\eta\in\mathbb{R}$

[TABLE]

On the phase space $\displaystyle\mathbb{R}^{2}$ we introduce the following measure

[TABLE]

where $\displaystyle x=(q,p)$. Let us denote by $\displaystyle\mathcal{L}$ the space of square integrable functions $\displaystyle f\in L^{2}(\mathbb{R}^{2},\mathrm{d}{l})$ such that for almost every $\displaystyle p\in\mathbb{R}$ the functions $\displaystyle q\mapsto f(q,p)$ extend to entire functions on $\displaystyle\mathbb{C}$ of exponential type $\displaystyle\leq\tfrac{\pi}{2\sqrt{\beta}}$, i.e. $\displaystyle f\in\mathcal{L}$ if and only if $\displaystyle f\in L^{2}(\mathbb{R}^{2},\mathrm{d}{l})$ and for almost every $\displaystyle p\in\mathbb{R}$ the function $\displaystyle q\mapsto f(q,p)$ extend to entire function on $\displaystyle\mathbb{C}$ satisfying for some constant $\displaystyle C(p)$ the following condition

[TABLE]

The space $\displaystyle\mathcal{L}$ is a Hilbert subspace of $\displaystyle L^{2}(\mathbb{R}^{2},\mathrm{d}{l})$.

For $\displaystyle f\in\mathcal{L}$ we define its generalized Fourier transform in position variable by

[TABLE]

If $\displaystyle f$ is not Lebesgue integrable, then the integral has to be considered as the improper integral

[TABLE]

This transform is a Hilbert space isomorphism of $\displaystyle\mathcal{L}$ onto $\displaystyle L^{2}(\mathbb{R}^{2},\mathrm{d}{\mu}\times\mathrm{d}{\mu})$ and its inverse is given by

[TABLE]

Indeed, if we denote by $\displaystyle\check{f}$ the usual Fourier transform of $\displaystyle f$ in position variable, then

[TABLE]

By Paley-Wiener theorem $\displaystyle\bar{p}\mapsto\check{f}(\bar{p},p)$ is square integrable with respect to the Lebesgue measure $\displaystyle\mathrm{d}{\bar{p}}$ and its support lies in \displaystyle\bigl{[}-\tfrac{\pi}{2\sqrt{\beta}},\tfrac{\pi}{2\sqrt{\beta}}\bigr{]}. Thus we can extend the integration with respect to $\displaystyle\bar{p}$ from $\displaystyle-\infty$ to $\displaystyle\infty$, which yields the inverse Fourier transform in position variable equal to $\displaystyle f(q,p)$ almost everywhere.

The generalized symplectic Fourier transform of a function $\displaystyle f\in\mathcal{L}$ will be denoted by $\displaystyle\mathcal{F}_{\beta}f$ and defined by the formula

[TABLE]

It is its own inverse: $\displaystyle\mathcal{F}_{\beta}^{-1}=\mathcal{F}_{\beta}$. The generalized symplectic Fourier transform has the following properties

[TABLE]

where

[TABLE]

is a generalized convolution.

4 Properties of the $\displaystyle\star$-product

Based on the formal star-product (12) we can get an integral formula for the star-product, which will describe a well defined non-formal star-product on a certain space of functions. In the rest of the paper we will develop a non-formal deformation quantization theory based on this integral formula for the star-product. The equation (12) will represent a formal expansion of this star-product in $\displaystyle\hbar$.

Theorem 1**.**

The star-product (12) can be written in the following integral forms

[TABLE]

Proof.

Taking the generalized symplectic Fourier transform of (12) and using (46b), (46c), (47) we find that

[TABLE]

Taking the generalized symplectic Fourier transform of (49) and performing the following change of variables under the integral sign

[TABLE]

gives (48a). By performing integration with respect to $\displaystyle q^{\prime\prime}$ and $\displaystyle q^{\prime}$ in (48a) we get (48b), and by expanding Fourier transforms in (48b) we receive (48c). ∎

Note, that by defining a twisted convolution $\displaystyle\diamond$ of functions $\displaystyle f$ and $\displaystyle g$ by the relation

[TABLE]

we get by virtue of (49) that

[TABLE]

Moreover, we can also define a twisted convolution with respect to position variables by the formula

[TABLE]

Then, we find from (48b) that

[TABLE]

Let us denote by $\displaystyle\widetilde{\mathcal{F}}(\mathbb{R}^{2})$ the space of complex-valued functions on $\displaystyle\mathbb{R}^{2}$ which extend to smooth functions on $\displaystyle\overline{\mathbb{R}}^{2}$. The space $\displaystyle\widetilde{\mathcal{F}}(\mathbb{R}^{2})$ is a Fréchet space with topology given by semi-norms

[TABLE]

Indeed, if $\displaystyle\Phi$ is the isomorphism (28) of the circle $\displaystyle\mathbb{T}^{1}$ onto $\displaystyle\overline{\mathbb{R}}$ and if we associate with every $\displaystyle f\in\widetilde{\mathcal{F}}(\mathbb{R}^{2})$ a function $\displaystyle F\in C^{\infty}(\mathbb{T}^{2})$ given by $\displaystyle F(p^{\prime},p)=f(\Phi(p^{\prime}),\Phi(p))$, then we can see that the semi-norms $\displaystyle\lVert\,\cdot\,\rVert_{n,m}$ can be written in a form

[TABLE]

Thus, the map $\displaystyle f\mapsto F$ is a topological isomorphism of the space $\displaystyle\widetilde{\mathcal{F}}(\mathbb{R}^{2})$ onto the space of smooth functions on the torus $\displaystyle\mathbb{T}^{2}=\mathbb{T}^{1}\times\mathbb{T}^{1}$ and endowed with a standard Fréchet topology of uniform convergence together with all derivatives.

We will denote by $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ the image of the space $\displaystyle\widetilde{\mathcal{F}}(\mathbb{R}^{2})$ with respect to the inverse generalized Fourier transform in position variable. The space $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ carries the Fréchet topology induced from $\displaystyle\widetilde{\mathcal{F}}(\mathbb{R}^{2})$.

Theorem 2**.**

If $\displaystyle f,g\in\mathcal{F}(\mathbb{R}^{2})$, then $\displaystyle f\star g\in\mathcal{F}(\mathbb{R}^{2})$ and $\displaystyle(f,g)\mapsto f\star g$ is a continuous bilinear operation on $\displaystyle\mathcal{F}(\mathbb{R}^{2})$.

Proof.

Using (33) and (54) we find that

[TABLE]

Hence, by induction on these formulas if $\displaystyle\tilde{f},\tilde{g}\in\widetilde{\mathcal{F}}(\mathbb{R}^{2})$ then also $\displaystyle\tilde{f}\odot\tilde{g}\in\widetilde{\mathcal{F}}(\mathbb{R}^{2})$. Thus $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ is closed with respect to the $\displaystyle\star$-product.

From (57) we get that

[TABLE]

and from (54)

[TABLE]

Therefore

[TABLE]

which shows that $\displaystyle(\tilde{f},\tilde{g})\mapsto\tilde{f}\odot\tilde{g}$ is jointly continuous for the topology of $\displaystyle\widetilde{\mathcal{F}}(\mathbb{R}^{2})$. Thus $\displaystyle(f,g)\mapsto f\star g$ is jointly continuous on $\displaystyle\mathcal{F}(\mathbb{R}^{2})\times\mathcal{F}(\mathbb{R}^{2})$. ∎

Note, that the generalized symplectic Fourier transform $\displaystyle\mathcal{F}_{\beta}$ is a topological isomorphism of $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ onto $\displaystyle\mathcal{F}(\mathbb{R}^{2})$. Thus, by (51) the space $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ is also closed under the twisted convolution $\displaystyle\diamond$.

Theorem 3**.**

The $\displaystyle\star$-product is associative, i.e.

[TABLE]

for $\displaystyle f,g,h\in\mathcal{F}(\mathbb{R}^{2})$.

Proof.

We have that

[TABLE]

Applying (51) yields the associativity of $\displaystyle\star$. ∎

Theorem 4**.**

For $\displaystyle f,g\in\mathcal{F}(\mathbb{R}^{2})$

[TABLE]

In particular, for $\displaystyle\lambda=\tfrac{1}{2}$

[TABLE]

Proof.

We have that

[TABLE]

Performing the following change of variables under the integral sign: $\displaystyle q^{\prime\prime}=-q^{\prime}$, $\displaystyle p^{\prime\prime}=-p^{\prime}$ we get (63). If $\displaystyle\lambda=\tfrac{1}{2}$, then

[TABLE]

which proves (64). ∎

Theorem 5**.**

For $\displaystyle f,g\in\mathcal{F}(\mathbb{R}^{2})$

[TABLE]

Proof.

These equalities follow immediately from (48c) and (33). ∎

For $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$ we define its conjugation $\displaystyle f^{*}$ by the formula

[TABLE]

where $\displaystyle\bar{f}$ denotes the complex-conjugation of $\displaystyle f$. For $\displaystyle\lambda=\tfrac{1}{2}$ it can be seen from (68) that the conjugation $\displaystyle*$ is the usual complex-conjugation. Formula (68) can be also written in the form

[TABLE]

From (69) we get that

[TABLE]

Thus, since $\displaystyle(p^{\prime},p)\mapsto(-p^{\prime},p\ominus(1-2\lambda)\circ p^{\prime})$ is smooth on $\displaystyle\overline{\mathbb{R}}^{2}$, $\displaystyle\widetilde{f^{*}}\in\widetilde{\mathcal{F}}(\mathbb{R}^{2})$ and hence $\displaystyle f^{*}\in\mathcal{F}(\mathbb{R}^{2})$. The conjugation $\displaystyle*$ has the following properties.

Theorem 6**.**

For $\displaystyle f,g\in\mathcal{F}(\mathbb{R}^{2})$

1. (i)

$\displaystyle f\mapsto f^{*}$* is anti-linear and continuous on $\displaystyle\mathcal{F}(\mathbb{R}^{2})$,* 2. (ii)

$\displaystyle(f\star g)^{*}=g^{*}\star f^{*}$, 3. (iii)

$\displaystyle(f^{*})^{*}=f$, 4. (iv)

$\displaystyle\int_{\mathbb{R}^{2}}f^{*}(q,p)\,\mathrm{d}{q}\,\mathrm{d}{\mu(p)}=\int_{\mathbb{R}^{2}}\bar{f}(q,p)\,\mathrm{d}{q}\,\mathrm{d}{\mu(p)}$, 5. (v)

$\displaystyle(\partial_{q}f)^{*}=\partial_{q}f^{*}$* and $\displaystyle(D_{p}f)^{*}=D_{p}f^{*}$.*

Proof.

(i) By virtue of (33) and (70)

[TABLE]

Hence

[TABLE]

and

[TABLE]

from which follows continuity of the conjugation $\displaystyle*$. Anti-linearity is an immediate consequence of the definition.

(ii) From (68) we find that

[TABLE]

With the help of (51), (52), and (74) we calculate that

[TABLE]

(iii) Using (74) we receive

[TABLE]

(iv) Again using (74) we find that

[TABLE]

(v) It follows directly from the property (46b) of the Fourier transform. ∎

From the above theorem it follows that the conjugation $\displaystyle*$ is an involution on the algebra $\displaystyle\mathcal{F}(\mathbb{R}^{2})$. As a consequence $\displaystyle*$ is an anti-automorphism on $\displaystyle\mathcal{F}(\mathbb{R}^{2})$.

5 Extension of the $\displaystyle\star$-product to the Hilbert space $\displaystyle\mathcal{L}$

In what follows we will extend the $\displaystyle\star$-product and involution $\displaystyle*$ to the Hilbert space $\displaystyle\mathcal{L}$ introduced at the end of Section 3. On the space $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ we define a scalar product by the following formula

[TABLE]

From (65) and (74) we get that

[TABLE]

thus $\displaystyle(\,\cdot\,,\,\cdot\,)$ is the usual scalar product from the Hilbert space $\displaystyle\mathcal{L}$. The norm corresponding to the scalar product $\displaystyle(\,\cdot\,,\,\cdot\,)$ will be denoted by $\displaystyle\lVert\,\cdot\,\rVert_{2}$. Immediately from (78) we get that

[TABLE]

from which it follows that the conjugation $\displaystyle*$ is continuous with respect to the $\displaystyle L^{2}$-norm. Moreover, the following inequality holds

[TABLE]

the consequent of which is continuity of the $\displaystyle\star$-product in the $\displaystyle L^{2}$-norm. The proof of (81) becomes straightforward if we make use of an operator representation introduced in Section 8.

Since $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ is dense in $\displaystyle\mathcal{L}$ and the conjugation $\displaystyle*$ and the $\displaystyle\star$-product are continuous in the $\displaystyle L^{2}$-norm, these operations can be uniquely extended to the whole space $\displaystyle\mathcal{L}$. Note, that $\displaystyle\mathcal{L}$ is a Banach algebra as well as a Hilbert algebra.

We will also introduce a trace functional

[TABLE]

for $\displaystyle f\in L^{1}(\mathbb{R}^{2},\mathrm{d}{l})$. The scalar product on $\displaystyle\mathcal{L}$ takes then the following form

[TABLE]

6 Extension of the $\displaystyle\star$-product to an algebra of distributions

In what follows we will extend the $\displaystyle\star$-product to a suitable space of distributions. The algebra $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ will play the role of the space of test functions. We will denote by $\displaystyle\mathcal{F}^{\prime}(\mathbb{R}^{2})$ the space of continuous linear functionals on $\displaystyle\mathcal{F}(\mathbb{R}^{2})$, i.e. distributions. The dual space $\displaystyle\mathcal{F}^{\prime}(\mathbb{R}^{2})$ is endowed with the strong dual topology, that of uniform convergence on bounded subsets of $\displaystyle\mathcal{F}(\mathbb{R}^{2})$. For $\displaystyle f\in\mathcal{F}^{\prime}(\mathbb{R}^{2})$ we will denote by $\displaystyle\langle{f,h}\rangle$ the value of the functional $\displaystyle f$ at $\displaystyle h\in\mathcal{F}(\mathbb{R}^{2})$. We will identify functions $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$ with the following functionals

[TABLE]

These functionals are continuous on $\displaystyle\mathcal{F}(\mathbb{R}^{2})$. Indeed, the integral is continuous on $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ since

[TABLE]

which together with the continuity of the $\displaystyle\star$-product implies the continuity of the functionals (84). Thus we may write $\displaystyle\mathcal{F}(\mathbb{R}^{2})\subset\mathcal{F}^{\prime}(\mathbb{R}^{2})$. The functionals (84), by virtue of (79), can be also written in the form

[TABLE]

where

[TABLE]

is a topological isomorphism of the vector space $\displaystyle\mathcal{F}(\mathbb{R}^{2})$. Note, that $\displaystyle S$ commutes with partial derivatives $\displaystyle\partial_{q}$, $\displaystyle D_{p}$ and that it reduces to the identity operator under the integral sign, i.e.

[TABLE]

Moreover, for $\displaystyle\lambda=\tfrac{1}{2}$ the operator $\displaystyle S$ is equal to the identity operator. If we explicitly denote dependence of the $\displaystyle\star$-product on $\displaystyle\lambda$ by writing $\displaystyle\star_{\lambda}$, then we get the following property of the operator $\displaystyle S$

[TABLE]

Formula (86) allows for identification of a broad class of functions with distributions. In particular, function identically equal to 1 can be identified with the following distribution

[TABLE]

The Fourier transform, partial differentiation, and ordinary multiplication by a function can be extended to the space of distributions $\displaystyle\mathcal{F}^{\prime}(\mathbb{R}^{2})$ in the following way

[TABLE]

where $\displaystyle f\in\mathcal{F}^{\prime}(\mathbb{R}^{2})$, $\displaystyle h\in\mathcal{F}(\mathbb{R}^{2})$, $\displaystyle\check{h}(q,p)=h(-q,-p)$, and $\displaystyle g$ is any function on $\displaystyle\mathbb{R}^{2}$ such that $\displaystyle gh\in\mathcal{F}(\mathbb{R}^{2})$ for every $\displaystyle h\in\mathcal{F}(\mathbb{R}^{2})$.

For $\displaystyle f\in\mathcal{F}^{\prime}(\mathbb{R}^{2})$ and $\displaystyle g\in\mathcal{F}(\mathbb{R}^{2})$ we define $\displaystyle f\star g\in\mathcal{F}^{\prime}(\mathbb{R}^{2})$ and $\displaystyle g\star f\in\mathcal{F}^{\prime}(\mathbb{R}^{2})$ by the formulas

[TABLE]

Note, that the maps $\displaystyle g\mapsto f\star g$ and $\displaystyle g\mapsto g\star f$ are continuous from $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ to $\displaystyle\mathcal{F}^{\prime}(\mathbb{R}^{2})$, since the maps $\displaystyle g\mapsto\langle{f,g\star h}\rangle$ and $\displaystyle g\mapsto\langle{f,h\star g}\rangle$ are continuous, uniformly for $\displaystyle h$ in a bounded subset of $\displaystyle\mathcal{F}(\mathbb{R}^{2})$.

Denote by $\displaystyle\mathcal{F}_{\star}(\mathbb{R}^{2})$ the following subspace of distributions:

[TABLE]

In particular, $\displaystyle\mathcal{F}(\mathbb{R}^{2})\subset\mathcal{F}_{\star}(\mathbb{R}^{2})$. For $\displaystyle f\in\mathcal{F}_{\star}(\mathbb{R}^{2})$ the maps $\displaystyle g\mapsto f\star g$ and $\displaystyle g\mapsto g\star f$ are continuous from $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ to $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ by the closed graph theorem. Thus, for $\displaystyle f,g\in\mathcal{F}_{\star}(\mathbb{R}^{2})$ we may define their $\displaystyle\star$-product by the formula

[TABLE]

The second equality in the above definition is indeed satisfied for every $\displaystyle h\in\mathcal{F}(\mathbb{R}^{2})$, which can be easily proved for $\displaystyle h=h_{1}\star h_{2}$ ($\displaystyle h_{1},h_{2}\in\mathcal{F}(\mathbb{R}^{2})$) and the general case follows from the fact that $\displaystyle\mathcal{F}(\mathbb{R}^{2})\star\mathcal{F}(\mathbb{R}^{2})$ is linearly dense in $\displaystyle\mathcal{F}(\mathbb{R}^{2})$. Straightforward calculations with the use of (92) and (94) verify that $\displaystyle f\star g\in\mathcal{F}_{\star}(\mathbb{R}^{2})$ and the associativity of the $\displaystyle\star$-product. Note, that $\displaystyle 1\in\mathcal{F}_{\star}(\mathbb{R}^{2})$ and $\displaystyle f\star 1=1\star f=f$ for every $\displaystyle f\in\mathcal{F}_{\star}(\mathbb{R}^{2})$.

The involution $\displaystyle*$ can be extended to the algebra $\displaystyle\mathcal{F}_{\star}(\mathbb{R}^{2})$ in a natural way:

[TABLE]

and $\displaystyle f\in\mathcal{F}_{\star}(\mathbb{R}^{2})$. In particular, $\displaystyle 1^{*}=1$. Thus, $\displaystyle\mathcal{F}_{\star}(\mathbb{R}^{2})$ is an involutive algebra with unity, being a natural extension of the algebra $\displaystyle\mathcal{F}(\mathbb{R}^{2})$.

Theorem 7**.**

If $\displaystyle\phi(p)$ is a smooth function on $\displaystyle\overline{\mathbb{R}}$, then $\displaystyle\phi\in\mathcal{F}_{\star}(\mathbb{R}^{2})$.

Proof.

For $\displaystyle g,h\in\mathcal{F}(\mathbb{R}^{2})$ we have

[TABLE]

Hence

[TABLE]

Since $\displaystyle\phi\star g$ is an inverse generalized Fourier transform in position variable of a smooth function on $\displaystyle\overline{\mathbb{R}}^{2}$, $\displaystyle\phi\star g\in\mathcal{F}(\mathbb{R}^{2})$. Similarly we can prove that $\displaystyle g\star\phi\in\mathcal{F}(\mathbb{R}^{2})$. Therefore $\displaystyle\phi\in\mathcal{F}_{\star}(\mathbb{R}^{2})$. ∎

Note, that $\displaystyle q$, $\displaystyle p$ and their natural powers are not elements of $\displaystyle\mathcal{F}^{\prime}(\mathbb{R}^{2})$, since the integral in (86) will not be well defined for every $\displaystyle h\in\mathcal{F}(\mathbb{R}^{2})$. However, the integral formula for the $\displaystyle\star$-product can still make sense for these and other functions which are not in $\displaystyle\mathcal{F}^{\prime}(\mathbb{R}^{2})$. In particular, for $\displaystyle f(q,p)=q^{n}\phi(p)$ where $\displaystyle n\in\mathbb{N}$ and $\displaystyle\phi$ is smooth on $\displaystyle\overline{\mathbb{R}}$ we can define

[TABLE]

for all $\displaystyle g\in\mathcal{F}(\mathbb{R}^{2})$ for which the integrals are convergent. Note, that the integral with respect to $\displaystyle q$, in general, will have to be an improper integral (42).

7 $\displaystyle C^{*}$-algebra of observables and states

On the space $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ we can introduce another norm according to the formula

[TABLE]

This is a $\displaystyle C^{*}$-norm, i.e. it satisfies

1. (i)

$\displaystyle\lVert f\star g\rVert\leq\lVert f\rVert\lVert g\rVert$, 2. (ii)

$\displaystyle\lVert f^{*}\rVert=\lVert f\rVert$, 3. (iii)

$\displaystyle\lVert f^{*}\star f\rVert=\lVert f\rVert^{2}$,

for $\displaystyle f,g\in\mathcal{F}(\mathbb{R}^{2})$. Indeed, this follows directly from the fact that $\displaystyle f\star{}$ is a bounded linear operator on the Hilbert space $\displaystyle\mathcal{L}$ defined on a dense domain $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ (the boundedness of $\displaystyle f\star{}$ can be seen from (81)). The norm $\displaystyle\lVert f\rVert$ is then defined as the operator norm of the operator $\displaystyle f\star{}$. Since $\displaystyle\lVert f\rVert$ is the smallest constant $\displaystyle C$ satisfying the inequality

[TABLE]

it is clear from (81) that $\displaystyle\lVert f\rVert\leq\lVert f\rVert_{2}$, and so convergence in $\displaystyle L^{2}$-norm implies convergence in the norm $\displaystyle\lVert\,\cdot\,\rVert$.

The space $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ is not complete with respect to the $\displaystyle C^{*}$-norm $\displaystyle\lVert\,\cdot\,\rVert$, thus $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ is only a pre-$\displaystyle C^{*}$-algebra. The completion of $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ to a $\displaystyle C^{*}$-algebra will be denoted by $\displaystyle\mathcal{A}(\mathbb{R}^{2})$. The algebra $\displaystyle\mathcal{A}(\mathbb{R}^{2})$ is a $\displaystyle C^{*}$-algebra of observables. With its help we can define states, in a standard way, as continuous positive linear functionals on $\displaystyle\mathcal{A}(\mathbb{R}^{2})$ normalized to unity, i.e. a continuous linear functional $\displaystyle\Lambda\colon\mathcal{A}(\mathbb{R}^{2})\to\mathbb{C}$ is a state if

1. (i)

$\displaystyle\lVert\Lambda\rVert=1$, 2. (ii)

$\displaystyle\Lambda(f^{*}\star f)\geq 0$ for every $\displaystyle f\in\mathcal{A}(\mathbb{R}^{2})$.

The set of all states is convex. Pure states are defined as extreme points of this set, i.e. as those states which cannot be written as convex linear combinations of some other states. In other words $\displaystyle\Lambda_{\text{pure}}$ is a pure state if and only if there do not exist two different states $\displaystyle\Lambda_{1}$ and $\displaystyle\Lambda_{2}$ such that $\displaystyle\Lambda_{\text{pure}}=p\Lambda_{1}+(1-p)\Lambda_{2}$ for some $\displaystyle p\in(0,1)$.

The expectation value of an observable $\displaystyle f\in\mathcal{A}(\mathbb{R}^{2})$ in a state $\displaystyle\Lambda$ is from definition equal

[TABLE]

If $\displaystyle f$ is self-adjoint, i.e. $\displaystyle f^{*}=f$, then $\displaystyle\langle{f}\rangle_{\Lambda}\in\mathbb{R}$.

The next two theorems provide characterization of states in terms of quasi-probabilistic distribution functions $\displaystyle\rho\in\mathcal{L}$. They follow directly from the operator representation introduced below.

Theorem 8**.**

If $\displaystyle\rho\in\mathcal{L}$ satisfies

1. (i)

$\displaystyle\rho^{*}=\rho$, 2. (ii)

$\displaystyle\int_{\mathbb{R}^{2}}\rho\,\mathrm{d}{l}=1$, 3. (iii)

$\displaystyle\int_{\mathbb{R}^{2}}f^{*}\star f\star\rho\,\mathrm{d}{l}\geq 0$* for every $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$,*

then the functional

[TABLE]

is a state. Vice verse, every state $\displaystyle\Lambda$ can be written in the form (102) for some $\displaystyle\rho\in\mathcal{L}$ and satisfying properties (i)–(iii). The representation (102) is unique.

Theorem 9**.**

A state $\displaystyle\Lambda_{\rho}$ is pure if and only if the corresponding function $\displaystyle\rho$ is idempotent, i.e.

[TABLE]

From (102) the expectation value of an observable $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$ in a state $\displaystyle\Lambda_{\rho}$ can be written in a form

[TABLE]

The above formula can be extended in a direct way to general observables $\displaystyle f\in\mathcal{F}_{\star}(\mathbb{R}^{2})$ provided that $\displaystyle f\star\rho$ is well defined. If $\displaystyle\rho\in\mathcal{F}(\mathbb{R}^{2})$ then the product $\displaystyle f\star\rho$ is a well defined function in $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ and we can treat $\displaystyle f\star{}$ as a densely defined operator on $\displaystyle\mathcal{L}$, which sometimes might be extended to a wider subspace of functions.

8 Operator representation

By virtue of Gelfand-Naimark theorem the $\displaystyle C^{*}$-algebra of observables $\displaystyle\mathcal{A}(\mathbb{R}^{2})$ can be represented as an algebra of bounded linear operators on a certain Hilbert space $\displaystyle\mathcal{H}$. In what follows we will present an explicit construction of this representation for $\displaystyle\mathcal{H}=L^{2}(\mathbb{R},\mathrm{d}{\mu})$. The Hilbert space $\displaystyle\mathcal{H}$ will play the role of the space of states from a standard description of quantum mechanics. The constructed representation will result, in fact, in a momentum representation of a quantum system.

Let $\displaystyle\mathcal{H}=L^{2}(\mathbb{R},\mathrm{d}{\mu})$ be a Hilbert space of square integrable functions on $\displaystyle\mathbb{R}$ with a scalar product given by

[TABLE]

where $\displaystyle\mathrm{d}{\mu}$ is a measure given by (37). For a function $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$ we define an operator $\displaystyle\hat{f}$ acting in $\displaystyle\mathcal{H}$ by the formula

[TABLE]

where $\displaystyle\lambda$ is the same as in the definition of the $\displaystyle\star$-product.

Let us denote by $\displaystyle\mathsf{f}$ the integral kernel of the operator $\displaystyle\hat{f}$, i.e.

[TABLE]

The function $\displaystyle f(q,p)$ can be expressed by the integral kernel of the corresponding operator $\displaystyle\hat{f}$ in the following way

[TABLE]

Indeed, from the above equation the generalized Fourier transform in position variable of the function $\displaystyle f$ is expressed by the integral kernel $\displaystyle\mathsf{f}$ in the following way

[TABLE]

Then

[TABLE]

Performing the following change of variables

[TABLE]

the integral kernel $\displaystyle\mathsf{f}$ is expressed by the function $\displaystyle f$ in the following way

[TABLE]

Note, that the transformation $\displaystyle(a,b)\mapsto(a\ominus b,(1-\lambda)\circ a\oplus\lambda\circ b)$ is a smooth bijection of $\displaystyle\overline{\mathbb{R}}^{2}$ onto $\displaystyle\overline{\mathbb{R}}^{2}$, which inverse is also smooth. Thus, $\displaystyle f\mapsto\hat{f}$ is a linear isomorphism of $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ onto the space of integral operators whose integral kernels $\displaystyle\mathsf{f}\in\widetilde{\mathcal{F}}(\mathbb{R}^{2})$.

Theorem 10**.**

For $\displaystyle f,g\in\mathcal{F}(\mathbb{R}^{2})$

1. (i)

$\displaystyle\widehat{f\star g}=\hat{f}\hat{g}$, 2. (ii)

$\displaystyle\widehat{f^{*}}=\hat{f}^{\dagger}$, 3. (iii)

$\displaystyle\hat{f}$* is a trace class operator and $\displaystyle\operatorname{Tr}(\hat{f})=\operatorname{tr}(f)$,* 4. (iv)

the Hilbert-Schmidt scalar product of operators $\displaystyle\hat{f}$ and $\displaystyle\hat{g}$ is equal $\displaystyle(\hat{f},\hat{g})\equiv\operatorname{Tr}(\hat{f}^{\dagger}\hat{g})=(f,g)$, 5. (v)

$\displaystyle\lVert\hat{f}\rVert=\lVert f\rVert$.

Proof.

(i) We have that

[TABLE]

Using the translational invariance of the measure $\displaystyle\mu$ when integrating over $\displaystyle p^{\prime}$ we can replace $\displaystyle p^{\prime}$ by $\displaystyle p^{\prime}\ominus p^{\prime\prime}$ under the integral sign receiving

[TABLE]

(ii) From (70) and translational invariance of the measure $\displaystyle\mu$ we get

[TABLE]

(iii) Since the Hilbert space $\displaystyle L^{2}(\mathbb{R},\mathrm{d}{\mu})$ is unitarily equivalent with a Hilbert space of square integrable functions on a circle and the integral kernel $\displaystyle\mathsf{f}$ of the operator $\displaystyle\hat{f}$ is smooth on $\displaystyle\overline{\mathbb{R}}^{2}$, then by [34, Proposition 3.5, page 174] the operator $\displaystyle\hat{f}$ is of trace class with the trace given by the formula

[TABLE]

Using (112) we calculate that

[TABLE]

(iv) This property immediately follows from (i)–(iii).

(v) The operator norm of a bounded operator $\displaystyle\hat{f}$ can be expressed in terms of the Hilbert-Schmidt norm $\displaystyle\lVert\,\cdot\,\rVert_{2}$ according to the formula

[TABLE]

Using properties (i) and (iv) and the fact that $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ is dense in $\displaystyle\mathcal{L}$ we get the result. ∎

The above theorem states that the map $\displaystyle f\mapsto\hat{f}$ is a faithful $\displaystyle*$-representation of the algebra $\displaystyle\mathcal{F}(\mathbb{R}^{2})$ on the Hilbert space $\displaystyle\mathcal{H}$. From property (iv) this representation can be extended to the algebra $\displaystyle\mathcal{L}$ resulting in a Hilbert algebra isomorphism of $\displaystyle\mathcal{L}$ onto the space of Hilbert-Schmidt operators $\displaystyle\mathcal{B}_{2}(\mathcal{H})$. Moreover, from property (v) we can further extend this representation to a representation of the $\displaystyle C^{*}$-algebra $\displaystyle\mathcal{A}(\mathbb{R}^{2})$, which will give us a $\displaystyle C^{*}$-algebra isomorphism of $\displaystyle\mathcal{A}(\mathbb{R}^{2})$ onto the $\displaystyle C^{*}$-algebra of compact operators $\displaystyle\mathcal{K}(\mathcal{H})$.

Let us consider the operator $\displaystyle\hat{f}=\psi(\varphi,\,\cdot\,)$ for $\displaystyle\varphi,\psi\in\mathcal{H}$. Its integral kernel is equal $\displaystyle\mathsf{f}(a,b)=\psi(a)\overline{\varphi(b)}$ and, therefore, the corresponding function on phase space $\displaystyle f(q,p)=W_{\lambda}(\varphi,\psi)(q,p)$ is equal

[TABLE]

The functions $\displaystyle W_{\lambda}(\varphi,\psi)$ are the generalized $\displaystyle\lambda$-Wigner functions and are elements of the Hilbert space $\displaystyle\mathcal{L}$. The following properties of the functions $\displaystyle W_{\lambda}(\varphi,\psi)$ are an immediate consequence of Theorem 10.

Theorem 11**.**

For $\displaystyle\varphi,\psi,\phi,\chi\in\mathcal{H}$ and $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$

1. (i)

$\displaystyle W_{\lambda}(\varphi,\psi)^{*}=W_{\lambda}(\psi,\varphi)$, 2. (ii)

$\displaystyle\int_{\mathbb{R}^{2}}W_{\lambda}(\varphi,\psi)\,\mathrm{d}{l}=(\varphi,\psi)$, 3. (iii)

\displaystyle\bigl{(}W_{\lambda}(\varphi,\psi),W_{\lambda}(\phi,\chi)\bigr{)}=\overline{(\varphi,\phi)}(\psi,\chi), 4. (iv)

$\displaystyle W_{\lambda}(\varphi,\psi)\star W_{\lambda}(\phi,\chi)=(\varphi,\chi)W_{\lambda}(\phi,\psi)$, 5. (v)

$\displaystyle f\star W_{\lambda}(\varphi,\psi)=W_{\lambda}(\varphi,\hat{f}\psi)$* and $\displaystyle W_{\lambda}(\varphi,\psi)\star f=W_{\lambda}(\hat{f}^{\dagger}\varphi,\psi)$.*

We can define a tensor product of the Hilbert space $\displaystyle\mathcal{H}$ and its dual $\displaystyle\mathcal{H}^{*}$ by the formula

[TABLE]

where $\displaystyle\varphi\mapsto\varphi^{*}$ is an anti-linear isomorphism of $\displaystyle\mathcal{H}$ onto $\displaystyle\mathcal{H}^{*}$ appearing in the Riesz representation theorem. The map $\displaystyle\otimes_{\lambda}\colon\mathcal{H}^{*}\times\mathcal{H}\to\mathcal{L}$ is clearly bilinear and from property (iii) from Theorem 11 it satisfies

[TABLE]

Moreover, since the set of generalized $\displaystyle\lambda$-Wigner functions $\displaystyle W_{\lambda}(\varphi,\psi)$ is linearly dense in $\displaystyle\mathcal{L}$ the map $\displaystyle\otimes_{\lambda}$ indeed defines a tensor product of $\displaystyle\mathcal{H}^{*}$ and $\displaystyle\mathcal{H}$ equal to $\displaystyle\mathcal{L}$.

For $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$ we can treat $\displaystyle f\star{}$ as an operator on the Hilbert space $\displaystyle\mathcal{L}$. Then, by property (v) from Theorem 11 we get that

[TABLE]

Note, that the operator representation of the algebra $\displaystyle\mathcal{L}$ gives a one to one correspondence between states $\displaystyle\rho\in\mathcal{L}$ and density operators $\displaystyle\hat{\rho}$, i.e. trace class operators satisfying

1. (i)

$\displaystyle\hat{\rho}^{\dagger}=\hat{\rho}$, 2. (ii)

$\displaystyle\operatorname{Tr}(\hat{\rho})=1$, 3. (iii)

$\displaystyle(\varphi,\hat{\rho}\varphi)\geq 0$ for every $\displaystyle\varphi\in\mathcal{H}$.

From this correspondence we can see that a function $\displaystyle\rho\in\mathcal{L}$ is a state if and only if it is in the form

[TABLE]

where $\displaystyle\varphi_{k}\in\mathcal{H}$, $\displaystyle\lVert\varphi_{k}\rVert=1$, $\displaystyle p_{k}\geq 0$, and $\displaystyle\sum_{k}p_{k}=1$. In particular, $\displaystyle\rho\in\mathcal{L}$ is a pure state if and only if

[TABLE]

for some normalized vector $\displaystyle\varphi\in\mathcal{H}$.

The generalized $\displaystyle\lambda$-Wigner functions obey the following probability interpretation

[TABLE]

The operator representation can be extended to the algebra $\displaystyle\mathcal{F}_{\star}(\mathbb{R}^{2})$ in the following manner. Let $\displaystyle\mathcal{D}$ denote a set of all functions on $\displaystyle\mathbb{R}$ which extend to smooth functions on $\displaystyle\overline{\mathbb{R}}$. Then $\displaystyle\mathcal{D}$ is a dense subspace of $\displaystyle\mathcal{H}$. If $\displaystyle\varphi,\psi\in\mathcal{D}$, then $\displaystyle W_{\lambda}(\varphi,\psi)\in\mathcal{F}(\mathbb{R}^{2})$. Hence, for $\displaystyle f\in\mathcal{F}_{\star}(\mathbb{R}^{2})$ we can define the operator $\displaystyle\hat{f}$ by the following bilinear form

[TABLE]

provided that this bilinear form is continuous with respect to the first variable. The formula (126) defines a unique possibly unbounded operator on $\displaystyle\mathcal{H}$ with a dense domain $\displaystyle\mathcal{D}$. In the particular case where $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$, this definition of the operator $\displaystyle\hat{f}$ coincides with the definition (106).

Let $\displaystyle\mathcal{S}$ denote a space of smooth functions on $\displaystyle\mathbb{R}$ which, together with all derivatives, decay faster than any power of $\displaystyle p$. This space is dense in $\displaystyle\mathcal{H}$. Let $\displaystyle\hat{q}=i\hbar D_{p}$ and $\displaystyle\hat{p}=p$ be operators on $\displaystyle\mathcal{H}$ defined on the domain $\displaystyle\mathcal{S}$. They are symmetric and satisfy the commutation relation (3). Moreover, as was discussed in [15], only $\displaystyle\hat{p}$ is essentially self-adjoint. In what follows we will show that these operators correspond to observables of position and momentum $\displaystyle q$, $\displaystyle p$, and that an operator $\displaystyle\hat{f}$ corresponding to a function $\displaystyle f$ is, in fact, a $\displaystyle\lambda$-ordered function $\displaystyle f$ of operators $\displaystyle\hat{q}$ and $\displaystyle\hat{p}$.

Theorem 12**.**

If $\displaystyle f(q,p)=q^{n}\phi(p)$ where $\displaystyle n\in\mathbb{N}$ and $\displaystyle\phi$ is smooth on $\displaystyle\overline{\mathbb{R}}$, then the corresponding operator $\displaystyle\hat{f}$ can be defined by an analogical formula to (126) and takes the form

[TABLE]

Proof.

For $\displaystyle\varphi\in\mathcal{H}$ and $\displaystyle\psi\in\mathcal{S}$ we have that $\displaystyle SW_{\lambda}(\varphi,\psi)=W_{1-\lambda}(\varphi,\psi)$. Hence

[TABLE]

Thus, the bilinear form $\displaystyle(\varphi,\psi)\mapsto\langle{f,W_{\lambda}(\varphi,\psi)}\rangle$ is continuous with respect to the first variable and defines an operator $\displaystyle\hat{f}$ through the formula

[TABLE]

By (33) we get

[TABLE]

∎

For $\displaystyle f\in\mathcal{F}(\mathbb{R}^{2})$ the corresponding operator $\displaystyle\hat{f}$ can be written in a form

[TABLE]

where $\displaystyle e^{\frac{i}{\hbar}\lambda(\hat{q},p^{\prime})}$ and $\displaystyle e^{-\frac{i}{\hbar}(q^{\prime},\hat{p})}$ are unitary operators which action on a function $\displaystyle\psi\in\mathcal{H}$ is given by the formulas

[TABLE]

9 Position eigenvectors

Solutions to the eigenvalue equations

[TABLE]

define position eigenvectors. They are in the form

[TABLE]

where $\displaystyle\operatorname{sinc}x=\frac{\sin\pi x}{\pi x}$. Indeed, since

[TABLE]

we get

[TABLE]

and similarly

[TABLE]

The elements of the Hilbert space $\displaystyle\mathcal{H}$ corresponding to $\displaystyle\rho_{\xi}$ are in the form

[TABLE]

A direct calculation shows that indeed $\displaystyle\rho_{\xi}=W_{\lambda}(\psi_{\xi},\psi_{\xi})$ and that $\displaystyle\psi_{\xi}$ are eigenvectors of the operator $\displaystyle\hat{q}$.

The position eigenvectors $\displaystyle\rho_{\xi}$ are not physical states as their uncertainty in position is smaller than the minimal uncertainty $\displaystyle\Delta q_{0}$ (cf. [15] where this is discussed in full detail).

10 Maximal localization states

In [15] authors calculated states of maximal localization around a position $\displaystyle\xi$, i.e. states with the following properties

[TABLE]

where $\displaystyle\Delta q_{0}=\hbar\sqrt{\beta}$ is the smallest uncertainty in position which can be reached. In momentum representation the received states were given by the formula

[TABLE]

To demonstrate the developed formalism we can calculate the form of the maximal localization states on phase space. By virtue of (124) we receive the following formula for the quasi-probability distribution function representing a state of maximal localization around a point $\displaystyle(\xi,0)$ in phase space:

[TABLE]

Indeed, we have that

[TABLE]

Since

[TABLE]

we get for $\displaystyle p^{\prime}=\frac{1}{\sqrt{\beta}}\tan\bar{p}$, after using various trigonometric identities, that

[TABLE]

Hence, the function $\displaystyle\rho_{\xi}^{\text{ML}}(q,p)$ is expressed by the following integrals

[TABLE]

which can be easily calculated to give (141).

Note, that the state $\displaystyle\rho_{\xi}^{\text{ML}}(q,p)$ is a shift in position by $\displaystyle\xi$ of the state $\displaystyle\rho_{0}^{\text{ML}}(q,p)$ localized around a point $\displaystyle(0,0)$ in phase space. For particular values of $\displaystyle\lambda$ the states $\displaystyle\rho_{\xi}^{\text{ML}}(q,p)$ take a simpler form. For instance, when $\displaystyle\lambda=\tfrac{1}{2}$ we have

[TABLE]

In this case the quasi-probability distribution functions $\displaystyle\rho_{\xi}^{\text{ML}}(q,p)$ are real-valued, which is the common property of every state. The plot of the state localized around a point $\displaystyle(0,0)$ is presented in Fig. 1. For $\displaystyle\lambda=0$ we have

[TABLE]

The plots of real and imaginary parts of the state localized around a point $\displaystyle(0,0)$ are presented in Fig. 2.

11 Conclusions and final remarks

In the paper we developed a formalism of non-formal deformation quantization exhibiting a minimal length scale. The theory was presented for the case of two dimensional phase space $\displaystyle\mathbb{R}^{2}$. The generalization to $\displaystyle 2n$ dimensions leading to the following commutation relations for the operators of position and momentum

[TABLE]

is straightforward. However, if we would like to consider different commutation relations, like the one found in [15]

[TABLE]

where $\displaystyle\hat{p}^{2}=\sum_{i=1}^{n}\hat{p}_{i}^{2}$, then the presented theory does not seem to generalize in a simple way.

An interesting topic of further investigation would be a development of deformation quantization theory with minimal length based on more general uncertainty relations exhibiting also a minimal uncertainty in momentum, like the one considered in [16]

[TABLE]

However, in this case there is neither position nor momentum operator representation, and one have to use Bergmann-Fock representation or only a deformation quantization approach.

The $\displaystyle\star$-product (12) is not the only product exhibiting a minimal uncertainty in position which can be introduced. Another example of such a star-product is

[TABLE]

where

[TABLE]

It can be easily verified that it satisfies the commutation relation (11). This star-product can be transformed to the product of the form (10) by the following noncanonical transformation of coordinates

[TABLE]

Note, however, that contrary to the star-product (12) for this star-product the property (13) will not hold. In fact, we can calculate that

[TABLE]

It seems that the star-products from the family (12) are the only products with the property (13). Thus, with respect to this property, the family (12) of star-products is distinguished from other products satisfying the commutation relation (11).