Coherent States for the Manin Plane via Toeplitz Quantization

TL;DR

This paper develops a novel approach to quantization on the non-commutative Manin plane using coherent states as eigenvectors of Toeplitz annihilation operators, creating a new layered quantization framework.

Contribution

It introduces a new quantization scheme based on eigenstates of Toeplitz operators, reversing classical order, and constructs a generalized Segal-Bargmann space for the Manin plane.

Findings

Defined coherent states as eigenvectors of Toeplitz annihilation operators.

Established a generalized Segal-Bargmann space with a reproducing kernel.

Compared the new quantization with that of the paragrassmann algebra.

Abstract

In the theory of Toeplitz quantization of algebras, as developed by the second author, coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the generically non-commutative Manin plane. In usual quantization schemes one starts with a classical phase space, then quantizes it in order to produce annihilation operators and then their eigenvectors and eigenvalues. But we do this in the opposite order, namely the set of the eigenvalues of the previously defined annihilation operator is identified as a generalization of a classical mechanical phase space. We introduce the resolution of the identity, upper and lower symbols as well as a coherent state quantization, which in turn quantizes the Toeplitz quantization. We thereby have a curious composition of quantization schemes. We proceed by identifying a generalized Segal-Bargmann space SB of square-integrable, anti-holomorphic functions as the image of a coherent state transform. Then SB has a reproducing kernel function which allows us to define a secondary Toeplitz quantization, whose symbols are functions. Finally, this is compared with the coherent states of the Toeplitz quantization of a closely related non-commutative space known as the paragrassmann algebra.

Equations

$$⟨{\theta }^i\overline{\theta }{}^j,{\theta }^k\overline{\theta }{}^l⟩:=w_{i+l}{\delta }_{i-j,k-l}\mathrm{for}i,j,k,l\in \mathbb{N},$$

$$⟨{\theta }^i,{\theta }^k⟩=w_i{\delta }_{i,k},$$

$$\mathcal{B}:=\Big{\{}\phi_{j}\equiv\phi_{j}(\theta):=\frac{1}{w_{j}^{1/2}}\theta^{j}~{}\Big{|}~{}j\geq 0\Big{\}}$$

$$P({\theta }^i\overline{\theta }{}^j):=\sum _{k=0}^{\infty }{w}_k^{-1}⟨{\theta }^k,{\theta }^i\overline{\theta }{}^j⟩{\theta }^k\mathrm{for}\mathrm{all}i,j\in \mathbb{N}.$$

$$T_{{\theta }^i\overline{\theta }{}^j}{\varphi }_n(\theta )=q^{-jn}\frac{w_{n+i}}{(w_n{w}_{n+i-j}{)}^{1/2}}{\varphi }_{n+i-j}(\theta ),$$

$$T_{{\theta }^i\overline{\theta }{}^j}{\varphi }_n(\theta )$$

$$\displaystyle=\big{\langle}\sum_{k}\phi_{k}(\overline{\theta})\otimes\phi_{k}(\eta),w_{n}^{-1/2}\eta^{i}\overline{\eta}\,\!^{j}\eta^{n}\big{\rangle}_{\eta}$$

$$\displaystyle=w_{n}^{-1/2}\big{\langle}\sum_{k}\phi_{k}(\overline{\theta})\otimes\phi_{k}(\eta),q^{-jn}\eta^{i+n}\overline{\eta}\,\!^{j}\big{\rangle}_{\eta}$$

$$\displaystyle=q^{-jn}w_{n}^{-1/2}\big{\langle}\sum_{k}w_{k}^{-1/2}\eta^{k},\,\eta^{i+n}\overline{\eta}\,\!^{j}\big{\rangle}\phi_{k}(\theta)$$

$$=q^{-jn}{w}_n^{-1/2}\sum _k{w}_k^{-1/2}{\delta }_{k+j,i+n}{w}_{k+j}{\varphi }_k(\theta )$$

$$=q^{-jn}{w}_n^{-1/2}{w}_{n+i-j}^{-1/2}{w}_{i+n}{\varphi }_{n+i-j}(\theta ).$$

$$T_{\overline{\theta }}{\varphi }_n=q^{-n}{\left(\frac{w_n}{w_{n-1}}\right)}^{1/2}{\varphi }_{n-1}\mathrm{for}\mathrm{every}\mathrm{integer}n\ge 1$$

$$\mathrm{D}(T_{\overline{\theta}}):=\Big{\{}\phi=\sum_{n=0}^{\infty}a_{n}\phi_{n}\in\mathcal{H}~{}\Big{|}~{}\sum_{n=1}^{\infty}|a_{n}|^{2}\,|q|^{-2n}\left(\dfrac{w_{n}}{w_{n-1}}\right)<\infty\Big{\}}$$

$$T_{\overline{\theta }}\varphi :=\sum _{n=1}^{\infty }{a}_n{q}^{-n}{\left(\frac{w_n}{w_{n-1}}\right)}^{1/2}{\varphi }_{n-1}$$

$$T_{\overline{\theta }}\varphi =\sum _{j=1}^{\infty }{a}_j{q}^{-j}{\left(\frac{w_j}{w_{j-1}}\right)}^{1/2}{\varphi }_{j-1}=\sum _{n=0}^{\infty }\lambda a_n{\varphi }_n.$$

$$\sum _{n=0}^{\infty }{a}_{n+1}{q}^{-(n+1)}{\left(\frac{w_{n+1}}{w_n}\right)}^{1/2}{\varphi }_n=\sum _{n=0}^{\infty }\lambda a_n{\varphi }_n.$$

$$a_{n+1}=\lambda q^{n+1}{\left(\frac{w_n}{w_{n+1}}\right)}^{1/2}{a}_n.$$

$$a_n={\lambda }^n{q}^{n(n+1)/2}{\left(\frac{w_0}{w_n}\right)}^{1/2}{a}_0\mathrm{for}\mathrm{all}\mathrm{integers}n\ge 0,$$

$$a_n={\lambda }^n{q}^{n(n+1)/2}{w}_n^{-1/2}\mathrm{for}\mathrm{all}\mathrm{integers}n\ge 0.$$

$${\varphi }_{\lambda }:=\sum _{n=0}^{\infty }{\lambda }^n{q}^{n(n+1)/2}{w}_n^{-1/2}{\varphi }_n$$

$$||{\varphi }_{\lambda }|{|}^2=\sum _{n=0}^{\infty }|\lambda {|}^{2n}|q{|}^{n(n+1)}{w}_n^{-1}<\infty ,$$

$$R_{w}=\Big{(}\limsup_{n\to\infty}|q|^{(n+1)}w_{n}^{-1/n}\Big{)}^{-1/2}\!\!=\,\,\Big{(}\liminf_{n\to\infty}|q|^{-(n+1)}w_{n}^{1/n}\Big{)}^{1/2}\!\!\!\!\!\!,$$

$$f(z):=\sum _{n=0}^{\infty }{z}^n|q{|}^{n(n+1)}{w}_n^{-1},$$

$$\sum _{n=1}^{\infty }|q{|}^{-2n}$$

$$=|\lambda {|}^2\sum _{n=1}^{\infty }\frac{|q{|}^{n(n-1)}}{w_{n-1}}|\lambda {|}^{2(n-1)}=|\lambda {|}^2||{\varphi }_{\lambda }|{|}^2<\infty ,$$

$$T_{\overline{\theta }}{\varphi }_{\lambda }$$

$$=\sum _{n=1}^{\infty }{\lambda }^n{q}^{n(n+1)/2}{w}_n^{-1/2}{q}^{-n}{\left(\frac{w_n}{w_{n-1}}\right)}^{1/2}{\varphi }_{n-1}$$

$$=\lambda \sum _{n=1}^{\infty }{\lambda }^{n-1}{q}^{n(n-1)/2}{w}_{n-1}^{-1/2}{\varphi }_{n-1}$$

$$=\lambda {\varphi }_{\lambda },$$

$$R_w=\underset{n\to \infty }{\lim }{\left((2\pi {)}^{1/2}{n}^{n+1/2}{e}^{-n}\right)}^{1/2n}=\underset{n\to \infty }{\lim }(2\pi {)}^{1/4n}{n}^{1/2+1/4n}{e}^{-1/2}=\infty .$$

$$I={\int }_{B_w}d\rho (\lambda )|{\varphi }_{\lambda }⟩⟨{\varphi }_{\lambda }|$$

$$⟨\varphi ,\psi ⟩={\int }_{B_w}d\rho (\lambda )⟨\varphi ,{\varphi }_{\lambda }⟩⟨{\varphi }_{\lambda },\psi ⟩.$$

$${\int }_{B_w}d\rho (\lambda )|\lambda {|}^{2n}|q{|}^{n(n+1)}{w}_n^{-1}=1\mathrm{for}\mathrm{every}\mathrm{integer}n\ge 0.$$

$$\big{|}\int_{B_{w}}d\rho(\lambda)w_{0}^{-1}\lambda^{2n}\big{|}\leq\int_{B_{w}}d\rho(\lambda)w_{0}^{-1}|\lambda|^{2n}=|q|^{-n(n+1)}w_{n}w_{0}^{-1}.$$

$${\int }_{B_w}d\rho (\lambda )⟨\varphi ,{\varphi }_{\lambda }⟩⟨{\varphi }_{\lambda },\psi$$

Full text

Keywords: Coherent states, Toeplitz operators, annihilation operators, coherent state quantization

MSC2010 codes Primary: 81R30, 47B35; Secondary: 81R60, 47B32

1. 1 Introduction
2. 2 The Setting
3. 3 Coherent States
4. 4 Resolution of the Identity
5. 5 Time Evolution
6. 6 Coherent State Transform
7. 7 Another Toeplitz Quantization
8. 8 Coherent State Quantization
9. 9 Upper and Lower Symbols
10. 10 Upper symbol of a Toeplitz operator
11. 11 A Comparison
12. 12 Concluding Remarks

1. Introduction

In [17] the second author has introduced a general formalism for defining Toeplitz operators whose symbols come from a not necessarily commutative algebra. One motivating example of this was presented in [16]. In the case when the algebra has a $*$-operation (also known as a conjugation) we can often identify in a natural way a sub-algebra, which is not closed under the $*$-operation, of holomorphic elements. Then its conjugate sub-algebra consists of the anti-holomorphic elements. The convention used here is that the common, invariant domain of the Toeplitz operators is the holomorphic sub-algebra, which is also a pre-Hilbert space. The Toeplitz operators whose symbols are holomorphic elements then play the role of creation operators while the Toeplitz operators whose symbols are anti-holomorphic elements play the role of annihilation operators.

In some examples there is a natural grading of the elements in the sub-algebra in which case the annihilation operators that lower degree by $1$ are analogous to the annihilation operator $\partial/\partial z$ acting on holomorphic functions of the Segal-Bargmann space. (For details see [4].) Such degree $-1$ annihilation operators can be used to define coherent states as their eigenvectors whose corresponding eigenvalues are the points in a classical phase space. Turning this understanding around, we may use an appropriate Toeplitz annihilation operator in order to define coherent states in the Toeplitz setting as its eigenvectors whose eigenvalues then define the classical phase space. We do exactly that in this paper in the setting given by the example in [16] of the non-commutative (or, as it is sometimes called, quantum) plane of Manin. However, we wish to emphasize our viewpoint that the Manin plane is not a quantum object in the sense of quantum theory, since among other things Planck’s constant $\hbar>0$ does not enter into its structure. On the other hand, the Weyl-Heisenberg (unital) algebra generated by elements $P$ and $Q$ and satisfying the commutation relation $i[P,Q]=\hbar\,1$ does come from quantum theory, explicitly involves Planck’s constant $\hbar$ and has a natural action of the Lie group $SL(2,\mathbb{C})\rtimes\mathbb{C}^{2}$ (the semi-direct product) that also acts naturally on the (complex) plane $\mathbb{C}^{2}$. So, the non-commutative Weyl-Heisenberg algebra deserves to be called the quantum plane. This point of view is also expressed in [20]. Nonetheless, the Manin plane is an interesting object studied in non-commutative geometry.

For background material on coherent states two excellent references with many examples are [1] and [11]. The recent book [3] has 14 review papers on a variety of topics in the field. We use standard notation. For example, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$ denote the sets of the non-negative integers, all the integers, the real numbers and the complex numbers, respectively. The complex conjugate of $\lambda\in\mathbb{C}$ is denoted by $\lambda^{*}$.

2. The Setting

We consider the Manin plane $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ which is the unital, complex algebra generated by two conjugate elements $\theta$ and $\overline{\theta}$ subject only to the commutation relation $\theta\overline{\theta}=q\overline{\theta}\theta$ for some non-zero $q\in\mathbb{C}$. This algebra is non-commutative except when $q=1$. The elements $\theta^{i}\overline{\theta}\,\!^{j}$ with $i,j\in\mathbb{N}$ form a basis of $\mathbb{C}Q_{q}(\theta,\overline{\theta})$. Toeplitz operators with symbols in $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ have been defined and studied in [16]. In particular, for any element $g\in\mathbb{C}Q_{q}(\theta,\overline{\theta})$ there is a linear operator, called the Toeplitz operator with symbol $g$, denoted by $T_{g}:\mathcal{P}\to\mathcal{P}$, where $\mathcal{P}$ is the holomorphic subalgebra generated by the holomorphic variable $\theta$. Of course, $\mathcal{P}=\mathbb{C}[\theta]$ is the polynomial algebra generated by $\theta$. So $\mathcal{P}$ has a grading given by the degree of a homogeneous polynomial and has a basis $\{\theta^{n}\,|\,n\geq 0\}$. We now sketch how this Toeplitz quantization is realized. See [16] for more details.

There is a sesqui-linear form determined on $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ by

[TABLE]

where the weights $w_{n}$ satisfy $w_{n}>0$ for every integer $n\geq 0$. Also, $\delta_{n,m}$ denotes the Kronecker delta for integers $n,m\in\mathbb{Z}$. Our convention is that a sesqui-linear form is linear is its second entry while being anti-linear in its first entry. The same convention holds for the inner product in a Hilbert space.

When restricted to $\mathcal{P}$ this sesqui-linear form (2.1) satisfies

[TABLE]

which clearly gives a positive definite inner product. So,

[TABLE]

is an orthonormal Hamel basis for the (incomplete) pre-Hilbert space $\mathcal{P}$. We let $\mathcal{H}$ denote the Hilbert space completion of $\mathcal{P}$. The inner product on $\mathcal{H}$, being the extension of that on $\mathcal{P}$, is also denoted as $\langle\cdot,\cdot\rangle$. We note that $\mathcal{B}$ is an orthonormal basis for $\mathcal{H}$. We use the sesqui-linear form to define a linear map $P$ on $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ determined by

[TABLE]

Due to the definition of the sesqui-linear form, this infinite sum has at most one non-zero term (when $k=i-j\geq 0$) and so it makes sense. Since the elements $\theta^{i}\overline{\theta}\,\!^{j}$ with $i,j\in\mathbb{N}$ form a basis of $\mathbb{C}Q_{q}(\theta,\overline{\theta})$, this uniquely determines the linear map $P$. In Dirac notation (which technically does not apply, since we are not in a Hilbert space setting) we have that $P=\sum_{k=0}^{\infty}|\phi_{k}(\theta)\rangle\langle\phi_{k}(\theta)|$. So it is not surprising that $P=P^{2}$ (that is, $P$ is a projection) with range $\mathcal{P}$, the algebraic span of the $\phi_{k}(\theta)$.

Now for any $g\in\mathbb{C}Q_{q}(\theta,\overline{\theta})$ we define the Toeplitz operator with symbol $g$, denoted as $T_{g}:\mathcal{P}\to\mathcal{P}$, by $T_{g}(\phi):=P(g\phi)$ for all $\phi\in\mathcal{P}$. Notice that the product $g\phi$ of the two elements $g,\phi\in\mathbb{C}Q_{q}(\theta,\overline{\theta})$ is again an element in the algebra $\mathbb{C}Q_{q}(\theta,\overline{\theta})$. Then the projection $P$ maps this product to an element of $\mathcal{P}$. We have chosen to multiply the symbol $g$ on the left in the definition of $T_{g}$. A similar, but not identical, theory entails if we multiply on the right, which was what was done in [16]. We will explain later on why we preferred using multiplication on the left in this paper.

As explained in detail in [17] this theory of densely defined Toeplitz operators, acting in a Hilbert space, gives a quantization scheme, called Toeplitz quantization, that includes Planck’s constant as well as creation and annihilation operators. So this is a quantum theory.

The operators $T_{\theta^{i}\overline{\theta}\,\!^{j}}$ were explicitly calculated in Theorem 4.4 in [16], where we had multiplication of the symbol on the right in the definition of Toeplitz operators. Since we have used here multiplication of the symbol on the left, we have a different result.

Theorem 2.1**.**

For integers $i,j,n\geq 0$ we have

[TABLE]

where for $m<0$ we put $\phi_{m}(\theta)=0$ and $w_{m}=1$.

Proof.

We use the reproducing kernel object $K(\theta,\eta)=\sum_{k}\phi_{k}(\overline{\theta})\otimes\phi_{k}(\eta)$, which is a tensor but not a function, to calculate the projection operator. Here $\eta,\overline{\eta}$ is another, independent pair of variables satisfying $\eta\overline{\eta}=q\overline{\eta}\eta$. Also, $\langle\cdot,\cdot\rangle_{\eta}$ denotes the same sesqui-linear form as above, but with respect to the new variables $\eta,\overline{\eta}$. See [16] for more details. We calculate

[TABLE]

Even though the sum is over all $k\in\mathbb{N}$, only at most one term is non-zero, namely when $k=n+i-j\geq 0$. ∎

Remark 2.1**.**

So $T_{\theta^{i}\overline{\theta}\,\!^{j}}$ changes the degree of the monomial $\phi_{n}$ by $i-j$ or annihilates it. Note that this result differs from Theorem 4.4 of [16] only by the factor of $q^{-jn}$. There are two special cases of interest. One is the annihilation operator $T_{\overline{\theta}}$ with anti-holomorphic symbol $\overline{\theta}$ when $i=0$ and $j=1$. And the other case is the creation operator $T_{\theta}$ with holomorphic symbol $\theta$ when $i=1$ and $j=0$.

We turn our attention to the annihilation operator about which (2.2) says that

[TABLE]

and that $T_{\overline{\theta}}\,\phi_{0}=0$. This defines $T_{\overline{\theta}}$ on the subspace $\mathcal{P}$, which is dense in the Hilbert space $\mathcal{H}$ and is invariant under the action of $T_{\overline{\theta}}$. Then $T_{\overline{\theta}}$ can be extended in the following way to the dense domain $\mathrm{D}(T_{\overline{\theta}})$ of elements $\phi=\sum_{n=0}^{\infty}a_{n}\phi_{n}$ such that $\sum_{n=0}^{\infty}a_{n}T_{\overline{\theta}}\,\phi_{n}$ is convergent in $\mathcal{H}$, that is to say,

[TABLE]

and

[TABLE]

for every $\phi=\sum_{n=0}^{\infty}a_{n}\phi_{n}\in\mathrm{D}(T_{\overline{\theta}})$. Actually, by standard functional analysis this extension is the closure of $T_{\overline{\theta}}$ defined on the domain $\mathcal{P}$.

Then we have $\mathcal{P}\subsetneq\mathrm{D}(T_{\overline{\theta}})\subset\mathcal{H}$. The last inclusion is an equality if and only if the sequence $\{|q|^{-2n}\,(w_{n}/w_{n-1})~{}|~{}n\geq 1\}$ is bounded if and only if $T_{\overline{\theta}}$ is a bounded operator. Also, $T_{\overline{\theta}}$ is compact if and only if $\lim_{n\to\infty}|q|^{-2n}\,(w_{n}/w_{n-1})=0$.

The choice $w_{n}=n!$ for the weights is motivated by the Segal-Bargmann space. (See [4].) With this choice $T_{\overline{\theta}}\,\phi_{n}=q^{-n}\,n^{1/2}\,\phi_{n-1}$ for $n\geq 1$. If also $q=1$, then $T_{\overline{\theta}}$ is an unbounded operator, which is unitarily equivalent to the annihilation operator $\partial/\partial z$ in the Segal-Bargmann space. Another choice is $w_{n}=1$ in which case $T_{\overline{\theta}}$ is a weighted backwards shift operator, a bounded operator if and only if $|q|\geq 1$.

3. Coherent States

We now arrive at our basic definition.

Definition 3.1**.**

A coherent state for this Toeplitz quantization of the Manin plane is an eigenvector of the annihilation operator $T_{\overline{\theta}}$. More explicitly, it is a non-zero vector $\phi_{\lambda}\in\mathrm{D}(T_{\overline{\theta}})$ such that $T_{\overline{\theta}}\,\phi_{\lambda}=\lambda\phi_{\lambda}$, where $\lambda\in\mathbb{C}$ is the eigenvalue. The set of eigenvalues of $T_{\overline{\theta}}$ is called the phase space.

Remark 3.2**.**

The terminology ‘phase space’ comes from the theory of classical mechanics. But in functional analysis the set of eigenvalues of a densely defined linear operator is called its point spectrum, at least by some authors. An important feature of this approach is that the (possibly empty!) phase space is characterized purely by properties of the quantum theory, that is to say, the phase space is a quantum object. N.B.: We do not start with a phase space and then quantize it.

We now find the coherent states by taking $\phi=\sum_{n=0}^{\infty}a_{n}\phi_{n}$, where the unknown coefficients must satisfy $\sum_{n=0}^{\infty}|a_{n}|^{2}<\infty$, and noting that $T_{\overline{\theta}}\,\phi=\lambda\phi$, where $\lambda\in\mathbb{C}$, becomes

[TABLE]

Putting $n=j-1$ in the first sum gives

[TABLE]

Using orthogonality we see that a necessary and sufficient condition for this equality to hold is that for all integers $n\geq 0$ we have

[TABLE]

Therefore, up to a non-zero multiplicative constant, there is at most one coherent state with eigenvalue $\lambda\in\mathbb{C}$. In fact, the recursion relation (3.1) is solved explicitly by

[TABLE]

where $a_{0}\in\mathbb{C}\setminus\{0\}$ is arbitrary. (We exclude $a_{0}=0$ since an eigenvector by definition must be non-zero.) Moreover, in the case when $\lambda\neq 0$ we see that $a_{n}\neq 0$ for all $n\geq 1$ and therefore $\phi=\sum_{0}^{\infty}a_{n}\phi_{n}$ can not be an element of $\mathcal{P}$. This is why we introduced the larger domain $\mathrm{D}(T_{\overline{\theta}})$. We will return to this point in a moment.

For convenience we simplify by taking $a_{0}=w_{0}^{-1/2}$ thereby getting

[TABLE]

And so, up to a multiplicative non-zero constant, the coherent state with eigenvalue $\lambda\in\mathbb{C}$ has to be

[TABLE]

provided at least that the series converges in $\mathcal{H}$. And if this series diverges, then there is no coherent state with eigenvalue equal to $\lambda$. Of course, this series is convergent by Hilbert space theory if and only if

[TABLE]

which is a power series in the variable $|\lambda|^{2}$. We let $w$ denote $\{w_{n}\,|\,n\geq 0\}$, the sequence of the weights. Then, using the theory of power series, the radius of convergence $R_{w}$ of this as a power series in $|\lambda|$ is given by the formula

[TABLE]

where we use the standard conventions $0^{-1}=\infty$ and $\infty^{-1}=0$. So, the formula $\eqref{coherent-state}$ defines a vector $\phi_{\lambda}\in\mathcal{H}$ for all $\lambda\in\mathbb{C}$ satisfying $|\lambda|<R_{w}$. Also, the series in (3.2) diverges if $|\lambda|>R_{w}$, in which case there is no coherent state with eigenvalue $\lambda$. It is worthwhile to note that the infinite series in (3.3) converges for some complex number $\lambda$ if and only if that series converges at every point on the circle $\lambda\,e^{i\theta}$ of radius $|\lambda|$, where $\theta\in\mathbb{R}$. The associated holomorphic function

[TABLE]

where $z$ is a complex variable, has radius of convergence $R_{w}^{2}$. Suppose that $R_{w}<\infty$. Since this series has positive coefficients, it converges absolutely at every point on the circle $|z|=R_{w}^{2}$ if and only if it converges absolutely at one point on that circle if and only if it converges (absolutely of necessity) at $z=R_{w}^{2}$.

It is not enough that $\phi_{\lambda}$ is an element of the Hilbert space $\mathcal{H}$, even though that is a necessary condition. It must also belong to $\mathrm{D}(T_{\overline{\theta}})$, the domain of the annihilation operator, as noted earlier, and satisfy $T_{\overline{\theta}}\,\phi_{\lambda}=\lambda\,\phi_{\lambda}$. We now explicitly prove this.

Proposition 3.1**.**

For all $\lambda$ satisfying (3.3) (in particular, if $|\lambda|<R_{w}$) we have that $\phi_{\lambda}\in\mathrm{D}(T_{\overline{\theta}})$. Moreover, $T_{\overline{\theta}}\,\phi_{\lambda}=\lambda\,\phi_{\lambda}$ and $\phi_{\lambda}\neq 0$ for all such $\lambda\in\mathbb{C}$.

Proof.

According to (2.4) and (3.2), proving $\phi_{\lambda}\in\mathrm{D}(T_{\overline{\theta}})$ is equivalent to showing that the following expression is finite:

[TABLE]

where the inequality holds because $\lambda$ satisfies (3.3). Since $\langle\phi_{0},\phi_{\lambda}\rangle=w_{0}^{-1/2}\neq 0$ by (3.2), we see that $\phi_{\lambda}\neq 0$. Finally, we prove $\phi_{\lambda}$ is an eigenvector whose eigenvalue is $\lambda$ by calculating

[TABLE]

which finishes what we wanted to prove. ∎

For $R_{w}>0$ the phase space consists of the open ball $|\lambda|<R_{w}$ plus possibly the points on its boundary for the case $R_{w}<\infty$. The case $R_{w}=0$ will be discussed in a moment. Clearly, (3.3) converges at some point on the boundary if and only if it converges at all points on the boundary. We let $B_{w}$ denote the phase space; this is either the open or closed ball of radius $R_{w}$. It seems reasonable to conjecture that the spectrum of $T_{\overline{\theta}}$ is the closure of $B_{w}$, that is, $\mathrm{Spec}(T_{\overline{\theta}})=\overline{B_{w}}$. We note the possibility that some elements on the boundary of $B_{w}$ may not be eigenvalues.

The case $w_{n}=n!$ and $|q|=1$ arises as noted above in analogy with the situation in the Segal-Bargmann space. Using the Stirling approximation for $n!$ we see for this case that

[TABLE]

Thus the phase space is $\mathbb{C}$, and the annihilation operator has point spectrum equal to the entire complex plane.

The case $w_{n}=1$ for all integers $n\geq 0$ and $q=1$ clearly leads to $R_{w}=1$, which is the spectral radius of the backwards shift operator. By picking other values for the weights and for $q$ we can find any value of $R_{w}$ in $[0,\infty]$.

In this paper the Manin plane $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ plays the role of the ‘classical phase space’ that is being quantized, but the phase space $B_{w}$ determined by the quantum theory is quite unlike $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ unless $R_{w}=\infty$ and $q=1$. The fact that the Manin plane $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ has the ball $B_{w}$ as its phase space is quite surprising. Of course, (3.2) always converges for $\lambda=0$. But in the case when $R_{w}=0$ that is the only complex number $\lambda$ for which (3.2) converges and so the phase space has exactly one point, a rather curious quantum situation with a trivial phase space. This motivates the following idea.

Definition 3.3**.**

We say that a quantum theory which has exactly one annihilation operator $A$ is an extreme quantum theory if the point spectrum of $A$ has at most one point, that is to say, the corresponding phase space has at most one point.

Remark 3.4**.**

Since one typically has $\ker\,A\neq 0$ for an annihilation operator $A$, the complex number [math] is then in the point spectrum of $A$. While it is mathematically possible to have $\ker\,A=0$, this may be undesirable from a physics viewpoint. Until Section 11 of this paper we will assume that $R_{w}>0$. So the interior of the phase space $B_{w}$ is non-empty. However, the case $R_{w}=0$ is fascinating, though we currently have few mathematical tools for studying it. For this reason, and this reason alone, we exclude it from consideration here.

4. Resolution of the Identity

In order to study the phase space $B_{w}$ we note that the coherent states define a parametrized family of rank-one projection operators $|\phi_{\lambda}\rangle\langle\phi_{\lambda}|$ in Dirac notation, where $\lambda\in B_{w}$. We might want to find a positive Borel measure $\rho$ on $B_{w}$ such that

[TABLE]

where $I$ is the identity operator acting in $\mathcal{H}$. This is a giant step beyond Toeplitz quantization whose virtue is that it does not use a measure. Equation (4.1) is called the resolution of the identity, and the integral in it is to be understood with respect to the weak operator topology, which means that for all $\phi,\psi\in\mathcal{H}$ we have

[TABLE]

The integrand is a measurable function of $\lambda\in B_{w}$. (See Remark 8.2 for why this is true.) We are requiring here that it is also absolutely integrable with respect to the unknown measure $\rho$ for all $\phi,\psi\in\mathcal{H}$. While this is the standard definition of a resolution of the identity, the reader should be aware that this is an extremely strong condition on $\rho$. In particular, this imposes a lot of necessary conditions on the measure $\rho$. For example, by putting $\phi=\psi=\phi_{n}$, a standard basis element, we see that

[TABLE]

The case $n=0$ says that $w_{0}^{-1}\rho$ is a probability measure, that is, $E\mapsto\int_{E}d\rho(\lambda)\,w_{0}^{-1}$ for Borel subsets $E\subset B_{w}$ is a probability measure on the phase space $B_{w}$. Also (4.3) gives us necessary conditions on the even complex moments of that probability measure, namely for every integer $n\geq 0$ we must have

[TABLE]

It may well be that no such measure $\rho$ exists, but even so the Toeplitz quantization has its own intrinsic interest. The failure of the necessary conditions would imply that is the case. Furthermore, even if such a measure exists, it may not be unique.

We expand any pair $\phi,\psi\in\mathcal{H}$ in the orthonormal basis to get $\phi=\sum_{j}a_{j}\phi_{j}$ and $\psi=\sum_{k}b_{k}\phi_{k}$ with complex numbers $a_{j},b_{k}$ satisfying $||\psi||^{2}=\sum_{j}|a_{j}|^{2}<\infty$ and $||\phi||^{2}=\sum_{k}|b_{k}|^{2}<\infty$. So we obtain

[TABLE]

The interchange of infinite sum and integral in the last equality needs to be justified. This is a delicate situation, which we now examine. We first estimate the partial sums of the infinite sum under the integral sign. We take integers $N,M\geq 0$ getting

[TABLE]

We got the last estimate by applying Cauchy-Schwarz twice, which accounts for the exponent $2(1/2)=2/2$ in the last factor. To apply the Lebesgue dominated convergence theorem now we would need to prove that $||\phi_{\lambda}||^{2}$ as a function of $\lambda\in B_{w}$ is integrable with respect to the measure $\rho$. But that is false! In fact,

[TABLE]

where we used (3.3), the monotone convergence theorem and (4.3) in that order. Therefore the interchange does not follow so easily from the Lebesgue dominated convergence theorem. One can change this argument in standard ways so that it only applies to $\phi,\psi$ in some dense subspace, and thereby justify the interchange for such $\phi,\psi$. The easiest change by far of this type is to suppose that $\phi,\psi\in\mathcal{P}$ so that the infinite sum collapses to a finite sum and therefore (4.4) is trivially true.

We next assume that the measure $\rho$ is radial and absolutely continuous with respect to Lesbegue measure on $B_{w}$. So it is equal to $\rho(r)\,rdrd\alpha$ in the standard polar coordinates $r,\alpha$. By a common abuse of notation we also let $\rho$ denote here a measurable function $\rho:[0,R_{w}]\to[0,\infty)$. The meaning of $\rho$ will be clear from context. Radial symmetry is an enormously severe restriction which could well eliminate from consideration many interesting cases which deserve further study. In particular if $B_{w}$ contains its boundary $\partial B_{w}$, this restriction together with the absolute continuity implies that $\rho(\partial B_{w})=0$. We make it in order to facilitate calculations such as the following which continues from (4.4).

We assume that $\phi,\psi\in\mathcal{P}$ and calculate that

[TABLE]

By the resolution of the identity (4.2) this has to be equal to $\langle\phi,\psi\rangle=\sum_{j=0}^{\infty}a_{j}^{*}b_{j}$ for all $\phi,\psi\in\mathcal{P}$, that is, for all sequences $\{a_{j}\,|\,j\geq 0\}$ and $\{b_{j}\,|\,j\geq 0\}$ with only finitely many non-zero terms. Therefore, by taking $\phi=\psi=\phi_{n}\in\mathcal{P}$, we have the following result.

Theorem 4.1**.**

Suppose that the resolution of the identity (4.1) holds for a measure $\rho$ on $B_{w}$ which is radial and absolutely continuous with respect to Lebesgue measure. Then necessary conditions on the ‘radial’ function $\rho:[0,R_{w}]\to[0,\infty)$ are:

[TABLE]

Remark 4.1**.**

We change variables in the integral (4.6) using $t=r^{2}$ getting the equivalent conditions

[TABLE]

To find a Borel measure on an interval with prescribed moments as in (4.7) is either a Hausdorff moment problem if $0<R_{w}<\infty$ or a Stieljes moment problem if $R_{w}=\infty$. These problems have been well studied in classical analysis as to the existence and uniqueness of solutions. (See the text [15].) For example, taking the case $w_{j}=j!$ and $|q|=1$ (for which $R_{w}=\infty$ as we have already seen), we can then take $\rho(r)=\pi^{-1}\exp(-r^{2})$ or, equivalently $\rho(t^{1/2})=\pi^{-1}e^{-t}$, by standard identities from integral calculus. See [10] for other examples in a non-commutative setting. Of course, after successfully finding a function $\rho$ solving the moment problem (4.6), one still has to check that the possibly stronger condition (4.2) holds.

Also, Equation (4.1), if it holds, allows us to identify readily all vectors $\psi\in\mathcal{H}$ that are orthogonal to all of the coherent states, which means that $\langle\phi_{\lambda},\psi\rangle=0$ for all $\lambda\in B_{w}$. One simply uses the equivalent equation (4.2) to see that $\langle\phi,\psi\rangle=0$ for all $\phi\in\mathcal{H}$, implying $\psi=0$. Another way of saying this is that $\mathrm{span}\{\phi_{\lambda}~{}|~{}\lambda\in B_{w}\}$ is a dense subspace of $\mathcal{H}$ provided that (4.1) holds.

5. Time Evolution

The degree of the homogeneous elements can be used to define an operator $N$ that for all integers $n\geq 0$ satisfies

[TABLE]

By standard techniques in functional analysis this has a unique extension as an unbounded, self-adjoint linear operator acting in a dense domain of the Hilbert space $\mathcal{H}$. One says that $N$ is the number operator. It also serves as a quantum Hamiltonian. The time evolution unitary group generated by $N$ is $e^{-itN}$, where $t\in\mathbb{R}$ is interpreted as the (dimensionless!) time. Then we have immediately by the functional calculus that

[TABLE]

Consequently, the time evolution of a coherent state $\phi_{\lambda}$ for $\lambda\in B_{w}$ is given by

[TABLE]

This shows that coherent states evolve under the flow of the unitary group $e^{-itN}$ into coherent states. Also the induced evolution in the phase space in time $t$ from the initial condition $\lambda$ is $\lambda\mapsto\lambda e^{-it}$, which is motion in a circle of radius $|\lambda|$ centered at the origin. So the operator $N$ generates a flow in the space of coherent states and hence in the phase space. Notice that the latter flow does indeed leave the phase space $B_{w}$ invariant.

It is curious that the (appropriately normalized) quantum harmonic oscillator is unitarily equivalent to $N+\frac{1}{2}I$, since neither the Manin plane nor the annihilation operator $T_{\overline{\theta}}$ have anything to do with it a priori. In fact, the phase space of the quantum harmonic oscillator is the entire complex plane $\mathbb{C}$, while for many choices of the weights the phase space of the Manin plane is $B_{w}\subsetneq\mathbb{C}$. For such choices of the weights, clearly the Toeplitz quantization of the Manin plane is not equivalent to the quantum harmonic oscillator.

6. Coherent State Transform

The coherent states can be used to define a transform from the Hilbert space $\mathcal{H}$ to a space of anti-holomorphic functions whose common domain is the interior of the phase space. For this we introduce the notations $D_{w}:=\{\lambda\in\mathbb{C}~{}|~{}|\lambda|<R_{w}\}$ and the complex vector space

[TABLE]

We recall that the open ball $D_{w}$ is a subset of the phase space $B_{w}$. We have the following standard definition.

Definition 6.1**.**

For $\psi\in\mathcal{H}$ we define $C\psi$, the coherent state transform of $\psi$, by

[TABLE]

Remark 6.2**.**

So, $C\psi:D_{w}\to\mathbb{C}$. Also, putting $\psi=\sum_{k}c_{k}\phi_{k}$ with $\sum_{k}|c_{k}|^{2}<\infty$, we see that

[TABLE]

a power series which is clearly anti-holomorphic in $\lambda\in D_{w}$. Hence, $C\psi\in\mathcal{A}(D_{w})$. Moreover, the mapping $C:\mathcal{H}\to\mathcal{A}(D_{w})$ given by $\psi\mapsto C\psi$ is linear.

We now would like to find an inner product on the range of $C$ so that the coherent state transform $C$ is unitary. Of course, if $C$ is injective, this can be done in a unique and trivial way. The point is to find an ‘intrinsic’ definition of that inner product, that is, a way of defining a subspace $\mathcal{C}$ of $\mathcal{A}(B_{w})$, then making $\mathcal{C}$ into a Hilbert space via an inner product on it and finally showing that the range of $C$ is $\mathcal{C}$ and that $C$ is unitary. The next lemma is well known, but bears on the present discussion.

Lemma 6.1**.**

$C$* is injective if and only if $\mathrm{span}\{\phi_{\lambda}\,|\,\lambda\in D_{w}\}$ is dense in $\mathcal{H}$*

Proof.

$\Rightarrow$: Suppose that $C$ is injective. Take $\psi\in\left(\mathrm{span}\{\phi_{\lambda}~{}|~{}\lambda\in D_{w}\}\right)^{\perp}$. It suffices to prove $\psi=0$. We have $C\psi(\lambda)=\langle\phi_{\lambda},\psi\rangle=0$ for all $\lambda\in D_{w}$. Therefore, $C\psi=0$. And then, by the hypothesis that $C$ is injective, we see that $\psi=0$.

$\Leftarrow$: Suppose that $\mathrm{span}\{\phi_{\lambda}~{}|~{}\lambda\in D_{w}\}$ is dense. Take $\psi\in\ker\,C$. It suffices to show that $\psi=0$ in order to prove that $C$ is injective. But then $0=C\psi(\lambda)=\langle\phi_{\lambda},\psi\rangle$ for all $\lambda\in D_{w}$. It follows that $\psi\in\left(\mathrm{span}\{\phi_{\lambda}~{}|~{}\lambda\in D_{w}\}\right)^{\perp}$, which is [math] by the hypothesis. So we see that $\psi=0$ as desired. ∎

The existence of a resolution of the identity is the key for the next result.

Theorem 6.1**.**

Suppose that there exists a positive measure $\rho$ on $B_{w}$ such that the resolution of the identity (4.1) holds. Suppose that $\rho(\partial B_{w})=0$ in the case that the boundary $\partial B_{w}\subset B_{w}$. Then the coherent state transform $C$ is a unitary transform from $\mathcal{H}$ onto its range $\mathrm{Ran}\,C$ in $L^{2}(D_{w},\rho)$. Consequently, $\mathrm{Ran}\,C$ is a closed subspace of $L^{2}(D_{w},\rho)$.

Notation:* $C:\mathcal{H}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{SB}\subset L^{2}(D_{w},\rho)$, where $\mathcal{SB}:=\mathrm{Ran}\,C$ is called a generalized Segal-Bargmann space. (See [4].)*

Moreover, $\mathcal{SB}$ is a reproducing kernel Hilbert space of anti-holomorphic functions with reproducing kernel function defined for $\mu,\lambda\in D_{w}$ by

[TABLE]

Remark: The existence of a reproducing kernel Hilbert space isomorphic to $\mathcal{H}$ is a nice property, but it depends on the existence of a resolution of the identity, which we will not have in many interesting cases. And, of course, there could be more than one resolution of the identity.

Proof.

Let $\psi_{1},\psi_{2}\in\mathcal{H}$ be given. Then we compute

[TABLE]

where we used the resolution of the identity (4.2) in the last equality. This shows exactly that $C$ is a unitary transform onto its range. In particular, $C$ is injective.

Next, let’s show that the function in (6.1) has the reproducing property. So we take an arbitrary element $f\in\mathcal{SB}$. Note that $f=C\psi$ for a unique element $\psi\in\mathcal{H}$. Then we calculate

[TABLE]

We used the hypothesis $\rho(\partial B_{w})=0$ in the first equality. This shows the reproducing property.

But we also have to show that $K(\cdot,\lambda)\in\mathcal{SB}$ for every $\lambda\in D_{w}$, this being the second defining property of a reproducing kernel function. But for all $\mu\in D_{w}$ we have that

[TABLE]

so that $K(\cdot,\lambda)=C\phi_{\lambda}\in\mathrm{Ran}\,C=\mathcal{SB}$ as desired.

So the function $K:D_{w}\times D_{w}\to\mathbb{C}$ in (6.1) satisfies the two defining properties of the (unique, it it exists) reproducing kernel function for $\mathcal{SB}$. ∎

There is an immediate corollary to this proof.

Corollary 6.1**.**

Assume the hypothesis of the previous theorem. Then the coherent state transform of a generic coherent state equals the reproducing kernel of the generalized Segal-Bargmann space.

Proof.

We just read (6.2) backwards, namely $C\phi_{\lambda}(\mu)=K(\mu,\lambda)$. This equality is what the corollary says clumsily in words. ∎

Remark 6.3**.**

The property in this corollary already appears in Bargmann’s 1961 seminal paper [4]. It is one of the characteristic properties of coherent states, though it is not always mentioned.

We can re-write one result of this theorem as $\mathcal{SB}\subset L^{2}_{AH}(D_{w},\rho)$, the subspace of $L^{2}(D_{w},\rho)$ of anti-holomorphic functions, namely,

[TABLE]

Another approach, more in line with [4], would be to define the Segal-Bargmann space as $L^{2}_{AH}(D_{w},\rho)$, which as far as we have shown at this point could be strictly larger than $\mathcal{SB}$. We would like to show that these spaces are actually equal. The following is a partial result in that direction.

Theorem 6.2**.**

Suppose that the resolution of the identity (4.1) holds for a measure that is radial and absolutely continuous with respect to Lebesgue measure. Then $\mathcal{SB}=L^{2}_{AH}(D_{w},\rho)$.

Proof.

Since $\mathcal{SB}=\mathrm{Ran}\,C$, it suffices to show that $\mathrm{Ran}\,C=L^{2}_{AH}(D_{w},\rho)$. We have shown already one inclusion. So it remains to show that any $f\in L^{2}_{AH}(D_{w},\rho)$ lies in $\mathrm{Ran}\,C$. Considering such a function $f$, we can express it as

[TABLE]

for certain coefficients $a_{n}\in\mathbb{C}$. We let $0<s<R_{w}$ and put $B_{s}:=\{\lambda\in\mathbb{C}~{}|~{}|\lambda|\leq s\}$, the closed ball of radius $s$. The condition that $f\in L^{2}(D_{w},\rho)$ implies that the first integral in the following calculation is finite:

[TABLE]

The interchange of the integral and the infinite sum in the second equality is justified since the double series converges uniformly on $D_{s}$ to its limit by standard properties of power series and by the hypothesis that $\rho$ is absolutely continuous with respect to Lebesgue measure. Taking the limit as $s\to R_{w}$ we get

[TABLE]

Here we used the Lebesgue monotone convergence theorem in the first, third and fourth equalities. The last equality follows from (4.6). We have shown that

[TABLE]

for all $f\in L^{2}_{AH}(D_{w},\rho)$, where $f$ is given by the series in (6.3).

We now consider the anti-holomorphic monomials $\lambda^{*j}$ for $j\geq 0$. Then one proves that $||\lambda^{*j}||^{2}=|q|^{-j(j+1)}w_{j}<\infty$ by evaluating the integral, and so we have that $\lambda^{*j}\in L^{2}_{AH}(D_{w},\rho)$. Consequently, all the anti-holomorphic polynomials are in $L^{2}_{AH}(D_{w},\rho)$. Another easy calculation shows that $\langle\lambda^{*j},\lambda^{*k}\rangle=0$ for all $j\neq k$. This combines to show that

[TABLE]

is an orthonormal set in the Hilbert space $L^{2}_{AH}(D_{w},\rho)$.

Next we claim that any $f\in L^{2}_{AH}(D_{w},\rho)$ as in (6.3) is the limit in the $L^{2}$-norm topology of the sequence of its partial sums $f_{N}(\lambda):=\sum_{j=0}^{N}a_{j}\lambda^{*j}$, which are anti-holomorphic polynomials. This is so since $f-f_{N}\in L^{2}_{AH}(D_{w},\rho)$ and therefore by (6.4) applied now to $f-f_{N}$ we have that

[TABLE]

which goes to [math] as $N\to\infty$ by (6.4) applied to $f$. The whole point of the proof so far is that we can conclude from this that $\mathcal{B}_{AH}$ is an orthonormal basis of the Hilbert space $L^{2}_{AH}(D_{w},\rho)$.

We next calculate the coherent state transforms of the standard basis elements $\phi_{j}\in\mathcal{H}$. For every integer $j\geq 0$ and $\lambda\in D_{w}$ this gives us

[TABLE]

Therefore, $C$ maps the orthonormal basis $\{\phi_{j}\,|\,j\geq 0\}$ of $\mathcal{H}$ onto the orthonormal basis $\mathcal{B}_{AH}$ of $L^{2}_{AH}(D_{w},\rho)$, which proves that $\mathrm{Ran}\,C=L^{2}_{AH}(D_{w},\rho)$ as desired. ∎

Remark 6.4**.**

In this argument we proved in (6.4) that for any anti-holomorphic function $f\in L^{2}(\rho)$ as in (6.3), we have $\sum_{j}|a_{j}|^{2}|q|^{j(j+1)}w_{j}$ is finite and is equal to $||f||^{2}_{L^{2}(\rho)}$. We did not prove the converse, which we now do.

Theorem 6.3**.**

Assume the same hypothesis as in the previous theorem. Suppose an anti-holomorphic function $f$ is given as in (6.3) and that $\sum_{j}|a_{j}|^{2}|q|^{j(j+1)}w_{j}<\infty$. Then $f\in L^{2}_{AH}(D_{w},\rho)$ and $||f||^{2}_{L^{2}(\rho)}=\sum_{j}|a_{j}|^{2}|q|^{j(j+1)}w_{j}$.

Proof.

We consider the partial sums $f_{N}(\lambda)=\sum_{j=0}^{N}a_{j}\lambda^{*j}$, which by the previous theorem are in $L^{2}_{AH}(D_{w},\rho)$ and satisfy $||f_{N}||^{2}=\sum_{j=0}^{N}|a_{j}|^{2}|q|^{j(j+1)}w_{j}$. Moreover, by (6.3) we have

[TABLE]

So, by the Fatou lemma we estimate

[TABLE]

This proves that $f\in L^{2}_{AH}(D_{w},\rho)$ but only gives an estimate on its $L^{2}$-norm. Next a similar argument using the Fatou lemma shows for every integer $N\geq 0$ that

[TABLE]

which goes to [math] as $N\to\infty$. This says that $f_{N}\to f$ in the topology of the $L^{2}$-norm. This in turn implies by the continuity of the $L^{2}$-norm that

[TABLE]

And that finishes the proof. ∎

As far as this analysis goes it still remains a logical possibility that the inclusion $\mathcal{SB}\subset L^{2}_{AH}(D_{w},\rho)$ is proper for other measures $\rho$. What is happening for such measures is an open problem.

7. Another Toeplitz Quantization

We continue to assume (4.1) holds for some measure $\rho$ on $B_{w}$ in this section. We proved in the last section that the Segal-Bargmann space $\mathcal{SB}$ is a closed subspace of $L^{2}(D_{w},\rho)$ and that it is a reproducing kernel Hilbert space. This gives the standard set-up for defining Toeplitz operators whose symbols are functions. First one uses the kernel function (6.1) as the kernel of an integral operator that defines an orthogonal projection $P_{K}:L^{2}(D_{w},\rho)\to L^{2}(D_{w},\rho)$ by

[TABLE]

This integral converges absolutely for $\mu\in D_{w}$ by the Cauchy-Schwarz inequality together with the fact that $K(\mu,\cdot)\in L^{2}(D_{w},\rho)$. By a standard argument $(P_{K}f)(\mu)$ is anti-holomorphic in $\mu\in D_{w}$. Moreover, by the resolution of the identity (4.1), $P_{K}$ acts as the identity on $\mathcal{SB}$. However, we only have shown that $P_{K}f\in L_{AH}^{2}(D_{w},\rho)$, which as we noted earlier might be strictly larger than $\mathcal{SB}$ for some measures $\rho$. It even seems possible to have

[TABLE]

To avoid such details for the rest of this section we assume that $\mathcal{SB}=L_{AH}^{2}(D_{w},\rho)$, which we know holds in many cases according to Theorem 6.2. Also $\mathcal{SB}$ has a lot of nice structure, such as an explicit reproducing kernel function and a standard orthonormal basis, making it a more preferable domain for Toeplitz operators.

Definition 7.1**.**

For $f\in L^{\infty}(D_{w},\rho)$ we define the (secondary) Toeplitz operator with symbol $f$ by

[TABLE]

Remark 7.2**.**

The notation $S_{f}$ distinguishes this from the Toeplitz operators $T_{g}$ which we introduced in Section 2. Clearly, $S_{f}:\mathcal{SB}\to\mathcal{SB}$ is linear. Moreover, we call the mapping

[TABLE]

given by $f\mapsto S_{f}$ the (secondary) Toeplitz quantization of the commutative algebra $L^{\infty}(D_{w},\rho)$. Also $S$ is linear. Recall that $D_{w}$ is the interior of the phase space associated to the Manin plane. So this is another quantization scheme associated with the Manin plane. Many standard properties hold for this theory. But we will leave the development of them for future research.

8. Coherent State Quantization

A general reference for coherent state quantization is Part II of the text [11] as well as the papers [8] and [9]. For an abstract approach see [2]. To start off this section we assume (4.1) holds for some measure $\rho$ on $B_{w}$. This is the key property in order to be able to define the coherent state quantization.

Definition 8.1**.**

Suppose $f:B_{w}\to\mathbb{C}$ is a measurable function. We then define the coherent state quantization of $f$ to be the linear operator

[TABLE]

where $|\phi_{\lambda}\rangle\langle\phi_{\lambda}|$ is Dirac notation for the rank one projection operator given by $\psi\mapsto\langle\phi_{\lambda},\psi\rangle\phi_{\lambda}$ for all $\psi\in\mathcal{H}$. This is also called the frame quantization of $f$. The integral in (8.1) is understood to mean the unique linear operator $Q_{cs}f:\mathcal{H}\to\mathcal{H}$ (if it exists) which satisfies

[TABLE]

for all $\phi,\psi\in\mathcal{H}$, where the right hand side of (8.2) is the Lebesgue integral of a complex valued, integrable function.

Also, define $\mathcal{L}_{cs}^{1}(B_{w},\rho)$ to be the set of all those measurable functions $f:B_{w}\to\mathbb{C}$ for which the integral in (8.2) exists for all $\phi,\psi\in\mathcal{H}$, that is to say, its integrand is absolutely integrable.

Remark 8.2**.**

The function $\langle\phi_{\lambda},\psi\rangle$ (resp., $\langle\phi,\phi_{\lambda}\rangle$) of $\lambda\in D_{w}$ is anti-holomorphic (resp., holomorphic) and therefore is a measurable function. Thus, the integrand in (8.2) is a measurable function of $\lambda\in D_{w}$. Also, it is clear that $\mathcal{L}_{cs}^{1}(B_{w},\rho)$ is a complex vector space with respect to the standard point-wise definitions of sum and scalar product. Another immediate property is $Q_{cs}(f^{*})=(Q_{cs}(f))^{*}$, the adjoint operator of $Q_{cs}(f)$. And the resolution of the identity (4.1) tells us that $Q_{cs}1=I$, where $1$ denotes the constant function.

At this point let us recall some standard notations. We let $\mathcal{B}(\mathcal{H})$ denote the Banach space of all linear, bounded maps $A:\mathcal{H}\to\mathcal{H}$ where the norm $||A||$ of such an $A$ is its operator norm $||A||:=\sup\{||A\psi||~{}|~{}\psi\in\mathcal{H},\,||\psi||\leq 1\}$.

Proposition 8.1**.**

Suppose $f:B_{w}\to\mathbb{C}$ is integrable with respect to the measure $||\phi_{\lambda}||^{2}\,d\rho=d\rho(\lambda)\,||\phi_{\lambda}||^{2}$ on $B_{w}$, that is,

[TABLE]

Then $f\in\mathcal{L}_{cs}^{1}(B_{w},\rho)$ and $Q_{cs}f:\mathcal{H}\to\mathcal{H}$ is a bounded operator whose operator norm satisfies $||Q_{cs}f||\leq||f||_{1}$, the $L^{1}$-norm of $f$ in the $L^{1}$-space in (8.3). Therefore, Q_{cs}:L^{1}\big{(}B_{w},||\phi_{\lambda}||^{2}\,d\rho\big{)}\to\mathcal{B}(\mathcal{H}) is a bounded linear map of Banach spaces with operator norm $||Q_{cs}||\leq 1$.

Proof.

We prove the absolute integrability of the integral in (8.2) for all $\phi,\psi\in\mathcal{H}$ as follows:

[TABLE]

where we used the Cauchy-Schwarz inequality twice and the definition of $||f||_{1}$. Referring back to (8.2) we see that $|\langle\phi,(Q_{cs}f)\psi\rangle|\leq||f||_{1}\,||\phi||\,||\psi||$ for all $\phi,\psi\in\mathcal{H}$, from which the bound on the operator norm $||Q_{cs}f||\leq||f||_{1}$ follows directly. Then we see immediately that $||Q_{cs}||\leq 1$. ∎

Remark 8.3**.**

We will not elaborate on the standard details needed to extend this definition to (possibly unbounded) operators $Q_{cs}f$ for $f$ in other spaces, including spaces of distributions. The condition (8.3) implies that $f$ has some decay that cancels the divergence of the integral (4.5). In particular the constant function $f\equiv 1$ does not satisfy (8.3). Nonetheless, the resolution of the identity says that $Q_{cs}1=I$, the identity map. So the condition (8.3) is not necessary for $Q_{cs}f$ to be bounded; it only is sufficient. We leave finding a nice necessary and sufficient condition as an open problem

9. Upper and Lower Symbols

We now discuss a standard topic in the theory of coherent states. This dates back to the seminal works of Berezin in [6] and [7], Glauber in [12] and [13] and Lieb in [14].

Definition 9.1**.**

Let $A$ be a densely defined linear operator acting in the Hilbert space $\mathcal{H}$. Suppose that for each $\lambda\in B_{w}$ we have that $\phi_{\lambda}\in\mathrm{D}(A)$, the domain of $A$. Then the unnormalized lower symbol $A^{\sharp}$ of $A$ is defined for all $\lambda\in B_{w}$ by

[TABLE]

One also says that $A^{\sharp}$ is the unnormalized covariant symbol of $A$.

Similarly, the (normalized) lower symbol $A^{\flat}$ is defined by

[TABLE]

again for all $\lambda\in B_{w}$. (Recall that $\phi_{\lambda}\neq 0$ so that the denominator is non-zero.) Also one says $A^{\flat}$ is the (normalized) covariant symbol of $A$ or the Berezin symbol.

Remark 9.2**.**

Therefore the lower symbol $A^{\flat}:B_{w}\to\mathbb{C}$, that is, $A^{\flat}$ is a function on the phase space. Also, if $A$ is self-adjoint, which says that it represents a quantum observable, then $A^{\flat}:B_{w}\to\mathbb{R}$, which says that it represents a classical observable. If $A$ is a bounded operator, then $A^{\flat}$ is a bounded function satisfying $||A^{\flat}||_{\infty}\leq||A||$, where $||\cdot||_{\infty}$ is the $L^{\infty}$ norm of a bounded function. An elementary example of a lower symbol is given by $(T_{1})^{\flat}(\lambda)=I^{\flat}(\lambda)=1$, the constant function. In quantum theory quantization is a (certain!) way of passing from functions on phase space to operators acting in a Hilbert space. Since the operation $A\mapsto A^{\flat}$ is a mapping in the opposite direction (namely, from operators to functions on phase space), one sometimes refers to it as a dequantization. Analogous comments hold for the unnormalized lower symbol $A^{\sharp}$.

Also, the unnormalized lower symbol is related to the coherent state transform and the reproducing kernel by

[TABLE]

The following property for self-adjoint operators is well known. We wish merely to emphasize that it comes from a property for a wider class of operators. Before stating this result, we recall that the adjoint of a densely defined operator $A$ acting in a Hilbert space is denoted as $A^{*}$.

Proposition 9.1**.**

Let $A$ be an operator as in Definition 9.1. Suppose that for each $\lambda\in B_{w}$ we have that $\phi_{\lambda}\in\mathrm{D}(A^{*})$, the domain of $A^{*}$. (This latter hypothesis guarantees that $(A^{*})^{\sharp}$ and $(A^{*})^{\flat}$ are defined. For example, it holds for symmetric operators.) Then $(A^{*})^{\sharp}=(A^{\sharp})^{*}$ and $(A^{*})^{\flat}=(A^{\flat})^{*}$.

In particular, if $A$ is self-adjoint, then both $A^{\sharp}$ and $A^{\flat}$ are real-valued functions.

Proof.

For all $\lambda\in B_{w}$ we see that

[TABLE]

The rest of the proposition is now immediate. ∎

Now we give new terminology to something we already have seen.

Definition 9.3**.**

Suppose that $f:B_{w}\to\mathbb{C}$ is a function for which the coherent state quantization $Q_{cs}f$ exists. Then we say that $f$ is the upper (or contravariant) symbol for the operator $Q_{cs}f$.

Remark 9.4**.**

Notice that both the upper and lower symbols are complex valued functions defined on $B_{w}$, that is, they are classical observables if they happen to be real valued. Moreover, each is associated with a linear operator, that is, a quantum observable if it happens to be self-adjoint. Our presentation of upper symbols is not standard. Typically one starts with a linear operator $A$ and looks for a function $f$ such that $A=Q_{cs}f$, in which case one writes $f=\hat{A}$. However, such an $f$ may not exist and, if it does, it may not be unique.

Due to the presence of two quantization schemes here, we can ask some questions about their relationship. For example, what are the lower symbols of the creation and annihilation operators?

Theorem 9.1**.**

The normalized lower symbol of the annihilation operator $T_{\overline{\theta}}$ is given by $(T_{\overline{\theta}})^{\flat}(\lambda)=\lambda$ for every $\lambda\in B_{w}$, that is, the identity function on $B_{w}$.

Proof.

For all $\lambda\in B_{w}$ we see that the unnormalized lower symbol is

[TABLE]

Recall from (3.3) that $||\phi_{\lambda}||^{2}=\sum_{k=0}^{\infty}|\lambda|^{2k}|q|^{k(k+1)}w_{k}^{-1}$. Consequently, the lower symbol of the annihilation operator $T_{\overline{\theta}}$ is given by the quotient of these expressions, namely by $(T^{r}_{\overline{\theta}})^{\flat}(\lambda)=\lambda$ ∎

Remark 9.5**.**

Notice that the unnormalized lower symbol $(T_{\overline{\theta}})^{\sharp}$ depends on both $q$ and the weights $w_{k}$. Remarkably, the normalized lower symbol $(T_{\overline{\theta}})^{\flat}$ is independent of these parameters.

We now use another consequence of (2.2), namely that the creation operator $T_{\theta}$ is given by

[TABLE]

Unlike the annihilation operator $T_{\overline{\theta}}$ this formula does not depend on $q$. Surprisingly, this means that for $q\neq 1$ the operators $T_{\theta}$ and $T_{\overline{\theta}}$ are not adjoints of each other. But this is simply a consequence of our definitions. Equation (9.1) defines $T_{\theta}$ on the subspace $\mathcal{P}$, which is dense in $\mathcal{H}$ and which is invariant under the action of $T_{\theta}$.

We now have gathered enough information for the following calculation for the lower symbol of the creation operator. This is a formal calculation, since we do not concern ourselves with domain considerations. So, we start with the unnormalized lower symbol:

[TABLE]

For $q\neq 1$ there is no apparent simplification of this formula. Also, the formula for the normalized lower symbol is equally unattractive. This is why we decided not to give the technical details to make this calculation rigorous. Also, this shows how the Toeplitz operators which use left multiplication differ from the Toeplitz operators which use right multiplication, since in the latter case all factors of $q$ disappear and everything works out too easily. For example, for the Toeplitz operators using right multiplication, $T_{\theta}$ and $T_{\overline{\theta}}$ are adjoint of each other (on the appropriate domain), while that is false for the Toeplitz operators using left multiplication if $q\neq 1$.

What this suggests is that instead of $T_{\theta}$ we should consider the adjoint operator $(T_{\overline{\theta}})^{*}$ to be the appropriate creation operator for this setting, even though it might not be a Toeplitz operator. An elementary calculation shows that

[TABLE]

which has degree $+1$. Then we can extend $(T_{\overline{\theta}})^{*}$ to $\mathrm{D}(T_{\overline{\theta}})$ by the obvious formula. Then a simple modification of the previous formal calculation shows rigorously for all $\lambda\in B_{w}$ and $q\in\mathbb{R}$ that

[TABLE]

We have proved the next result.

Theorem 9.2**.**

Suppose that the parameter $q$ of the Manin plane is real. Then the lower symbol of $(T_{\overline{\theta}})^{*}$ is given by $(T_{\overline{\theta}})^{*\flat}(\lambda)=\lambda^{*}$ for all $\lambda\in B_{w}$.

10. Upper symbol of a Toeplitz operator

We now consider another relation between these two quantizations. We take a Toeplitz operator and ask whether it has the form $Q_{cs}f$ for some upper symbol $f$. If f\in L^{1}\big{(}B_{w},||\phi_{\lambda}||^{2}\,d\rho\big{)}, we have that $Q_{cs}f$ is bounded, and so $\phi_{n}$ is in its domain. When considering the more general case of an unbounded $Q_{cs}f$ we will only consider the case when $\phi_{n}$ is in its domain for all $n\geq 0$. We start off with a calculation for integers $k,n\geq 0$ of a matrix element,

[TABLE]

One question is whether this can give us the annihilation operator $T_{\overline{\theta}}$. But we know from (2.3) that the matrix elements of $T_{\overline{\theta}}$ for $n\geq 1$ are

[TABLE]

This formula is also valid for $n=0$ provided that we put $w_{-1}=1$, say. Two operators are equal on $\mathcal{P}$ if and only if their matrix elements are equal on $\mathcal{P}$, and so we see that $T_{\overline{\theta}}=Q_{cs}(f)$ on $\mathcal{P}$ if and only if the unknown upper symbol $f$ satisfies

[TABLE]

for all $k,n\geq 0$. Therefore the question reduces to whether there exists an upper symbol $f$ satisfying (10.2). We have the following partial answer.

Theorem 10.1**.**

Let $\rho$ be a radial measure that satisfies (4.6). Define $f:B_{w}\to\mathbb{C}$ by $f(\lambda)=\lambda$, the identity map. Suppose that $T_{\overline{\theta}}$ is a bounded operator. Then $T_{\overline{\theta}}=Q_{cs}f$, that is, the operator $T_{\overline{\theta}}$ has $f$ as an upper symbol.

Proof.

We evaluate the integral in (10.2) for this choice of $f$ as follows:

[TABLE]

where we used (4.6) in the next to last equality. And so (10.2) is proved, which shows that on $\mathcal{P}$ we have $T_{\overline{\theta}}=Q_{cs}f$, a bounded operator. So, $T_{\overline{\theta}}=Q_{cs}f$ on $\mathcal{H}$. ∎

We also have a partial answer for the creation operator $(T_{\overline{\theta}})^{*}$.

Corollary 10.1**.**

With the same hypotheses as in Theorem 10.1 there exists an upper symbol $g:B_{w}\to\mathbb{C}$ for $(T_{\overline{\theta}})^{*}$, which is a bounded operator. This means that $(T_{\overline{\theta}})^{*}=Q_{cs}g$.

Proof.

Again taking $f(\lambda)=\lambda$ we have $(T_{\overline{\theta}})^{*}=(Q_{cs}f)^{*}=Q_{cs}(f^{*}),$ where we used Theorem 10.1 in the first equality and a basic property of the coherent state quantization $Q_{cs}$ in the second equality. So we take $g(\lambda)=f^{*}(\lambda)=\lambda^{*}$. The boundedness of $(T_{\overline{\theta}})^{*}$ follows immediately from the hypothesis of Theorem 10.1. ∎

Remark 10.1**.**

The hypothesis in Theorem 10.1 that the measure $\rho$ is radial is quite restrictive, of course. We do not know nor venture to conjecture what happens if that hypothesis is dropped. We also leave as open problems whether the number operator $N$ or the creation operator $T_{\theta}$ has an upper symbol.

11. A Comparison

We will now make a comparison with a finite dimensional algebra called the paragrassmann algebra, whose coherent states and their corresponding quantization were introduced and studied in [5]. The reproducing kernel object and the Toeplitz operators for this non-commutative space were investigated in [18] and [19]. Coherent states were not introduced in [16], but in that setting they would have been defined analogously to the coherent states introduced in [5] as a finite sum, namely as a formal infinite sum in Dirac notation

[TABLE]

where $\{e_{n}~{}|~{}n\in\mathbb{N}\}$ is an orthonormal basis for an auxiliary Hilbert space. This corresponds to (3.2) if we put $\lambda=1$ there and take that also to be a formal infinite sum. But here we have what appears to be a better approach, since we do not fuss with making sense of formal infinite sums but rather use convergent infinite series. This also has the advantage of providing a naturally defined phase space arising from the quantum theory. However, in [5] there is no phase space presented and that is in accord with our approach here as we now discuss.

We start with a brief review of material in [19] where more details can be found. First, fix $q\in\mathbb{C}\setminus\{0\}$ and an integer $l\geq 2$. Then the paragrassmann algebra $PG_{l,q}$ is defined as the quotient of the Manin plane $\mathbb{C}Q_{q}(\theta,\overline{\theta})$ by putting $\theta^{l}=0$ and $\overline{\theta}\,^{l}=0$. Then using a finite sequence of weights $w_{n}>0$ for $0\leq n\leq l-1$ one introduces a sesqui-linear form by $\langle\theta^{i}\overline{\theta}\,\!^{j},\theta^{k}\overline{\theta}\,\!^{l}\rangle:=w_{i+l}\,\delta_{i-j,k-l}\chi_{l}(i+l),$ where $\chi_{l}(n)=1$ for $0\leq n\leq l-1$ and $\chi_{l}(n)=0$ for $n\geq l$. Here we have $i,j,k,l\in\{1,2,\dots,l-1\}$. (Technically, we also have to put $w_{n}=0$ for $n\geq l$.) Then this sesqui-linear form, when restricted to the finite-dimensional sub-algebra $\mathcal{F}$ of $PG_{l,q}$ generated by $\theta$, satisfies

[TABLE]

and so is a positive definite inner product, thereby making the sub-algebra $\mathcal{F}$ into a Hilbert space. There is an associated projection operator $P:PG_{l,q}\to PG_{l,q}$, given in Dirac notation as $P=\sum_{k=0}^{l-1}w_{k}^{-1}|\theta^{k}\rangle\langle\theta^{k}|$, whose range is $\mathcal{F}$. Then a Toeplitz operator $T_{g}$ with symbol $g\in PG_{l,q}$ is defined as expected: $T_{g}\,\phi:=P(\phi g)$ for all $\phi\in\mathcal{F}$. And it turns out that $T_{g}:\mathcal{F}\to\mathcal{F}$ is a linear map. All of this is reminiscent of the Manin quantum plane except for details having to do with the fact that the variables $\theta$ and $\overline{\theta}$ are nilpotents. And a similar argument (see [19]) also shows that the annihilation operator $T_{\overline{\theta}}$ satisfies

[TABLE]

But we use $\theta^{l}=0$ and $\theta^{l-1}\neq 0$ to see that $(T_{\overline{\theta}})^{l}=0$ and $(T_{\overline{\theta}})^{l-1}\neq 0$. So, $T_{\overline{\theta}}$ is a nilpotent operator of nilpotency $l$ acting on the Hilbert space $\mathcal{F}$ whose dimension is $l$. In fact (11.1) shows that $T_{\overline{\theta}}$ is equivalent to one $l\times l$ Jordan block with zeros along the diagonal. And its spectrum is $\mathrm{Spec}(T_{\overline{\theta}})=\{0\}$, with $\theta^{0}=1$ being an eigenvector of multiplicity one. Also, there are no other eigenvectors. So the phase space is the one-point set $\{0\}$, a truly trivial situation. And $\phi_{\lambda=0}=1$ is the only coherent state for the paragrassmann algebra, again a trivial situation.

So our approach gives rather curious results for the paragrassmann algebra $PG_{l,q}$ in contrast to the more conventional example of the Manin plane. We wish to emphasize that the paragrassmann algebra $PG_{l,q}$ for $q\neq 1$ is a non-commutative quantum theory whose phase space is trivial. Moreover, the Hilbert space $\mathcal{F}$ is not spanned by the coherent states, but far from it. According to Definition 3.3 the paragrassmann algebra $PG_{l,q}$ is an extreme quantum theory. On the other hand, $PG_{l,q}$ (even for the commutative case $q=1$) is a quantum space with only one point, that is, with exactly one unital algebra morphism $\alpha:PG_{l,q}\to\mathbb{C}$, since the nilpotency conditions force $\alpha(\theta)=\alpha(\overline{\theta})=0$.

12. Concluding Remarks

Here we have introduced the coherent states associated with just one Toeplitz annihilation operator. Rather analogous results should hold in settings where there is a commuting family of Toeplitz annihilation operators whose coherent states are defined as their common eigenvectors and whose phase space consists of their ordered $n$-tples of eigenvalues. A more interesting situation arises if one is dealing with a family of non-commuting Toeplitz annihilation operators and their coherent states. Other possibilities for further research on this topic include studying the semi-classical limit and the minimal uncertainty of the coherent states. Also, the role of the Manin plane can be played by other non-commutative planes, and we will consider that topic in a forthcoming paper.