ABSTRACT

After developing the mathematical means for the correspondence of classical phase-space function to quantum-mechanical operators with symmetrical ordering of the basic canonical operators in the sense of Weyl the approach is applied to an infinite series of classical monomial functions of the canonical variables. These include as well as pure powers of the amplitude ${\left|\alpha \right|}^k\mathrm{,}\left(k=\mathrm{0,1,2,}\cdots \right)$ as also basic periodic functions ${\text{e}}^{\text{i}l\phi }\mathrm{,}\left(l=\pm \mathrm{1,}\pm \mathrm{2,}\cdots \right)$ of the phase $\phi$ with their quantum-mechanical correspondence. In the representation by number states $|n\rangle \mathrm{,}\left(n=\mathrm{0,1,}\cdots \right)$ , all the considered operators involve the Jacobi polynomials as the essential formative element. Whereas the quantity $\overline{\mathcal{N}\left\{a^{†2}{a}^2\right\}}-{\overline{\mathcal{N}\left\{a^{†}a\right\}}}^2$ in normal ordering due to its indeterminacy leads to the introduction of the notions of sub- and super-Poissonian statistics the analogous quantity in (Weyl) symmetrical ordering $\overline{\mathcal{S}\left\{a^{†2}{a}^2\right\}}-{\overline{\mathcal{S}\left\{a^{†}a\right\}}}^2$ is positive definite and satisfies an inequality. The notions of sub- and super-Poissonian statistics are problematic when they are used for the definition of nonclassicality of states since the mentioned measure in normal ordering does not determine the Poisson statistics in their middle in unique way but determines only a large set of statistics which may be very far in the sense of the Hilbert-Schmidt distance from a Poisson statistics that is discussed.

Keywords:

Wigner Quasiprobability, Symmetrical (Weyl) Ordering, Nonclassicality of Steates, Distance of States, Sub- and Super-Poissonian Statistics, Phase Operator, Laguerre 2D Polynomials, Jacobi Polynomials

1. Introduction

The basic part of quantum mechanics originated from classical Hamilton mechanics and was developed in the twenties and thirties of last century. The canonical variables $\left(q\mathrm{,}p\right)$ of the phase space in the Hamilton function $H\left(q\mathrm{,}p\right)$ were substituted by operators $\left(Q\mathrm{,}P\right)$ which obey the commutation relations $\left[Q\mathrm{,}P\right]=\text{i}\hslash I$ , ( $\hslash$ Planck’s action quantum $h$ divided by $2\text{π}$ and $I$ identity operator of the Hilbert space of the representation of the operator algebra). In general, a classical function $A\left(q\mathrm{,}p\right)$ of the canonical variables $\left(q\mathrm{,}p\right)$ cannot be translated into a corresponding quantum-mechanical function $A\left(Q\mathrm{,}P\right)$ without additional rules for operator ordering since $QP\ne PQ$ . Luckily, this did not play a role in the first very successful applications of the Schrödinger equation with a Hamilton operator $H$ with additively separated classical kinetic energy $T\left(p\right)\to T\left(P\right)$ from potential energy $U\left(q\right)\to U\left(Q\right)$ with $q\equiv r$ the position vector and $p$ the (canonical) momentum vector. Thus it was translated into the quantum-mechanical Hamilton operator $H=T\left(P\right)+U\left(Q\right)$ where $\left(q,p\right)$ in classical mechanics are, in general, three-dimensional vectors and $\left(Q,P\right)$ corresponding vector operators with independent components and independent commutation relations $\left[Q_j\mathrm{,}{P}_k\right]=\text{i}\hslash {\delta }_{j\mathrm{,}k}$ . Luckily also, this did not play a role in the translation of the classical angular momentum $L\left(q,p\right)\equiv \left[q,p\right]$ (vector product of $q$ with $p$ ) or $L_i\left(q,p\right)\equiv {\epsilon }_{ijk}{q}_j{p}_k$ into a quantum-me- chanical operator since there are only combined independent (commutating) components of $Q_j$ and $P_k\mathrm{,}\left(k\ne j\right)$ , in the quantum-mechanical operator $L_i={\epsilon }_{ijk}{Q}_j{P}_k$ . In quantum field theories such as quantum electrodynamics and optics where one usually speaks about independent modes the canonical variables $\left(q\mathrm{,}p\right)\to \left(Q\mathrm{,}P\right)$ are internal field variables whereas the position $r$ and the time $t$ are classical variables as parameters of each mode determining the shape of the field and $p$ is not the momentum of the field, for example, of a wave packet or beam¹. In present paper, however, we mainly use the transition from canonical coordinates $\left(q\mathrm{,}p\right)$ to complex coordinates $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ and from canonical operators $\left(Q\mathrm{,}P\right)$ to annihilation and creation operators $\left(a\mathrm{,}{a}^{†}\right)$ , correspondingly.

¹Therefore, in quantum optics it is usually unfavorable to denote the canonical variables of the phase space by $\left(x\mathrm{,}p\right)$ that may lead to confusion with spatial variables. Energy and momentum of a light-wave packet are connected with quadratic combinations of the canonical variables $q$ and $p$ involving frequency and wave vector as parameters.

Hermann Weyl in [1] and in his book [2] from 1928 (chap. IV, §14) proposed a general rule for the translation of arbitrary functions $A\left(q,p\right)$ of the canonical phase-space variables in a unique way into quantum-mechanical operator-ordered functions $A\left(Q,P\right)$ of the operators $\left(Q,P\right)$ . His way was via the Fourier transform of the classical function $A\left(q,p\right)$ (denoted there in the way $f\left(p\mathrm{,}q\right)={\displaystyle \int }{\displaystyle {\int }_{-\infty }^{+\infty }}e\left(\sigma p+\tau q\right)\xi \left(\sigma \mathrm{,}\tau \right)\text{d}\sigma \text{d}\tau$ where $e\left(x\right)\equiv {\text{e}}^{\text{i}x}$ and $\xi \left(\sigma \mathrm{,}\tau \right)$ is the Fourier transform of $f\left(p\mathrm{,}q\right)$ and $\left(p\mathrm{,}q\right)$ are here operators). On the opposite our preliminarily written “function” $A\left(Q,P\right)$ is not well and uniquely defined without an ordering rule. The form proposed by Weyl is called the (Weyl) symmetrical ordering and we denote it by $\mathcal{S}\left\{A\left(Q,P\right)\right\}$ . In this symmetrical ordering of the operators $\left(Q,P\right)$ in a function $A\left(Q,P\right)$ we may consider $\left(Q,P\right)$ first as the classical variables $\left(q,p\right)$ with respect, for example, of Taylor series expansions and may order then their sum terms in corres- ponding way but we can make this also in the form of the Fourier integral as a whole. Our way of definition of symmetrical ordering (in the sense of Weyl) in the following is fully equivalent to that of Weyl but looks only a little different at the first glance. We explain this later in detail. To a classical function $A\left(q,p\right)$ over the phase space corresponds then a uniquely defined operator $A\equiv \mathcal{S}\left\{A\left(Q,P\right)\right\}$ . The calculation of expectation values of symmetrically ordered operators is best suited to the Wigner function $W\left(q\mathrm{,}p\right)$ which is a quasiprobability over the phase space introduced in 1932 by Eugen P. Wigner [3] (republished in [4] ). Together with the Weyl ordering this is often called the Weyl-Wigner formalism of correspondence between classical and quantum mechanics [5] . Other corres- pondences of classical phase-space functions to quantum-mechanical operators are possible, in particular, the normally-ordered correspondence for which another quasiprobability $P\left(q\mathrm{,}p\right)$ called the Glauber-Sudarshan quasipro- bability [6] [7] [8] [9] is best suited for the calculation of expectation values of such operators if we know only the corresponding classical phase-space function. For anti-normally-ordered operators, the Husimi-Kano quasiprobability $Q\left(q\mathrm{,}p\right)$ takes on this place. Other quasiprobabilities for the calculation of the expecta- tion values of arbitrarily ordered operators are also appropriate, however with more complicated formulae in this case.

The symmetrical ordering in the sense of Weyl possesses the highest degeneracy of the operator kernel in the integral transform defining it and every change of this kernel removes this degeneracy in different possible directions [10] . From the theoretical point of view the symmetrical ordering is the most aesthetical and attractive one but does nature also prefer it? The zero-point energy of the modes in quantum optics and its consequences, for example, in the theoretically derived and experimentally observed (?) Casimir effect gives some evidence that the symmetrical ordering of operators is, at least, in quantum optics likely the correct correspondence between classical and quantum physics. An early and well organized representation of many problems concerning the different quasiprobabilities and ordering used in quantum optics is given by Peřina [11] . Problems of the determination of a phase operator in quantum optics are discussed and referred in detail by Peřinová, Lukš and Peřina [12] .

There are some technical difficulties to implement the explicit calculation of the symmetrically ordered operators corresponding to given classical phase-space functions in general cases, in particular, in the Fock or number state representation. A basic result for operators of the form

$\mathcal{S}\left\{{\left(\sqrt{a{a}^{†}}\right)}^k{\left(\sqrt{\frac{a}{a^{†}}}\right)}^l\right\}\equiv \mathcal{S}\left\{a^m{a}^{†n}\right\}$ for integers $\left(k\mathrm{,}l\right)$ , $\left(k=m+n,l=m-n\right)$ ,

was communicated in [13] with some promise to give its detailed derivation in another paper. We discuss this in present paper but the more technical details of this calculation we shift to the Appendices. We connect the results with other already known more special results. In particular, we consider from Section 6 on a set of classical basic functions $A^{\left(k\mathrm{,}l\right)}\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ and determine the corresponding basic operators $A^{\left(k\mathrm{,}l\right)}$ in number representation using the Jacobi polynomials for the representation of the coefficients. In Section 8 we generalize this to smoothing of the operators by means of normalized Gaussian bell functions. The results for the special case $A^{\left(k\mathrm{,0}\right)}$ can be also obtained by integration of the general Wigner quasiprobability over the angle and the special case $A^{\left(\mathrm{0,}l\right)}$ by integration of the Wigner quasiprobability over the radius that for this last case was first made by Garraway and Knight [14] (see also [12] ). One has to liberate oneself in these cases from the general density operator $\rho$ in the Wigner quasiprobability $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ in complex representation and obtain then in last case the quantum equivalents $A^{\left(\mathrm{0,}l\right)}$ to the basic classical periodic phase functions ${\text{e}}^{\text{i}l\phi }\mathrm{,}\left(l=\mathrm{0,}\pm \mathrm{1,}\pm \mathrm{2,}\cdots \right)$ .

In Section 12 we derive the connection of the symmetrically ordered operators $\mathcal{S}\left\{a^{†k}{a}^k\right\}$ to powers of the number operator $N\equiv a^{†}a$ . This suggests to use the expectation values (notation by overlining operators) $\overline{\mathcal{S}\left\{a^{†2}{a}^2\right\}}-{\overline{\mathcal{S}\left\{a^{†}a\right\}}}^2$ which are positive definite as alternative to the corresponding normally ordered quantity $\overline{\mathcal{N}\left\{a^{†2}{a}^2\right\}}-{\overline{\mathcal{N}\left\{a^{†}a\right\}}}^2$ which is indefinite and leads in dependence of its negativity or positivity to the definition of sub- and super-Poissonian quantum statistics which are problematic when they are used for the definition of non-classicality of states.

2. Basic Notions and Displacement Operator

Note: The trace of an operator $A$ is denoted by $\langle A\rangle$ and the expectation value of $A$ by overlining the operator $\bar{A}\equiv \langle \varrho A\rangle$ if $\varrho$ is the density operator.

In this Section, we prepare the description of the symmetrical (Weyl) correspondence of classical to quantum mechanics by some, in principle known, basic notions and explain our notations. We consider a Hamilton system of one degree of freedom in canonical variables $\left(q\mathrm{,}p\right)$ . Additionally, we introduce complex variables $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ in the following way

$\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\frac{1}{\sqrt{2\hslash }}\left(q+\text{i}p\mathrm{,}q-\text{i}p\right)\mathrm{,}\iff \left(q\mathrm{,}p\right)=\sqrt{2\hslash }\left(\frac{\alpha +{\alpha }^{\mathrm{<mo>\; *\; </mo>}}}{2}\mathrm{,}-\text{i}\frac{\alpha -{\alpha }^{\mathrm{<mo>\; *\; </mo>}}}{2}\right)$

$\left(\frac{\partial }{\partial \alpha }\mathrm{,}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)=\sqrt{2\hslash }\left(\frac{1}{2}\left(\frac{\partial }{\partial q}-\text{i}\frac{\partial }{\partial p}\right)\mathrm{,}\frac{1}{2}\left(\frac{\partial }{\partial q}+\text{i}\frac{\partial }{\partial p}\right)\right)\mathrm{,}\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}=\frac{\text{d}q\wedge \text{d}p}{2\hslash }$ (2.1)

with correspondence to the basic quantum-mechanical operators $\left(q\mathrm{,}p\right)\iff \left(Q\mathrm{,}P\right)$ and their combinations $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\iff \left(a\mathrm{,}{a}^{†}\right)$ which become the annihilation $a$ and creation $a^{†}$ operator for a harmonic oscillator

$\left(a\mathrm{,}{a}^{†}\right)=\frac{1}{\sqrt{2\hslash }}\left(Q+\text{i}P\mathrm{,}Q-\text{i}P\right)\iff \left(Q\mathrm{,}P\right)=\sqrt{2\hslash }\left(\frac{a+a^{†}}{2}\mathrm{,}-\text{i}\frac{a-a^{†}}{2}\right)$ (2.2)

They obey the commutation relations ( $I$ identity operator of the represen- tation space (Hilbert space))

$\left[Q\mathrm{,}P\right]=\text{i}\hslash I\mathrm{,}\left[a\mathrm{,}{a}^{†}\right]=I$ (2.3)

The reason that we introduce the complex substitutes of the canonical variables by (2.1) and not simply by $\left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)=\left(q+\text{i}p\mathrm{,}q-\text{i}p\right)$ is that we are interested in the following mainly in the expectation values of ordered functions of $\left(a\mathrm{,}{a}^{†}\right)$ which are (boson) annihilation and creation operators for harmonic oscillators of modes of the electromagnetic field. However, in transitions from quantum to classical mechanics and optics setting $\hslash =0$ one has to be cautious

using the operators $\left(a\mathrm{,}{a}^{†}\right)$ . For example, in the operator $N\equiv a^{†}a=\frac{Q^2+P^2}{2\hslash }-\frac{1}{2}I$

one cannot set $\hslash =0$ since N (number operator) does not possess a classical analogue and only after multiplication of this operator with $\hslash$ we get an

operator with a classical analogue in form of the function $\frac{q^2+p^2}{2}=\hslash \alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}$ proportional to the intensity or to energy and momentum of the field.

Starting from the well-defined operator function $exp\left(\mu a+\nu a^{†}\right)$ with arbitrary parameters $\left(\mu \mathrm{,}\nu \right)$ the symmetrical Weyl ordering can be defined by (e.g., [15] )

$exp\left(\mu a+\nu a^{†}\right)={\displaystyle \underset{j=0}{\overset{\infty }{\sum }}}\frac{{\left(\mu a+\nu a^{†}\right)}^j}{j\mathrm{<mo>\; !\; </mo>}}\equiv {\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{l=0}{\overset{\infty }{\sum }}}\frac{{\mu }^k{\nu }^l}{k\mathrm{<mo>\; !\; </mo>}l\mathrm{<mo>\; !\; </mo>}}\mathcal{S}\left\{a^k{a}^{†l}\right\}\equiv \mathcal{S}\left\{exp\left(\mu a+\nu a^{†}\right)\right\}$

${\left(\mu a+\nu a^{†}\right)}^j={\displaystyle \underset{k=0}{\overset{j}{\sum }}}\frac{j\mathrm{<mo>\; !\; </mo>}}{k\mathrm{<mo>\; !\; </mo>}\left(j-k\right)\mathrm{<mo>\; !\; </mo>}}{\mu }^k{\nu }^{j-k}\mathcal{S}\left\{a^k{a}^{†j-k}\right\}$ (2.4)

From this follows for integer $\left(k\mathrm{,}l\right)$

$\mathcal{S}\left\{a^k{a}^{†l}\right\}={\left\{\frac{{\partial }^{k+l}}{\partial {\mu }^k\partial {\nu }^l}exp\left(\mu a+\nu a^{†}\right)\right\}}_{\mu =\nu =0}$ (2.5)

or using the binomial formula for ${\left(\mu a+\nu a^{†}\right)}^{k+l}$ with observation of the non-commutativity of $a$ and $a^{†}$

$\mathcal{S}\left\{a^k{a}^{†l}\right\}=\frac{1}{\left(k+l\right)\mathrm{<mo>\; !\; </mo>}}\frac{{\partial }^{k+l}}{\partial {\mu }^k{\nu }^l}{\left(\mu a+\nu a^{†}\right)}^{k+l}$ (2.6)

²Some authors write the corresponding classical variables within an ordering symbol (here symbol $\mathcal{S}\{\dots \}$ ) that means, e.g., $\mathcal{S}\left\{A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right\}$ instead of our $\mathcal{S}\left\{A\left(a\mathrm{,}{a}^{†}\right)\right\}$ .

The symbol $\mathcal{S}\{\dots \}$ is not a linear operator since²

$\mathcal{S}\left\{a^k{a}^{†l}\right\}=\mathcal{S}\left\{a^{†l}{a}^k\right\}=\mathcal{S}\left\{\mathcal{P}\left(a^k{a}^{†l}\right)\right\}$ (2.7)

where $\mathcal{P}\left(a^k{a}^{†l}\right)$ means here the product of k operators a and $l$ operators $a^{†}$ in arbitrary order (i.e., permutations) but linear combinations of them are understood in the sense of the distributive law

$\mathcal{S}\left\{\kappa A\left(a\mathrm{,}{a}^{†}\right)+\lambda B\left(a\mathrm{,}{a}^{†}\right)\right\}=\kappa \mathcal{S}\left\{A\left(a\mathrm{,}{a}^{†}\right)\right\}+\lambda \mathcal{S}\left\{B\left(a\mathrm{,}{a}^{†}\right)\right\}$ (2.8)

This gives the possibility to determine linear spaces of ordered operators if one introduces a system of basis operators. Clearly, two operators $\mathcal{S}\left\{A\left(a\mathrm{,}{a}^{†}\right)\right\}$ and $\mathcal{S}\left\{B\left(a\mathrm{,}{a}^{†}\right)\right\}$ are, in general, noncommutative.

In the following we use the displacement operator $D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ defined by (e.g., [7] [8] [11] )

$\begin{array}{c}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=exp\left(\alpha a^{†}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}a\right)\equiv \mathcal{S}\left\{exp\left(\alpha a^{†}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}a\right)\right\}\\ =exp\left(-\frac{\alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}{2}\right)exp\left(\alpha a^{†}\right)exp\left(-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}a\right)\\ \equiv exp\left(-\frac{\alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}{2}\right)\mathcal{N}\left\{exp\left(\alpha a^{†}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}a\right)\right\}\end{array}$ (2.9)

where the symbol $\mathcal{N}\{\dots \}$ means normal ordering of the content in braces (all powers of $a^{†}$ in front of powers of $a$ ). From (2.9) follows for its Hermitean adjoint operator

${\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}=exp\left({\alpha }^{\mathrm{<mo>\; *\; </mo>}}a-\alpha a^{†}\right)={\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{-1}=D\left(-\alpha \mathrm{,}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ (2.10)

To obtain the normally ordered form of the displacement operator on the right-hand side of (2.9) we applied here the well-known theorem (e.g., [7] )

$exp\left(A\right)exp\left(B\right)=exp\left(A+B+\frac{1}{2}\left[A\mathrm{,}B\right]\right)\mathrm{,}\text{if}\left[A\mathrm{,}\left[A\mathrm{,}B\right]\right]=\left[B\mathrm{,}\left[A\mathrm{,}B\right]\right]=0$ (2.11)

³Louisell gives two proofs, the first using the diagonal matrix elements with coherent states and the second by a differential equation. One may add a further easy one using the matrix elements in the basis of number states $|n\rangle ,\left(n=0,1,\cdots \right)$ as the eigenstates of the number operator $N\equiv a^{†}a$ , with $N|n\rangle =n|n\rangle$ , $a|n\rangle =\sqrt{n}|n-1\rangle$ , $a^{†}|n\rangle =\sqrt{n+1}|n+1\rangle$ and the completeness relation ${\displaystyle {\sum }_{n=0}^{\infty }}|n\rangle \langle n|=I$
$\begin{array}{l}\mathcal{N}\left\{exp\left(\left({\text{e}}^{\kappa }-1\right)a^{†}a\right)\right\}\equiv {\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}\frac{{\left({\text{e}}^{\kappa }-1\right)}^k}{k\mathrm{<mo>\; !\; </mo>}}{a}^{†k}\left({\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}|n\rangle \langle n|\right)a^k\end{array}$ $={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}\frac{{\left({\text{e}}^{\kappa }-1\right)}^k}{k\mathrm{<mo>\; !\; </mo>}}{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{\left(\sqrt{\frac{\left(n+k\right)\mathrm{<mo>\; !\; </mo>}}{n\mathrm{<mo>\; !\; </mo>}}}\right)}^{\text{}2}|n+k\rangle \langle n+k|={\displaystyle \underset{n^{\prime }=0}{\overset{\infty }{\sum }}}\left({\displaystyle \underset{k=0}{\overset{n^{\prime }}{\sum }}}\frac{n^{\prime }\mathrm{<mo>\; !\; </mo>}}{k\mathrm{<mo>\; !\; </mo>}\left(n^{\prime }-k\right)\mathrm{<mo>\; !\; </mo>}}{\left({\text{e}}^{\kappa }-1\right)}^k\right)|n^{\prime }\rangle \langle n^{\prime }|$ $={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{\left({\text{e}}^{\kappa }-1+1\right)}^n|n\rangle \langle n|={\text{e}}^{\kappa N}$ .

which is true for arbitrary operators $A$ and $B$ which commute with their commutator $\left[A\mathrm{,}B\right]$ . This is a special case of the general Baker-Campbell-Hausdorff-Dynkin formula for the product of the exponentials of two operators which for the case in (2.11) can be proved more directly, e.g., [7] . In the following we apply this theorem repeatedly.

The displacement operator possesses the property

$D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)a{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}=a-\alpha I\mathrm{,}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)a^{†}{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}=a^{†}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}I$ (2.12)

For the product of two displacement operators one finds applying (2.11)

$D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)D\left(\beta \mathrm{,}{\beta }^{\mathrm{<mo>\; *\; </mo>}}\right)=exp\left(\frac{1}{2}\left(\alpha {\beta }^{\mathrm{<mo>\; *\; </mo>}}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\beta \right)\right)D\left(\alpha +\beta \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}+{\beta }^{\mathrm{<mo>\; *\; </mo>}}\right)$ (2.13)

Furthermore, we need the normal ordering of operators of the form $exp\left(\kappa a^{†}a\right)$ with parameter $\kappa$ . The following relation is well known, e.g., [16] (chap. 3.3., pp. 156/157)³

$exp\left(\kappa N\right)=exp\left(\kappa a^{†}a\right)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}\frac{{\left({\text{e}}^{\kappa }-1\right)}^k}{k\mathrm{<mo>\; !\; </mo>}}{a}^{†k}{a}^k\equiv \mathcal{N}\left\{exp\left(\left({\text{e}}^{\kappa }-1\right)a^{†}a\right)\right\}$ (2.14)

The conversion of (2.14) with respect to the parameter $\kappa$ as it is easily seen is

$\mathcal{N}\left\{exp\left(\lambda a^{†}a\right)\right\}\equiv {\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}\frac{{\lambda }^k}{k\mathrm{<mo>\; !\; </mo>}}{a}^{†k}{a}^k={\left(1+\lambda \right)}^{a^{†}a}={\left(1+\lambda \right)}^N$ (2.15)

For $\lambda =1,0,-1,-2$ one obtains the operators

$\mathcal{N}\left\{exp\left(a^{†}a\right)\right\}=2^N\mathrm{,}\mathcal{N}\left\{exp\left(0a^{†}a\right)\right\}=\mathcal{N}\left\{I\right\}=I$

$\mathcal{N}\left\{exp\left(-a^{†}a\right)\right\}=\underset{\epsilon \to 0}{lim}{\epsilon }^N=|0\rangle \langle 0|\mathrm{,}\mathcal{N}\left\{exp\left(-2a^{†}a\right)\right\}={\left(-1\right)}^N=\Pi$ (2.16)

All these special operators play a certain role in quantum optics, in particular, $\mathcal{N}\left\{exp\left(0a^{†}a\right)\right\}$ is the unity operator $I$ , $\mathcal{N}\left\{exp\left(-a^{†}a\right)\right\}$ is the vacuum-state operator $|0\rangle \langle 0|$ and $\mathcal{N}\left\{exp\left(-2a^{†}a\right)\right\}$ the parity operator $\Pi$ which we consider more in detail in next Section.

3. Basic Relations of the Weyl Formalism and the Parity Operator

The general formula for the transition from arbitrary classical phase-space functions $A\left(q\mathrm{,}p\right)$ in the symmetrical Weyl ordering to quantum-mechanical operators $A\equiv \mathcal{S}\left\{A\left(Q\mathrm{,}P\right)\right\}$ can be written

$A\left(q\mathrm{,}p\right)\to A={\displaystyle \int }\text{d}q\wedge \text{d}pA\left(q\mathrm{,}p\right)exp\left(-Q\frac{\partial }{\partial q}-P\frac{\partial }{\partial p}\right)\delta \left(q\right)\delta \left(p\right)$ (3.1)

or in complex representation by complex variables $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ (see (2.1)) as follows (we do not use a new function symbol and set $A\left(q\mathrm{,}p\right)\equiv A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ )

$\begin{array}{l}A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\to A={\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)exp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ ={\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\mathcal{S}\left\{\delta \left(\alpha I-a\mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}I-a^{†}\right)\right\}\\ \equiv \mathcal{S}\left\{A\left(a\mathrm{,}{a}^{†}\right)\right\}\end{array}$ (3.2)

where $\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ is the two-dimensional delta function in complex representa- tion according to

${\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)f\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=f\left(\mathrm{0,0}\right)\mathrm{,}\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=2\hslash \delta \left(q\right)\delta \left(p\right)$ (3.3)

The integration goes in both representations over the whole phase-plane in real or complex coordinates $\left(q\mathrm{,}p\right)\leftrightarrow \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ . After partial integration in the first line we find equivalently to (3.2)

$\begin{array}{c}A={\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)exp\left(a\frac{\partial }{\partial \alpha }+a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ ={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{l=0}{\overset{\infty }{\sum }}}\frac{\mathcal{S}\left\{a^k{a}^{†l}\right\}}{k\mathrm{<mo>\; !\; </mo>}l\mathrm{<mo>\; !\; </mo>}}{\left\{\frac{{\partial }^{k+l}}{\partial {\alpha }^k\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}l}}A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right\}}_{\alpha ={\alpha }^{\mathrm{<mo>\; *\; </mo>}}=0}\end{array}$ (3.4)

For the trace $\langle A\rangle$ of the operator $A$ we find from (3.2) or (3.7)

$\langle A\rangle =\frac{1}{\text{π}}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\mathrm{,}\left(\langle A\rangle \equiv \text{Trace}\left(A\right)\right)$ (3.5)

The factor $\frac{1}{\text{π}}$ in (3.5) in front of the integral ${\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ is

already an indication that non-orthogonal (“overlapping”) states are involved in the definitions (these are the displaced number states; see below). The form of the classical-quantum correspondence in the second line of (3.2) is very near to the form given by Weyl if we make in addition the Fourier transformation of $\mathcal{S}\left\{\delta \left(\alpha I-a\mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}I-a^{†}\right)\right\}$ but our form has the advantage that we do not have to discuss the exact form of this transform (i.e., coefficient in front and factors in the exponent).

The inversion of (3.1) is (we denote the trace of an arbitrary operator $B$ by $\langle B\rangle$ )

$A\left(q\mathrm{,}p\right)=2\text{π}\hslash \langle Aexp\left(-Q\frac{\partial }{\partial q}-P\frac{\partial }{\partial p}\right)\rangle \delta \left(q\right)\delta \left(p\right)$ (3.6)

or of the complex form (3.2)

$A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\text{π}\langle Aexp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)\rangle \delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ (3.7)

The transformation $A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\to A$ in (3.2) together with the inverse transformation $A\to A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ in (3.7) is a mapping which preserves the distribution law for arbitrary complex numbers $\mu$ and $\nu$

$A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\leftrightarrow A\mathrm{,}B\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\leftrightarrow B\iff \mu A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)+\nu B\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\leftrightarrow \mu A+\nu B$ (3.8)

due to the linearity of the transformation.

We now make the normal ordering of the operator which plays a role in (3.2) in the transition from classical phase-space functions $A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ to quantum- mechanical operators $A$ which are symmetrically ordered equivalents in the sense of Weyl

$\begin{array}{l}exp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ =exp\left(-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)exp\left(-a\frac{\partial }{\partial \alpha }\right)exp\left(\frac{1}{2}\frac{{\partial }^2}{\partial \alpha \partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ =exp\left(-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)exp\left(-a\frac{\partial }{\partial \alpha }\right)\frac{2}{\text{π}}exp\left(-2\alpha {\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ =\frac{2}{\text{π}}\mathcal{N}\left\{exp\left(-2\left(a-\alpha I\right)\left(a^{†}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}I\right)\right)\right\}\\ =\frac{2}{\text{π}}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\mathcal{N}\left\{exp\left(-2a{a}^{†}\right)\right\}{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}\\ =\frac{2}{\text{π}}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right){\left(-1\right)}^{a^{†}a}{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}\end{array}$ (3.9)

In the derivation we used the identity

$exp\left(\frac{r}{2}\frac{{\partial }^2}{\partial z\partial z^{\mathrm{<mo>\; *\; </mo>}}}\right)\delta \left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)=\frac{2}{\text{π}r}exp\left(-\frac{2z{z}^{\mathrm{<mo>\; *\; </mo>}}}{r}\right)$ (3.10)

specialized to $r=1$ , which generally can be proved by two-dimensional Fourier transformation. The operator $exp\left(\frac{r}{2}\frac{{\partial }^2}{\partial z\partial z^{\mathrm{<mo>\; *\; </mo>}}}\right)$ with positive values of the para- meter $r$ applied to a function $f\left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)$ makes a smoothing of this function.

The operator $\Pi$ defined by (see also (2.15) and (2.16) for the different representations)

$\Pi \equiv {\left(-1\right)}^{a^{†}a}={\left(-1\right)}^N=exp\left(\pm \text{iπ}N\right)=\mathcal{N}\left\{exp\left(-2a{a}^{†}\right)\right\}={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}\frac{{\left(-2\right)}^k}{k\mathrm{<mo>\; !\; </mo>}}{a}^{†k}{a}^k$ (3.11)

is called the parity operator and the operator $\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ defined by

$\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\equiv D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi {\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}$ (3.12)

the displaced parity operator, correspondingly. Thus we obtained in (3.9) the basic relation

$\begin{array}{c}\frac{2}{\text{π}}\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=exp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ ={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{l=0}{\overset{\infty }{\sum }}}\frac{{\left(-1\right)}^{k+l}}{k!l!}\mathcal{S}\left\{a^k{a}^{†l}\right\}\frac{{\partial }^{k+l}}{\partial {\alpha }^k\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}l}}\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\end{array}$ (3.13)

which plays a role in the following. We see here that the limiting transition $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\to \left(\mathrm{0,0}\right)$ on the right-hand side in (3.13) is not possible in this way and the representation of the parity operator $\Pi =\Pi \left(\alpha \to \mathrm{0,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\to 0\right)$ is not locally possible and the representation of $\Pi$ by the symmetrically ordered operators $\mathcal{S}\left\{a^k{a}^{†l}\right\}$ does not exist in contrast to the representation by the normally ordered operators $\mathcal{N}\left\{a^k{a}^{†l}\right\}=a^{†k}{a}^l$ (see (3.11)).

The parity operator $\Pi$ is a Hermitean and at once an idempotent operator (squared it is equal to the identity operator)

$\Pi ={\Pi }^{†}={\Pi }^{-1},{\Pi }^2=I,$ (3.14)

with the following interesting properties of transformation of $\left(a\mathrm{,}{a}^{†}\right)$ justifying its name

$\begin{array}{l}\Pi a{\Pi }^{†}={\text{e}}^{\text{iπ}N}a{\text{e}}^{-\text{iπ}N}=a+\frac{\text{iπ}}{\mathrm{1\; <mo>\; !\; </mo>}}\underset{=-a}{\underset{︸}{\left[N\mathrm{,}a\right]}}+\frac{{\left(\text{iπ}\right)}^2}{\mathrm{2\; <mo>\; !\; </mo>}}\underset{=+a}{\underset{︸}{\left[N\mathrm{,}\left[N\mathrm{,}a\right]\right]}}+\cdots =\left({\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}\frac{{\left(-\text{iπ}\right)}^n}{n\mathrm{<mo>\; !\; </mo>}}\right)a\\ ={\text{e}}^{-\text{iπ}}a=-a\mathrm{,}\Pi a^{†}{\Pi }^{†}=-a^{†}\Rightarrow \Pi D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right){\Pi }^{†}=D\left(-\alpha \mathrm{,}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\end{array}$ (3.15)

From these commutation properties follows using (3.15)

$\begin{array}{c}\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi {\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}=D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi \\ =D\left(2\alpha \mathrm{,2}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi =\Pi D\left(-\alpha \mathrm{,}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)D\left(-\alpha \mathrm{,}-{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\Pi D\left(-2\alpha \mathrm{,}-2{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\end{array}$ (3.16)

The parity operator $\Pi$ possesses only the two eigenvalues $+1$ and $-1$ to even and odd number states $|n\rangle$ and $\langle n|$ as right-hand and corresponding left-hand eigenstates

${\left(-1\right)}^N|n\rangle ={\left(-1\right)}^n|n\rangle \mathrm{,}\langle n|{\left(-1\right)}^N={\left(-1\right)}^n\langle n|\mathrm{,}\left(N\equiv a^{†}a\right)$ (3.17)

that leads to the following possible representations by eigenstates

$\Pi ={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{\left(-1\right)}^n|n\rangle \langle n|\mathrm{,}\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{\left(-1\right)}^{n}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|n\rangle \langle n|{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}$ (3.18)

It is a highly degenerate operator which therefore admits many other representations by linear combinations of the eigenstates to the eigenvalues $+1$ and $-1$ separately. The displaced parity operator $\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ defined in (3.12) possesses the same eigenvalues $+1$ and $-1$ but to displaced number states $|\alpha \mathrm{,}n\rangle$ defined by

$|\alpha \mathrm{,}n\rangle \equiv D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|n\rangle \mathrm{,}\left(n=\mathrm{0,1,2,}\cdots \right)$ (3.19)

as right-hand eigenstates of $\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ according to

$\begin{array}{c}\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|\alpha \mathrm{,}n\rangle =D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi {\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{-1}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|n\rangle \\ =D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi |n\rangle \equiv {\left(-1\right)}^{n}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|n\rangle ={\left(-1\right)}^n|\alpha \mathrm{,}n\rangle \end{array}$ (3.20)

and similar for the left-hand eigenstates $\langle \alpha \mathrm{,}n|$ . The displaced number states $|\alpha \mathrm{,}n\rangle$ are ortho-normalized for discrete $\left(m\mathrm{,}n\right)$ and arbitrary fixed $\alpha$ according to

$\langle \alpha \mathrm{,}m|\alpha \mathrm{,}n\rangle =\langle m|{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|n\rangle =\langle m|n\rangle ={\delta }_{m\mathrm{,}n}$ (3.21)

and they obey the following relation (see below (3.25))

$\frac{1}{\text{π}}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}|\alpha \mathrm{,}n\rangle \langle \alpha \mathrm{,}m|=\langle m|n\rangle I={\delta }_{m\mathrm{,}n}I$ (3.22)

This means that the states $|\alpha \mathrm{,}n\rangle$ are mutually orthogonal for $m\ne n$ and that they are (over-) complete for fixed $m=n$ in the quantum phase space of variables $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ such as the coherent states $|\alpha \rangle$ which are their special case $m=n=0$ with the well-known completeness (over-)relation

$\frac{1}{\text{π}}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}|\alpha \rangle \langle \alpha |=I\mathrm{.}$ (3.23)

Relation (3.22) is a consequence of the more general relation for arbitrary operators $A$ (remind that $\langle A\rangle$ denotes the trace of $A$ )

$\frac{1}{\text{π}}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)A{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}=\langle A\rangle I$ (3.24)

We do not derive it here (see, e.g., [9] (chap. I: A coherent state primer) and [10] ). In the special cases $A=|n\rangle \langle m|\mathrm{<mo>\; ;\; </mo>}\left(m\mathrm{,}n=\mathrm{0,1,2,}\cdots \right)$ follows from (3.25)

$\frac{1}{\text{π}}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|n\rangle \langle m|{\left(D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right)}^{†}=\langle m|n\rangle I={\delta }_{m\mathrm{,}n}I$ (3.25)

In (3.19) we defined by $D\left(\alpha ,{\alpha }^{*}\right)|n\rangle \equiv |n,\alpha \rangle$ as the displaced number states. For $n=0,m\ne 0$ relation (25) expresses the overcompleteness of the coherent states $|0,\alpha \rangle \equiv |\alpha \rangle$ .

The displaced parity operators possess the trace equal to $\frac{1}{2}$ and are (over-)

complete in the quantum phase space in the sense described by the resolution of the identity operator $I$

$\langle \Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\rangle =\langle \Pi \rangle =\frac{1}{2}\mathrm{,}\frac{2}{\text{π}}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=I$ (3.26)

and they are mutually orthogonal expressed by

$\langle \Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi \left(\beta \mathrm{,}{\beta }^{\mathrm{<mo>\; *\; </mo>}}\right)\rangle =\frac{\text{π}}{4}\delta \left(\alpha -\beta \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}-{\beta }^{\mathrm{<mo>\; *\; </mo>}}\right)$ (3.27)

Therefore, operators A can be expanded in integrals over phase-space functions as given in (3.2) and the function $A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ in dependence on $A$ is then determined by the operator $A$ and vice versa.

4. The Wigner Quasiprobability and Reconstruction of Density Operators

The Wigner quasiprobability $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ in the complex variables $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ can be defined in (not full) analogy to (3.7) by (remind, $\langle \cdots \rangle$ means trace of content)

$\begin{array}{l}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\langle \varrho exp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)\rangle \delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ =\frac{2}{\text{π}}\langle \varrho \Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\rangle \mathrm{,}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=1\end{array}$ (4.1)

The reconstruction of the density operator $\varrho$ from it is then determined by

$\begin{array}{c}\varrho =2{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\Pi \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ =\text{π}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)exp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\end{array}$ (4.2)

that after partial integration leads to

$\begin{array}{c}\varrho =\text{π}{\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)exp\left(a\frac{\partial }{\partial \alpha }+a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\\ =\text{π}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}}{\displaystyle \underset{l=0}{\overset{\infty }{\sum }}}\frac{\mathcal{S}\left\{a^k{a}^{†l}\right\}}{k!l!}{\left\{\frac{{\partial }^{k+l}}{\partial {\alpha }^k\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}l}}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)\right\}}_{\alpha \mathrm{<mo>\; =\; </mo>}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mo>\; =\; </mo>0}}\mathrm{,}\langle \varrho \rangle =1\end{array}$ (4.3)

The calculation of expectation values $\bar{A}\equiv \langle \varrho A\rangle$ of operators $A$ for density operators $\varrho$ using the Wigner quasiprobability $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ has to be made by the formula

$\bar{A}\equiv \langle \varrho A\rangle ={\displaystyle \int }\frac{\text{i}}{2}\text{d}\alpha \wedge \text{d}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)A\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)={\displaystyle \int }\text{d}q\wedge \text{d}pW\left(q\mathrm{,}p\right)A\left(q\mathrm{,}p\right)$ (4.4)

in analogy to the classical probability theory.

If one compares the relations between classical phase-space functions $A\left(q\mathrm{,}p\right)$ with the quantum-mechanical equivalent operators $A$ in the Weyl- Wigner formalism in (3.2) and (3.7) with that of the Wigner quasiprobability $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ and the reconstruction of the density operator $\varrho$ from $W\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ in (4.1) and in (4.2) then we find a difference in the factors in front. One may be astonished about this but it is not very principal and finds a simple historical explanation⁴.

For the transition to the representation by real canonical variable $\left(q\mathrm{,}p\right)$ and operators $\left(Q\mathrm{,}P\right)$ one has to use, in particular, the relation

$exp\left(-a\frac{\partial }{\partial \alpha }-a^{†}\frac{\partial }{\partial {\alpha }^{\mathrm{<mo>\; *\; </mo>}}}\right)=exp\left(-Q\frac{\partial }{\partial q}-P\frac{\partial }{\partial p}\right)$ (4.5)

and, furthermore, the relation

$\delta \left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)=\delta \left(\frac{q+\text{i}p}{\sqrt{2\hslash }}\mathrm{,}\frac{q-\text{i}p}{\sqrt{2\hslash }}\right)=2\hslash \delta \left(q+\text{i}p\mathrm{,}q-\text{i}p\right)=2\hslash \delta \left(q\right)\delta \left(p\right)$ (4.6)

Then from (4.1) for the Wigner quasiprobability $W\left(q\mathrm{,}p\right)$ follows (remind that $\langle \dots \rangle$ means trace)

$W\left(q\mathrm{,}p\right)=\langle \varrho exp\left(-Q\frac{\partial }{\partial q}-P\frac{\partial }{\partial p}\right)\rangle \delta \left(q\right)\delta \left(p\right)\mathrm{,}{\displaystyle \int }\text{d}q\wedge \text{d}pW\left(q\mathrm{,}p\right)=1$ (4.7)

and the reconstruction of the density operator $\varrho$ from the Wigner quasi- probability is possible by

$\varrho =2\text{π}\hslash {\displaystyle \int }\text{d}q\wedge \text{d}pW\left(q\mathrm{,}p\right)exp\left(-Q\frac{\partial }{\partial q}-P\frac{\partial }{\partial p}\right)\delta \left(q\right)\delta \left(p\right)\mathrm{,}\langle \varrho \rangle =1$ (4.8)

One may prove then after some calculation that the definition of $W\left(q\mathrm{,}p\right)$ in (4.7) is fully equivalent to the definition by Wigner [3] (see also [4] ) (Wigner de- notes it $P\left(x\mathrm{<mo>\; ;\; </mo>}p\right)$ and generalizes it for several variables to $P\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_n\mathrm{<mo>\; ;\; </mo>}{p}_1\mathrm{,}\cdots \mathrm{,}{p}_n\right)$ ). In the following, however, we will stay at the representation by complex vari- ables $\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)$ and will now discuss representations by number and by displaced number states.

5. Number-State Representations of Displaced Number States Using Laguerre 2D Polynomials

⁴As a quasi probability one is not obliged to accept the normalizations ${\displaystyle \int \frac{i}{2}}\text{d}\alpha \wedge \text{d}{\alpha }^{*}W\left(\alpha ,{\alpha }^{\ast }\right)={\displaystyle \int \text{d}q}\wedge \text{d}pW\left(q,p\right)=1$ such as for genuine probability densities since $W\left(\alpha ,{\alpha }^{\ast }\right)$ involves non-orthogonal states of the variables $\left(\alpha ,{\alpha }^{\ast }\right)$ in its definition but one has in this case also to change the calculation of expectation values in corresponding way. For the transition from the Wigner quasiprobability to a classical distribution function by the limiting procedure $\hslash \to 0$ it is even favorable to use this normalization but one has before this to make the transition to real canonical variables $\left(q,p\right)$ and thus to $W\left(q,p\right)$ . Therefore, it seems to be unreasonable and not useful to change the established normalization.

In the following we derive number-state representations of the relations of the Weyl-Wigner formalism and as a preparation for our next aim we derive the number-state representation of displaced number states. It is advantageous to use for this purpose the Laguerre 2D polynomials ${\text{L}}_{m\mathrm{,}n}\left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)$ defined as follows (see [17] and citations therein)

$\begin{array}{c}{\text{L}}_{m\mathrm{,}n}\left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)=exp\left(-\frac{{\partial }^2}{\partial z\partial z^{\mathrm{<mo>\; *\; </mo>}}}\right)z^m{z}^{\mathrm{<mo>\; *\; </mo>}n}\\ ={\left(-1\right)}^{m+n}exp\left(z{z}^{\mathrm{<mo>\; *\; </mo>}}\right)\frac{{\partial }^{m+n}}{\partial z^{\mathrm{<mo>\; *\; </mo>}m}\partial z^n}exp\left(-z{z}^{\mathrm{<mo>\; *\; </mo>}}\right)\end{array}$ (5.1)

and explicitly

${\text{L}}_{m\mathrm{,}n}\left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)={\displaystyle \underset{j=0}{\overset{\left\{m\mathrm{,}n\right\}}{\sum }}}\frac{{\left(-1\right)}^{j}m!n!}{j!\left(m-j\right)!\left(n-j\right)!}{z}^{m-j}{z}^{\mathrm{<mo>\; *\; </mo>}n-j}$ (5.2)

with the following relation to generalized Laguerre (or Laguerre-Sonin) poly- nomials ${\text{L}}_n^{\nu }<mi>(\; u\; )</mi>$

${\text{L}}_{m\mathrm{,}n}\left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)={\left(-1\right)}^{n}n\mathrm{<mo>\; !\; </mo>}{z}^{m-n}{\text{L}}_n^{m-n}\left(z{z}^{\mathrm{<mo>\; *\; </mo>}}\right)={\left(-1\right)}^{m}m\mathrm{<mo>\; !\; </mo>}{z}^{\mathrm{<mo>\; *\; </mo>}n-m}{\text{L}}_m^{n-m}\left(z{z}^{\mathrm{<mo>\; *\; </mo>}}\right)$ (5.3)

The definition of ${\text{L}}_{m\mathrm{,}n}\left(z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)$ may be generalized to polynomials ${\text{L}}_{m\mathrm{,}n}\left(U\mathrm{<mo>\; ;\; </mo>}z\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)$ where $U$ is a two-dimensional unimodular matrix which makes a linear transformation of the two variables [18] that, however, we do not need here.

First, we calculate the expansion of the displaced number states $|\alpha \mathrm{,}n\rangle \equiv D\left(\alpha \mathrm{,}{\alpha }^{\mathrm{<mo>\; *\; </mo>}}\right)|n\rangle$ defined in (3.19) in number states $|m\rangle$ using the completeness of the number states ${\displaystyle {\sum }_{m=0}^{\infty }}|m\rangle \langle m|=I$

$|\alpha ,n\rangle \equiv {\displaystyle \underset{m=0}{\overset{\infty }{\sum }}}|m\rangle \langle m|D\left(\alpha ,{\alpha }^{*}\right)|n\rangle$ (5.4)