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 of the phase space in the Hamilton function were substituted by operators which obey the commutation relations , ( Planck’s action quantum divided by and identity operator of the Hilbert space of the representation of the operator algebra). In general, a classical function of the canonical variables cannot be translated into a corresponding quantum-mechanical function without additional rules for operator ordering since . Luckily, this did not play a role in the first very successful applications of the Schrödinger equation with a Hamilton operator with additively separated classical kinetic energy from potential energy with the position vector and the (canonical) momentum vector. Thus it was translated into the quantum-mechanical Hamilton operator where in classical mechanics are, in general, three-dimensional vectors and corresponding vector operators with independent components and independent commutation relations . Luckily also, this did not play a role in the translation of the classical angular momentum (vector product of with ) or into a quantum-me- chanical operator since there are only combined independent (commutating) components of and , in the quantum-mechanical operator . In quantum field theories such as quantum electrodynamics and optics where one usually speaks about independent modes the canonical variables are internal field variables whereas the position and the time are classical variables as parameters of each mode determining the shape of the field and 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 to complex coordinates and from canonical operators to annihilation and creation operators , correspondingly.
1Therefore, in quantum optics it is usually unfavorable to denote the canonical variables of the phase space by 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 and involving frequency and wave vector as parameters.
Hermann Weyl in  and in his book  from 1928 (chap. IV, §14) proposed a general rule for the translation of arbitrary functions of the canonical phase-space variables in a unique way into quantum-mechanical operator-ordered functions of the operators . His way was via the Fourier transform of the classical function (denoted there in the way where and is the Fourier transform of and are here operators). On the opposite our preliminarily written “function” 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 . In this symmetrical ordering of the operators in a function we may consider first as the classical variables 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 over the phase space corresponds then a uniquely defined operator . The calculation of expectation values of symmetrically ordered operators is best suited to the Wigner function which is a quasiprobability over the phase space introduced in 1932 by Eugen P. Wigner  (republished in  ). Together with the Weyl ordering this is often called the Weyl-Wigner formalism of correspondence between classical and quantum mechanics  . Other corres- pondences of classical phase-space functions to quantum-mechanical operators are possible, in particular, the normally-ordered correspondence for which another quasiprobability called the Glauber-Sudarshan quasipro- bability     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 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  . 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  . Problems of the determination of a phase operator in quantum optics are discussed and referred in detail by Peřinová, Lukš and Peřina  .
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
for integers , ,
was communicated in  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 and determine the corresponding basic operators 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 can be also obtained by integration of the general Wigner quasiprobability over the angle and the special case by integration of the Wigner quasiprobability over the radius that for this last case was first made by Garraway and Knight  (see also  ). One has to liberate oneself in these cases from the general density operator in the Wigner quasiprobability in complex representation and obtain then in last case the quantum equivalents to the basic classical periodic phase functions .
In Section 12 we derive the connection of the symmetrically ordered operators to powers of the number operator . This suggests to use the expectation values (notation by overlining operators) which are positive definite as alternative to the corresponding normally ordered quantity 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 is denoted by and the expectation value of by overlining the operator if 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 . Additionally, we introduce complex variables in the following way
with correspondence to the basic quantum-mechanical operators and their combinations which become the annihilation and creation operator for a harmonic oscillator
They obey the commutation relations ( identity operator of the represen- tation space (Hilbert space))
The reason that we introduce the complex substitutes of the canonical variables by (2.1) and not simply by is that we are interested in the following mainly in the expectation values of ordered functions of 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 one has to be cautious
using the operators . For example, in the operator
one cannot set since N (number operator) does not possess a classical analogue and only after multiplication of this operator with we get an
operator with a classical analogue in form of the function proportional to the intensity or to energy and momentum of the field.
Starting from the well-defined operator function with arbitrary parameters the symmetrical Weyl ordering can be defined by (e.g.,  )
From this follows for integer
or using the binomial formula for with observation of the non-commutativity of and
2Some authors write the corresponding classical variables within an ordering symbol (here symbol ) that means, e.g., instead of our .
The symbol is not a linear operator since2
where means here the product of k operators a and operators in arbitrary order (i.e., permutations) but linear combinations of them are understood in the sense of the distributive law
This gives the possibility to determine linear spaces of ordered operators if one introduces a system of basis operators. Clearly, two operators and are, in general, noncommutative.
In the following we use the displacement operator defined by (e.g.,    )
where the symbol means normal ordering of the content in braces (all powers of in front of powers of ). From (2.9) follows for its Hermitean adjoint operator
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.,  )
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
as the eigenstates of the number operator
and the completeness relation
which is true for arbitrary operators and which commute with their commutator . 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.,  . In the following we apply this theorem repeatedly.
The displacement operator possesses the property
For the product of two displacement operators one finds applying (2.11)
Furthermore, we need the normal ordering of operators of the form with parameter . The following relation is well known, e.g.,  (chap. 3.3., pp. 156/157)3
The conversion of (2.14) with respect to the parameter as it is easily seen is
For one obtains the operators
All these special operators play a certain role in quantum optics, in particular, is the unity operator , is the vacuum-state operator and the parity operator 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 in the symmetrical Weyl ordering to quantum-mechanical operators can be written
or in complex representation by complex variables (see (2.1)) as follows (we do not use a new function symbol and set )
where is the two-dimensional delta function in complex representa- tion according to
The integration goes in both representations over the whole phase-plane in real or complex coordinates . After partial integration in the first line we find equivalently to (3.2)
For the trace of the operator we find from (3.2) or (3.7)
The factor in (3.5) in front of the integral 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 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 by )
or of the complex form (3.2)
The transformation in (3.2) together with the inverse transformation in (3.7) is a mapping which preserves the distribution law for arbitrary complex numbers and
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 to quantum- mechanical operators which are symmetrically ordered equivalents in the sense of Weyl
In the derivation we used the identity
specialized to , which generally can be proved by two-dimensional Fourier transformation. The operator with positive values of the para- meter applied to a function makes a smoothing of this function.
The operator defined by (see also (2.15) and (2.16) for the different representations)
is called the parity operator and the operator defined by
the displaced parity operator, correspondingly. Thus we obtained in (3.9) the basic relation
which plays a role in the following. We see here that the limiting transition on the right-hand side in (3.13) is not possible in this way and the representation of the parity operator is not locally possible and the representation of by the symmetrically ordered operators does not exist in contrast to the representation by the normally ordered operators (see (3.11)).
The parity operator is a Hermitean and at once an idempotent operator (squared it is equal to the identity operator)
with the following interesting properties of transformation of justifying its name
From these commutation properties follows using (3.15)
The parity operator possesses only the two eigenvalues and to even and odd number states and as right-hand and corresponding left-hand eigenstates
that leads to the following possible representations by eigenstates
It is a highly degenerate operator which therefore admits many other representations by linear combinations of the eigenstates to the eigenvalues and separately. The displaced parity operator defined in (3.12) possesses the same eigenvalues and but to displaced number states defined by
as right-hand eigenstates of according to
and similar for the left-hand eigenstates . The displaced number states are ortho-normalized for discrete and arbitrary fixed according to
and they obey the following relation (see below (3.25))
This means that the states are mutually orthogonal for and that they are (over-) complete for fixed in the quantum phase space of variables such as the coherent states which are their special case with the well-known completeness (over-)relation
Relation (3.22) is a consequence of the more general relation for arbitrary operators (remind that denotes the trace of )
We do not derive it here (see, e.g.,  (chap. I: A coherent state primer) and  ). In the special cases follows from (3.25)
In (3.19) we defined by as the displaced number states. For relation (25) expresses the overcompleteness of the coherent states .
The displaced parity operators possess the trace equal to and are (over-)
complete in the quantum phase space in the sense described by the resolution of the identity operator
and they are mutually orthogonal expressed by
Therefore, operators A can be expanded in integrals over phase-space functions as given in (3.2) and the function in dependence on is then determined by the operator and vice versa.
4. The Wigner Quasiprobability and Reconstruction of Density Operators
The Wigner quasiprobability in the complex variables can be defined in (not full) analogy to (3.7) by (remind, means trace of content)
The reconstruction of the density operator from it is then determined by
that after partial integration leads to
The calculation of expectation values of operators for density operators using the Wigner quasiprobability has to be made by the formula
in analogy to the classical probability theory.
If one compares the relations between classical phase-space functions with the quantum-mechanical equivalent operators in the Weyl- Wigner formalism in (3.2) and (3.7) with that of the Wigner quasiprobability and the reconstruction of the density operator from 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 and operators one has to use, in particular, the relation
and, furthermore, the relation
Then from (4.1) for the Wigner quasiprobability follows (remind that means trace)
and the reconstruction of the density operator from the Wigner quasi- probability is possible by
One may prove then after some calculation that the definition of in (4.7) is fully equivalent to the definition by Wigner  (see also  ) (Wigner de- notes it and generalizes it for several variables to ). In the following, however, we will stay at the representation by complex vari- ables 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 such as for genuine probability densities since involves non-orthogonal states of the variables 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 it is even favorable to use this normalization but one has before this to make the transition to real canonical variables and thus to . 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 defined as follows (see  and citations therein)
with the following relation to generalized Laguerre (or Laguerre-Sonin) poly- nomials
The definition of may be generalized to polynomials where is a two-dimensional unimodular matrix which makes a linear transformation of the two variables  that, however, we do not need here.
First, we calculate the expansion of the displaced number states defined in (3.19) in number states using the completeness of the number states
Using the normally ordered representation of the displacement operator in (2.9) we find
and by substitution of the summation indices according to
By definition of the Laguerre 2D polynomials in (5.1) this can be written
The expansion (5.4) of displaced number states becomes
For using one obtains the corresponding expansion of coherent states .
For the product of two displacement operators and one finds from (2.13)
Applying this one obtains for the general scalar product of displaced number states
One sees immediately that for using from this relation follows the orthonormality (3.21) of the displaced number states and for using the well-known relation for the scalar product of coherent states.
We now derive the representation of the displaced parity operator by number states. From (3.16) and (3.18) follows
and using (5.8) with corresponding substitutions
Thus we found the following basic number-state representation of the operator
and the Formulas (3.2) for the transition from a classical phase-space function to a quantum-mechanical operator in the Weyl-Wigner formalism takes on the form
The representation of the displaced parity operator and its consequence (5.13) leads also to a convenient representation of the Wigner quasiprobability in the number-state representation. We express the general density operator by its matrix elements as follows
Then the Wigner quasiprobability (4.1) using (5.13) can be represented by the following expansion
For its importance we will translate this here also into the representation by canonical variables (see (2.1)) with the result
in the given normalization .
For a displaced number state the Wigner quasiproba- bility follows immediately from (5.16) by argument transformation
as the displaced Wigner quasiprobability for a number state . Therefore, more generally, if is the density operator of a displaced state to density operator according to
then the corresponding Wigner quasiprobabilities and are related by
that means by a displacement of the arguments.
6. Quantum-Mechanical Operators Corresponding to Classical Monomial Phase-Space Functions
We calculate and discuss now the operators which correspond in the Weyl formalism to the basic “classical” phase-space functions according to
in the number representation and express the result by means of the Jacobi polynomials. The functions (6.1) are chosen to include besides the amplitude functions the basic periodic functions of the phase of a harmonic oscillator. Expressed by the real canonical variables according to (2.1) this corresponds to the complex functions
Due to factors in the denominator the functions are strictly speaking for not genuine classical functions without multipli- cation by these factors. For positive integer and one has the identity
With we denote the Jacobi polynomials in the now generally accepted definition by Szegö  (chap IV) in which they are also programmed in Wolfram’s “Mathematica”. An older definition with direct reference to Jacobi can be found in  . Formulas (6.3) suggests that working with the canonical variables one may choose the functions as a basis of a space of functions which one may translate into quantum-mechanical symmetrically ordered operators in the sense of Weyl plus distri- butive law for arbitrary functions of this space. This space of functions, however, is narrower than the space of functions built with the basis functions (6.1). We come back to this at the end of this Section.
We now calculate the quantum-mechanical Weyl equivalents to the basic functions (6.1). Some formal part of these calculations we delegate to Appendix A where we also give the most necessary formulae for the Jacobi polynomials by means of which we represent the results. The calculation of the double integral in (5.14) in Appendix A leads to the following number-state representation of the operator (see (A.2))
This may be represented using the Jacobi polynomials in two alternative forms as follows (already communicated without detailed derivation in  )
Explicit representations of the Jacobi polynomials in two different forms are given in (A.3) of Appendix A. The general transformation relation of the Jacobi polynomials specialized from a transformation relation of the Hypergeometric function which lead in our case from argument to argument are written down in (A.4). A more detailed treatment of Jacobi polynomials we find besides cited  , e.g., also in vol. 2 of the monographs of Bateman and Erdélyi  , in the article of Koornwinder et al.  in the NIST Handbook  and in our article  .
There is another transformation relation of the Jacobi polynomials with integer upper index or specialized from a corresponding transfor- mation relation of the Hypergeometric function and given in (A.8). It leads from (6.5) to the following essentially different representations
In comparison to (6.5) it establishes some symmetry by transformations and changing then the summation index between functions which are involved in these relations as coefficients of and if one makes the Hermitean conjugation of these relations. All 4 forms (6.5) and (6.6) for are useful since in special cases parts in these formulae become singular and using then the other representations one may avoid limiting considerations. Furthermore, by transformation of the forms (6.5) are transformed into the forms (6.6) and vice versa and one easily proves the conjugation relation
With the Formulas (6.5) and (6.6) we gave four essentially different number- state representations of the quantum-mechanical (Weyl) equivalents to the classical functions (6.1) by means of the Jacobi polynomials. In some cases one or two of these formulae are not equally appropriate for the calculation of these equivalents because they do not provide the results directly without limit considerations.
We may consider the operators as basis of a linear space of symmetrically ordered operators with the possibility to add such operators and to multiply them by numbers under validity of the distributive law. Before we discuss special cases of the relations (6.5) and (6.6) we make in generalization of them in next Section a smoothing of the classical
functions by a normalized Gaussian function and calculate their quantum-mechanical equivalents.
Before implementing the announced programme we establish now the connection between symmetrically ordered powers of operators and of operators . According to (6.3) we have
with the inversion
The symmetrically ordered operators can be represented in standard and anti-standard ordering according to
where means an arbitrary permutation of the operators in analogy to (2.7).
In the special case using the following special values of the Jacobi polynomials of argument
we obtain from (6.8)
and from (6.9)
with only powers of the squared operators or , respectively, within the ordering symbol on the right-hand sides. In particular, for we find
The special values (6.11) for the Jacobi polynomials follow easily from the general expansion (6.3) in case of using the binomial formula.
7. Quantum-Mechanical Equivalents of Smoothed Classical Functions
We calculate in this Section the transition from a smoothed classical function of the canonical variables in representation by the complex variables to its equivalent quantum-mechanical operator in the sense of Weyl. The smoothing of the classical function is made by convolution with a normalized Gaussian function as follows
where “ ” denotes the convolution. It is a smoothing of the function for . For it is the opposite of smoothing for which we do not find an appropriate word. The equivalence of the right-hand sides in (7.1) is related to the equivalence (3.10) by representing as convolution with a two-dimensional delta function.
According to (5.14) one has now to calculate the equivalent operator according to
where we applied partial integration.
Let us first make a remark. The smoothing with an operator does not lead in all cases of to a new function . Due to and we have
Therefore, the smoothed quantum-mechanical equivalents and for are not different from and , respectively. One may look to this also in the following way. The operators
form an Abelian (i.e., commutative) one-parameter Lie group and as basis for (in general, reducible) representations may serve functions of (very general) function spaces. The functions and in (7.3) form a basis of the function space for the identical representation of the mentioned group. Products of genuine powers of functions and do not belong to this last function space since, for example
More generally, we find for the smoothed functions to the functions by Taylor-series expansion of the operator
Without more detailed discussion (see, e.g.,  ) we mention that for we get antinormally ordered operators if the corresponding operators are the symmetrically ordered operators and similarly the symmetrically ordered operators if the corresponding operators are the normally ordered operators according to
These formulae can be represented by the Laguerre 2D polynomials (2) but with the imaginary unit “ ” in their arguments.
In quantum optics there is often used a class of smoothed ( ) quasiprobabilities (  ( there), and, e.g.,  ) according to ( mostly restricted to )
The class of quasiprobabilities does not belong to function classes which for different may take on the same functions since and cannot be quasiprobabilities to density operators with trace equal to 1 (the traces of the operators and are 0). Expectation values of smoothed operators can be calculated with the “smoothed” quasiprobability according to (remind that means the trace)
where we used partial integration. This provides with definition (7.7)
On the other side the expectation value of the smoothed operator can be calculated using the function together with the Wigner quasiproba- bility according to
The second form is obtained from the first form by partial integration. Formulas (7.9) and (7.10) equip us with different possibilities to calculate the expectation values and .
8. Explicit Expressions for the Quantum-Mechanical Weyl Equivalents to Smoothed Classical Monomial Phase-Space Functions
We now calculate the equivalent quantum-mechanical operators to the smoothed classical functions according to (7.1) by means of the Formulas (7.2). The detailed calculations are represented in Appendix B. The smoothed functions possess the explicit form as a series expansions
For both integer and integer the right-hand side can be repre-
sented by the Laguerre 2D polynomials (5.2). It is easy to check that for
one obtains the function and for and
the functions and , respectively, which are independent of the parameter as discussed in the previous Section.
However, Formulas (8.1) is problematic for cases when or is not a
non-negative integer which restricts the sum over to a finite sum since in the other cases one has to investigate the character of the convergence of the infinite sum over which in the neighborhood of and for is not guaranteed.
As the first step we obtained in Appendix B as generalization of (6.4)
This can be represented using the Jacobi polynomials in the following two equivalent ways
in generalization of (6.6). Alternatively, using the relation (A.8) this can be also represented in the form
showing some symmetry of (8.3) to (8.4) under substitutions and Hermitean adjunction. All alternative forms in (8.3) and (8.3) are useful because some in special cases become undetermined but the others in these cases, as a rule, can be used without limiting procedures.
In special case we get the Formulas (6.5) and (6.6), respectively, setting .
In special case Formulas (8.2) and the first parts in (8.3) and (8.4) become indeterminate and have to be dealt with by the limiting transition but from the second parts we find without limiting procedure
Using the Formulas (A.6) for the Jacobi polynomials of argument observing the decomposition and apply- ing the general relation for integer we obtain from both relations (8.5)
The same result can be also calculated by applying the Formulas (B.9) with the specialization (6.1) of the functions or simpler by limiting transition in (8.2). We mention yet that for the above formulae (8.2), (8.3) and (8.4) possess a singularity and becomes genuinely singular.
In (7.3) we found that smoothing of the functions and with normalized Gaussian functions does not influence these functions. It is interesting to consider this from the point of view of our general formulae (8.3) and (8.4). Therefore, we consider now the special case of the classical function
which shows some interesting aspects. As result for the corresponding operators which do not depend on the parameter we find in number represen- tation
On the other side from (8.3) follows
and, in analogous way, from (8.4)
We see here that from the knowledge of the result (8.8) we find identities for special classes of Jacobi polynomials which are representable in these cases by simple expressions and, clearly, can be derived also in pure mathematical way from explicit representations of the Jacobi polynomials by finite sums (e.g., second line in (A.3) or third line in (A.3) together with the symmetry (A.5)).
In a widely analogous way we may consider the special case of the classical functions
which leads to
which is independent on the smoothing parameter and provides us evaluations for special cases of the Jacobi polynomials if we do not know them already from direct considerations.
9. Classical Amplitude Functions and Their Equivalent Quantum-Mechanical Weyl Operators
We now investigate the equivalent quantum-mechanical operators to amplitude functions according to
Due to in the denominator of the transition of Planck’s constant is not possible and the quantum-mechanical opera-
tors do not possess a direct correspondence to classical functions of the canonical variables without multiplication by . From (8.2) follows for and arbitrary smoothing parameter
with the special case of the identity operator for
for arbitrary smoothing parameters . The operators (9.2) are diagonal in the number states and are therefore functions of the number operator alone. Expressed by the Jacobi polynomials both Formulas (8.3) and (8.4) provide in this case the same representations as follows
For smoothing of with parameter follows (see also (8.6))
The two cases and are illustrated in Figure 1.
For even we have the following decomposition of in powers of the operator
where are the Stirling numbers of first kind (e.g,   , from decomposition of in powers of ). For odd numbers it is
Figure 1. C oefficients in and of from (9.4). The upper points are for the smoothed operators but apart from the first they are difficult to distinguish for from the points for in the chosen scale. It is easy to generate these figures from the formulae in a larger scale.
not possible to find a finite decomposition only in powers of .
The case (that means without smoothing) of the Weyl correspondence follows from (9.2) ( )
and expressed by the Jacobi polynomials, alternatively
The Formulas (9.7) and (9.8) can be extended from integer to arbitrary real since the upper indices of the general Jacobi polynomials can be arbitrary real (or even complex) numbers  .
For even one obtains from (9.7) and (9.8)
and, in particular, for
More general relations for are given in Section 12 and in Sequence 1 and Sequence 2 of Appendix C.
In the special case we find for the Jacobi polynomials where we have to distinguish the even case and the odd case
and from (9.8) follows
In the special case we find for the Jacobi polynomials where again we have to distinguish the even case and the odd case
and from (9.8) follows
It is difficult to find such explicit forms for higher odd .
One may even calculate the case according to the Formulas (9.8). For the Jacobi polynomials we find
and we obtain
We have here only the even number states involved.
10. Classical Periodic Phase Functions and Their Equivalent Quantum-Mechanical Weyl Operators
After the amplitude functions we now investigate the equivalent quantum- mechanical operators to periodic phase functions according to
The expression does not contain Planck’s constant and the limiting transition setting is possible but number states are nonclassical. From (8.2) follows for and arbitrary
For it is “non-diagonal” in the number-state representation.
If we express (10.2) by the Jacobi polynomials we find from (8.3) the representations
or alternatively from (8.4)
For the Weyl correspondence which corresponds to we find from (10.3)
The Jacobi polynomials to argument or to with the present upper indices in (10.5) or in (10.6) can be expressed by simple formulae of multiplicative type (see Appendix A, Equation (A.10)) but we have to distinguish the cases of even and of odd of the degree of the polynomials. This leads to the following general formula for arbitrary integer and (i.e., not smoothed)
The two cases and are illustrated in Figure 2.
The Formulas (10.2) for can also be obtained by integrating the Wigner quasiprobability over the radius in polar coordinates . One obtains in this way observing the generality of the density operator
where denotes the trace of the content in brackets and are the operators explicitly given in equivalent representations in (10.5) and (10.6). The right-hand side of (10.8) possesses the form of the Fourier decomposition of the 2π-periodic function with Fourier coefficients determined by
Figure 2. C oefficients in and in . This corresponds to classical phase functions and , correspondingly. The formulae for the quantum-mechanical equivalents are given in (10.7) (for ) and in (10.10) (for smoothed case ). In the Susskind- Glogower formalism (see (11.9)) this corresponds to the operators and .
This is the way in which it was first derived by Garraway and Knight  (see also Peřinová, A. Lukš and J. Peřina  ). In our derivation the operators are embedded together with the operators into the representation of a more general class of operators with a more general number-state representation by means of the Jacobi polynomials.
In special case of smoothed functions (10.1) (see also (8.1)) follows from the second of the representations in (10.3) or from (8.6)
This formula can also be obtained by integration of the Husimi-Kano quasiprobability over the radius in polar coor- dinates coordinates according to
in analogy to (10.8) as the Fourier series of the 2p-periodic function with the Fourier coefficients
in analogy to (10.9).
The phase space distributions and are normalized as given in (10.8) and (10.11) but they are quasiprobabilities. The function can take on negative values depending on the states whereas is non-negative but, nevertheless, it is a quasiprobability because it involves the non-orthogonal coherent states for its definition. As an example, we calculate their explicit forms for coherent states with the quasiprobabilities
with and . From this one finds by integration over the modulus of for coherent states with
with the special cases and where the last becomes singular and has to be considered as a generalized function. Since for is positive definite the functions are also positive definite for these parameter values.
We mention here that the information contained in cannot be directly obtained from the function and that for this purpose the complete function is necessary since a part of this information is already destroyed in . The relation between the Wigner quasiprobability and the quasiprobabilities is (see (7.7), “ ” means convolution)
One finds from this by integration with
For one obtains the special case . Formulas (10.16) shows that it is not possible to get a direct relation between, for example, and without knowing the more general function . This is different from the functions and which both contain the same complete information over the state only coded in different way.
11. About the Algebra of the Weyl Correspondences to Classical Phase-Space Functions
As system of basis operators for a quantum-mechanical harmonic oscillator the operators defined in (6.4) are overcomplete since already each set of operators , , with is appropriate as basic set for the expansion of arbitrary non-singular operators in connection with the distributive law.
For the products of classical functions (6.1) we have the following relations
The quantum-mechanical equivalent operators do not satisfy analogous relations and instead we find from (6.6)
that means that the products are, in general, noncommutative and therefore also in contrast to the classical equality (11.1) that they are not equal, in general, to that means
plus possible further representations using the alternative representations for . However these products are associative according to
This follows from the associativity of the products in the arising triple sum over .
In general, the operators and do not commute. Apart from the trivial cases the operators and commute also for according to
In these cases the operators and and thus their products are diagonal in the number representation but generally their products are not equal to . According to (8.8) and (8.11) in the special cases and the operators are and , respectively. Since smoothing of these operators does not change them we can extend this behavior to arbitrary smoothing parameter and taking into account (8.9) or (8.10)
for arbitrary and and, analogously, taking into account (8.12)
In cases when and do not commutate one may calculate the commutator from the given relations and may express it by means of the Jacobi polynomials. For example, for the commutator of corresponding to classical with follows from (10.7)
In the Susskind-Glogower formalism    , for comparison, we have for the analogous operators and (correspondence and )
For explicit calculation this formalism is often simpler than using the formulae in Section 10 resulting from the Weyl correspondence of classical to quantum optics. However, one cannot make in this formalism a distinction between symmetrical Weyl ordering and normal ordering. To find a simple general mathematical relation between these two approaches seems to be difficult. It is also easier to deal with the eigenvalue problems (right-hand eigenstates) for the operators than the corresponding eigenvalue problems for the operators .
12. Powers of the Classical Intensity and Their Equivalent Quantum-Mechanical Weyl Operators
The classical intensity is by definition if is the complex amplitude of a considered process (e.g., harmonic oscillator). We made in (6.1), (6.2) and (9.1) the agreement (not also with some disadvantages) to “normalize” it using the Planck constant to get in the Weyl correspondence directly the symmetrized product of annihilation and creation operator connected with the number operator . For its k-th powers we have according to (9.1) the “classical” function in representation by canonical variables and in complex variables (substitute in (9.9))
with the quantum-mechanical equivalent (smoothing parameter is here )
According to the meaning of the symbol for symmetrical ordering the ordering of the annihilation and creation operators within the braces is arbitrary.
We now derive the relations between symmetrical ordering and normal ordering for products of equal numbers of annihilation and creation operators that means for combinations where the phase of these opera- tors is fully eliminated. From (7.6) follows for the special case
The inversion of this relation is
This can be proved analogously to corresponding more general formulae for by inserting one of the Formulas (12.3) and (12.4) into the other one and using after a simple transformation of the arising double sum the binomial formula. In particular, we find from (12.3)
The explicit form for more initial special cases is given in Appendix C. The inversion of (12.5) could be immediately written down from the analogous structure of the relations (12.3) and (12.4) with changing signs. In Appendix C we also derive general representations of the symmetrically ordered operators by powers of the number operator N and by powers of the operator
and give them explicitly for a few initial cases.
We now derive from (12.5) an inequality for expectation values. For this purpose we use the Cauchy-Bunyakovski-Schwarz inequality for states and in Hilbert space or for operators A and B in a Hilbert space of operators in the forms
From the second equation in (12.5) follows (remind that overlining means forming the expectation value and forming of the trace of an arbitrary operator C)
The operator is here defined as a positive semi-definite Hermitean operator that is possible since the density operator itself is also a positive semi-definite Hermitean operator.
From (12.7) we see the inequality
with the usual definition of the operator
One may consider the expression on the left-hand side of (12.8) as quantum- mechanical analogue of the variance of a classical function
proportional to the intensity. Accepting this, in quantum optics this quantity possesses a minimum of uncertainty which cannot be undercut in contrast to classical optics where it can be equal to zero. However, one may simply consider as analogue to a classical uncertainty where, however, we remind that
does not admit a limiting transition to a finite classical corresponding quantity (see also remark after Eq. (2.3)).
The minimum of the left-hand side of (12.8) where the inequality makes the transition to an equality is obtained for all number states
For coherent states one does not obtain this minimum value on the right-hand side and find instead
For thermal states to a harmonic oscillator of frequency with density operator according to
with abbreviation to temperature T and with the Boltzmann constant and with the relations
Thus we have illustrated the inequality (12.8) for three important categories of states.
Instead of symmetrical (Weyl) ordering one may consider normal ordering in analogy to (12.5) with
In a way which is analogous to (12.7) one derives the well-known inequality
The left-hand side of this inequality may take on positive as well as negative values. It becomes equal to zero for coherent states which obey a Poisson statistics defined in classical probability theory by the probabilities
with as a parameter and in quantum optics with respect
to the eigenvectors and of the operator N by
as it is well known. The probabilities alone do not determine the coherent states since information about the phase is absent and much less the normally ordered moments and do this alone. In next Section we consider shortly the reconstruction of a (one-mode) state from its normally ordered moments.
If one looks to the quantity (12.16) not only as to a pure definition but as a quantity which can be measured and which, moreover, is the quantum- mechanical analogue of a classical quantity which last can take on only non-negative values then this becomes highly problematic.
Both quantities on the left-hand sides of (12.8) and in the middle (12.16) cannot directly be measured but can only be calculated from measured quantities of (or ) and of . Glauber in the measurement theory within his lectures  considered the following two cases: 1. measurement by one-atom photon detector (chap. 4) and 2. measurement by multi-atom photon detector (chap. 5). The conclusion was that since the detectors are basically in the ground state the expectation values of powers of normally ordered annihilation and creation operators are measured. On the basis of the inequality (12.16) Mandel  (see also  ) defined sub-Poissonian and super-Poissonian statistics in quantum optics in dependence on the sign of this quantity, “sub”-Poissonian if and “super”-Poissonian if . With effort to the difficult task to implement the measurement theory to photon statistics     Paul calculated and discussed anti-bunching of states as a typical non-classical property with no correspondence in classical optics in  and in  (anti-bunching occasionally renamed there in anti-correlations). In a short paper of Zou and Mandel  these authors reclaimed that Paul  does not consider anti-bunching and bunching but instead of this sub- and super-Poissonian statistics and that anti-bunching is not a property of a state but a property of the time evolution of a state when the time derivative of the quantity becomes positive. We are not of the opinion that the notions of sub- and super-Poissonian statistics of Mandel are much better since the prefixes “sub” and “super” are misleading and suggest too much that the statistics of states in quantum optics can be linearly ordered with the Poisson statistics of the coherent states in the “middle”. Besides the coherent states a very large category of different states possess for arbitrary given the same expectation values as given in (12.17) (exception: which is uniquely only possible for the vacuum state ). This becomes clear from the reconstruction of states by their normally ordered moments. These states can be very far from coherent states and may possess even the greatest possible distance to the nearest coherent state as can be calculated using the Hilbert-Schmidt distance, for example, for sets of some squeezed coherent states or what is the same of displaced squeezed vacuum states in the limiting procedure to maximal squeezing but with the same values and as the considered coherent states. This means that one cannot establish a linear ordering by means of the parameter (12.16) under fixed and that one may continuously go from sub-Poissonian to super-Poissonian statistics without touching the Poisson statistics of coherent states and one cannot expect a very unique behavior of states with sub- and super-Poissonian statistics5. The separation of sub- and super-Poissonian statistics goes amidst within the set of squeezed coherent states. Such orderings which are not a full linear ordering are called semi-orderings.
5In the table 1 on p. 187 in  such a subdivision is made with respect to the sign of the quantity (16) but the case of its vanishing is identified with Poisson statistics and it was forgotten to mention that this does not necessarily mean the coherent states with their Poisson statistics.
Besides the quantity (12.16) there are often used corresponding relative quantities obtained by division of (12.16) by  or by  where only the last corresponds to approaches in classical theory if the investigated quantity possesses a dimension. In case of the number operator N such a division by the squared expectation value enlarges without any further changes the importance of an effect for small expectation values , in particular, in this case for suggesting its highly quantum character for very small expectation values . This was estimated in  (in the middle of p. 187) as a pleasant agreement with Bohr’s correspondence principle according to which in the limiting case of high excitations (here mean photon numbers ) the quantum-mechanical description should make the transition to a classical one. On the other side, the smaller the nearer the state is to the vacuum state and in the limiting case it becomes the vacuum state. This even can be described by an inequality (Section 14) for the distance to the vacuum state which continuously is reached for . The vacuum state is a coherent state with vanishing complex displacement parameter and according to the usual opinion, the coherent states are the “most classical states”.
13. Reconstruction of Density Operator from Normally Ordered Moments
A general quantum-mechanical state (here of the free electromagnetic field) is fully characterized by its density operator . If it is known one may determine from it, for example, the matrix element with the number states and is already the reconstruction formula for the density operator (suppose that it is only one mode). One may also determine the moments of the density operator with powers of the annihilation and creation operators, in simplest case in normal ordering as another kind of “coordinate representation” of the density operator (analogously to repre- sentation of vectors by coordinates). The reconstruction of a density operator from its moments is more complicated than from matrix elements of the number states since it corresponds to a non-orthogonal basis system.
The reconstruction formula of the density operator from its normally ordered moments was derived in  with the result (is correct also for arbitrary operators A if involved quantities exist)
where is the abbreviation for a set of auxiliary operators necessary for the reconstruction and defined as follows (remind that means the trace of an operator and and are number states)
shows that the two sets of “coordinates” and are related to each other similarly as covariant and contravariant components of a vector. From (13.3) follows as special case for the traces of the operators
and using this together with (13.1) one may check the normalization
In contrast, the traces of are also vanishing for but do not possess finite values for .
Usually it is assumed that a density operator expresses the maximum knowledge for an ensemble of states which individual members are in states described by different exact wave functions that means by averaging over density operators for pure states with probability as coefficients in front of them, i.e. . Then arise problems of determination of the possible pure states together with a discrete averaging function leading to the diagonal form of the density operator.
For a full reconstruction of the density operator according to the basic Formulas (13.1) we need all normally ordered moments . If one determines only the diagonal values as in photon statistics then one can reconstruct only a part of the density operator of the form
where the factorial moments are connected with the expectation values of the number operator N together with its inversion by
with and the Stirling numbers of first and second kind, respectively. Writing down explicitly the first four sum terms of (13.6) we have
and we see that it is absolutely insufficient to conclude that in case of we have a Poisson distribution to density operator
in particular, for high expectation values , apart from the full absence of information about the phases (expectation values with ). For a “sufficient” reconstruction of a density operator we need, at least, expectation values up to values  . For the density operator is near to that for the vacuum state but this is better to see from the Hilbert-Schmidt distance to the vacuum state (next Section).
The treatment of reconstruction of the density operator reveals many interesting problems partially not solved up to now. Within which limits can a density operator reconstructed with incomplete knowledge of necessary quantities (expectation values, moments) if only a small part of them is known or if it is assumed that the density operator possesses only one, two, three and so far eigenvalues different from zero. If only one eigenvalue is different from zero (then equal to 1) we have the problem of reconstruction of pure states. Also in this case one has an arbitrariness, for example, if and is known which admits as well as coherent states as different squeezed states and also other states. We do not consider this here (some elements of treatment of squeezed states we developed already in former publications, e.g.,  ).
14. Hilbert-Schmidt Distance of Two Quantum-Mechanical States
A measure how near or far are two quantum-mechanical states is their distance. Distances are determined by a few axioms. They have to be non-negative and should be zero for equal states and they should obey the triangular inequality (to find in almost every monograph about functional analysis). From the many possible distances the only distance in quantum theory with which one can calculate in convenient way is the Hilbert-Schmidt distance determined by the Hilbert-Schmidt norm of the elements of the Hilbert space of states. Since a general state is described by a density operator this is in our case the Hilbert space of all (normalizable) operators according to their scalar products . Thus the Hilbert-Schmidt distance of two states described by the density operators and and their special cases of pure states or (and) we define the distances denoted by