Most representations of probability theory begin with the discussion of some examples where probabilities play a main role and introduce then axiomatically the probability as a positively semi-definite and “normalized” function over a set of events. The main purpose of this function is to allow us to calculate mean values and their variances or more generally expectation values for arbitrary functions over the set of events when the initial conditions or the prehistory of the events are not fully under control of the experimenter or observer. The results of such calculations are then true in the mean for great ensembles of “equal” events and of their dynamics which are made under the same uncontrolled or uncontrollable initial conditions. In Hamilton dynamics of a system of one degree of freedom a trajectory is fully determined by a pair of canonical coordinate and momentum in two-dimensional phase space as initial condition and the probability function is given by a positive semi-definite function (called distribution or partition function) and from the dynamics of single trajectories, in principle, can be determined by the time evolution of the function (the lately discovered cases of possible chaotic dynamics destroy this assumption in some way). After foundation of probability theory in 17-th century mainly by Blaise Pascal and great contributions by Jakob B. 1654-1705 and Daniel (I.) Bernoulli 1700-1782 and by Huygens in 17-th century and by Laplace and Bayes in 18-th century a culminating point was reached by the axiomatic foundation which in the now finally accepted form was given in 1933 by Kolmogorov.
Quantum mechanics gave birth to a new kind of probabilities which are not definitely non-negative and which are called quasiprobabilities. The first and, for some reason, forever the most important such quasiprobability is the Wigner quasiprobability  (see also  ), often called Wigner function. Due to substitution of the classical canonical variables by non-commuting operators (commutator ; I identity operator in representation
space; , h Planck’s constant) in quantum mechanics a single event is not
determined as in classical theory by a (displaced) delta function and its trajectory in phase space is not defined and one may best have moving “spots” of events with minimal uncertainty products equivalent to Gaussian distribution functions with
with all possible points (displacements and ) in phase space. The functions to different displacements are overlapping and, therefore, are not independent from each other. The canonical operators are determined up to unitary transformations which preserve the commutation relations and correspond to canonical transformations in classical theory  .
2. The Wigner Quasiprobability in Different Representations and Their Equivalence
In quantum mechanics it is not possible to find a probability function over the phase space which satisfies all requirements for such functions from classical theory for the calculation of expectation values, in particular, positive semi-definiteness. One also cannot translate in a unique way a classical function over the phase space into a quantum-mechanical function of the operators since such a function is ill-defined without giving a rule for the ordering of the non-commuting operators in this function. Thus one has two problems, first to give an ordering rule for operator functions which substitute classical functions and then to find a quasidistribution function which allows to calculate all expectation values of well-defined operator functions and which should be in best agreement with the classical distribution function. This is only possible by compromises.
The functions of the irreducible representation of the Heisenberg-Weyl (Lie) algebra of the canonical commutation relations in an infinite-dimensional Hilbert space form the deeper level discovered by quantum mechanics to which we do not possess a direct access with macroscopic instruments and in almost all measurable expectation values we have it to do with the representation of objects of combinations of or, more generally, .
For functions of Q or of P alone one does not have the ordering problem and expectation values of such operator functions for pure states can be calculated with two (different!) normalized functions or , correspondingly, according to
The two (different) functions and satisfy all requirements for genuine positively semi-definite probability densities.
Wigner, apparently, looked for a quasidistribution function normalized according to
1Wigner denotes it by and generalizes it to n dimensions . He remarks that it was found together with L. Szilard. We apply here Dirac’s notation for states with and write it for density operators with for pure states .
which integrated over one of the canonical variables possesses the property ( density operator)
and which is the best compromise for such a function in quantum theory and found   1
where are the eigenstates of the operator Q to eigenvalues q. By Fourier transformation according to
one finds the analogous formula by the wave functions in momentum representation
In this transformation we used the scalar products
and the completeness relations
The definitions (2.4) and (2.6) are fully equivalent to the following definition ( denotes the trace of an operator A)
showing explicitly the symmetry between the canonical variables in the definition of the Wigner quasiprobability. The equivalence of (2.9) to (2.4) and (2.6) can be proved, for example, as follows. Starting from (2.9) and using the
well-known relation if A and B commute with the commutator
, i.e. , and the completeness of the states according to (2.8), one finds
where we made the substitution with
of the integration variables. In a last step using the completeness of the states according to (2.8) and their scalar products given in (2.7), one calculates
Thus the relation (2.10) in connection with (2.11) proves the equivalence of (2.9) to (2.4).
We now give a representation of the Wigner quasiprobability by the displaced parity operator. With respect to the complete sets of eigenstates or of the Hermitean operators Q or P the parity operator acts according to
and the parity operator itself can be represented by the following equivalent integrals
From this follows
where is the unitary displacement operator defined by2
Factorizations (or disentanglements) of the operator are obtained by the already mentioned theorem used to derive Equation (2.10) and are
The product of two displacement operators is also a displacement operator (group property) multiplied by a phase factor
with the trace
The displaced parity operator is defined as a Hermitean operator by
Using (2.10), (2.18) and (2.19) and one derives the following properties of the displaced parity operator
and according to definition (2.20)
where we used the property given in (2.13).
The Wigner quasiprobability is up to a factor the expectation
value of the displaced parity operator in the following sense
From this follows taking into account that and are right- and left-hand eigenstates of Q
and it can now be seen that in (2.23) is identical with the Wigner definition (2.4).
3. Further Properties of the Wigner Quasiprobability
We refer now further properties of the Wigner quasiprobability (e.g.,         ). Since the density operator and the displaced parity operator are Hermitean operators the Wigner quasiprobability according to (2.23) is a real-valued function
However, it is not positively semi-definite and may possess regions of negativity. It is normalized if the trace of the density operator is normalized (integrations without writing the limits go over the whole phase space)
The Wigner quasiprobability contains the complete information of the density operator and it can be reconstructed from the Wigner quasiprobability by the formula
This can be derived by inserting according to (3.3) into (2.23) and using the properties (2.21) of the displaced parity operator.
The Wigner quasiprobability is restricted by
This follows using the Cauchy-Bunyakovski-Schwarz inequality in operator form ( can be defined as positively semi-definite Hermitean operator) from
and taking into account one has proved (3.4). It can be also derived from (2.4). In comparison to the bound (3.4) a classical probability function of two real variables apart from being non-negative must not possess such a bound.
From (3.3) follows for the product of two density operators and to the Wigner quasiprobabilities and using
or if we apply the product formula (2.18) for displacement operators
For the trace over the product of two density operators and corresponding to two Wigner quasiprobabilities and one obtains (e.g.,      )
If the two states are mutually orthogonal that means if vanishes then the integral over the two Wigner functions also vanishes and vice versa, i.e.
that can be expressed as orthogonality of the corresponding Wigner quasiprobabilities. In particular, the Wigner quasiprobabilities for the discrete orthonormalized eigenstates of a Hermitean operator are orthogonal to each other and in orthormalized form one finds
This is only possible if with exception of the ground state of a Hamiltonian the Wigner quasiprobability to all other states possesses regions of negativities.
In case of one finds from (3.8)
For one has a pure state that requires the equality sign in (3.11) for the integral over the squared Wigner quasiprobability that can be used as indicator for a pure state.
4. The Wigner Quasiprobability for a Particle in a Square Well with Infinitely High Impenetrable Walls
As an example, we consider the stationary Schrödinger equation for a particle in a square well with width a symmetrically to the coordinate origin and with infinitely high impenetrable walls (e.g.,  ) (Figure 1)
The wave functions of the eigenstates to energy ( mass of particle)
as it is well known, are ( Heaviside jump function)
In representation by the coordinate of the canonical momentum (see (2.5)) they are (Figure 1)
Figure 1. Square well potential with infinitely high walls and first 6 eigenfunctions in position and momentum representation.
For the Wigner quasiprobability of an arbitrary state with density operator
calculated from (2.4) we find
For the eigenstates of the Hamilton operator to this system this leads to
They are illustrated in Figure 2.
We emphasize here that the calculation of expectation values from the Wigner quasiprobability is by no means in every case the simplest way. Often it is much simpler to calculate them from the wave functions in position or momentum representation. However, the principal possibility to calculate them from the Wigner quasiprobability shows the way of correspondences between classical and quantum mechanics including the transition from the last to the classical phase space by approximations.
A Hamilton system of one degree of freedom with a general quadratic energy spectrum with constants such as (4.2)3 can be considered by the group with the basis operators of its Lie algebra 
Figure 2. Wigner quasiprobabilities for first 6 eigenstates of square well with infinitely high impenetrable walls . vanishes for . We set ; (made by “Mathematica 6” in 2011).
which satisfy the commutation relations
The Casimir operator C as invariant of the considered irreducible representation is
Thus the considered system corresponds to the index of the
irreducible representation. The completeness relation is here .
The Hamilton operator H of the system can be represented by
In addition we introduce the operators and by
Their commutation relations with the operator are
These relations may be considered as a possible equivalent to the quantum-mechanical commutation relations for classical action j and phase with the classical to quantum correspondences if the action operator J is determined by
Hamilton operator H and action operator J are here not generally proportional as it is the case for a harmonic oscillator and, therefore, is omnipresent in quantum optics. Due to
The operators and are not unitary operators and it is not possible to determine a Hermitean phase operator from them by transition to the Logarithms or . This is related to the fact that in classical theory the transition from canonical variables to action-angle variables possesses a defect as a canonical transformation for with undetermined there.
For systems such as here a quantum-mechanical particle in a potential well or, more generally for example, in other potentials (e.g. Pöschl-Teller potentials  ) the Wigner quasiprobability plays a subsidiary role for the calculation of expectation values since for mixed powers of operators Q and P they are mostly not of great interest. The operators Q and P are here not involved in a symmetrical way in the Hamiltonian and, intuitively, play an “absolute” role independently of possible canonical transformations mixing the variables.
5. Transition to Complex Conjugate Coordinates in Quantum Optics and Coherent States
The following well-known considerations serve at once for the further introduction of our notations. The Hamilton operator to a one-dimensional harmonic oscillator of frequency in quantum mechanics of a (charged)
particle in a potential , ( its second derivative), and with kinetic energy (m mass) can be represented by
By a unitary transformation the
Hamiltonian is transformed to
which is the usual form for a member (mode) of the sum of harmonic oscillators into which a bounded electromagnetic field in quantum optics can be decomposed. Canonical coordinate and momentum are then in “symmetric way” related to the electric and magnetic field and have nothing to do with “coordinate” and moment of the mode as wave packet in free space.
We introduce now the pair of boson annihilation and creation operators and corresponding complex conjugate variable by
with the inversion
From this results the representation of the Wigner quasiprobability by the following substitution in (2.9)
plus the substitution of by with
. The Hamilton operator (5.2) takes on
The transition from (5.2) (or equivalently from (5.6)) to (5.1) and inversely is a special squeezing transformation with real squeezing parameter and
with the correspondence and, therefore, a canonical
The eigenstates (number states) of the number operator N to eigenvalues are
The vacuum state as ground state of the Hamilton operator H in (5.6) for a single mode of the electromagnetic field is in quantum optics considered as the genuine vacuum to this single mode in contrast to possible squeezed vacua but, apparently, in agreement with experiments.
Coherent states are the displaced states (or excitations) of the vacuum state as follows
The displacement operator in variables was introduced by Glauber  and is defined by
The coherent states are the right-hand eigenstates of the operator a to complex eigenvalues
and they are mutually non-orthogonal and (over-)complete. This is well known   and, e.g., , and is written here for convenience. It is, however, less known that the two-dimensional Fourier transformation of coherent states provides again coherent states in the following sense
with specialization to the vacuum state (vacuum state possesses even parity, i.e., )
Relation (5.11) is one of the possible forms to show the over-completeness of the coherent states. This can be also proved using the group integral over the displacement operators (see Appendix B)
providing the parity operator as “sum” (integral) over all weighted elements of the Heisenberg-Weyl group.
A general group relation for arbitrary operators A which includes the (over-)completeness of coherent states is the following ( is trace of A)
It is written in  in another but equivalent form (Equation (3.9) there) and is the special case for the Heisenberg-Weyl group (with usual basis of its Lie algebra) of a general relation for unitary irreducible representations of a group with an invariant measure and corresponds to the lemmas of Schur (properly to the first; the second lemma of Schur includes also inequivalent irreducible representations)4. In Appendix B we give a derivation of this relation for the case of the Heisenberg-Weyl group.
A more special form is obtained if we insert in (5.14) the number-state operators . With the following two equivalent definitions of the displaced number states (see Section 7 for more details)
and find from (5.14) the special form
For and results from this the well-known completeness relation of the coherent states
and for the following series of special forms
This operator relation can be multiplied by arbitrary states and we obtain different forms of linear dependence between the coherent states.
The parity operator using the completeness of the coherent states (5.17) can be represented now by
Using the action of the displacement operators from this relation follows
where are displaced number states . Thus the displaced parity
operator is the Fourier transformation of the operator with
respect to variables .
6. The Wigner Quasiprobability in Complex Conjugate Coordinates and Related Quasiprobabilities of Quantum Optics
The coherent states form the basis of the representations of the quasiprobabilities used in quantum optics. The representation of in complex conjugate coordinates by the displaced parity operator and its representation by displaced number states (5.15) possesses the form (see Section 2, Equation (2.23) and also , Equation (4.1) there)
Inserting the representation (5.20) for the displaced parity operator into this relation immediately follows
In its structure this is up to a constant factor a two-dimensional Fourier transformation of the matrix elements where the involved states are the coherent states5. In this sense it is widely in analogy two the one-dimensional Fourier transformation of the matrix elements in the definition (2.4) given by Wigner (or equivalently in (2.6)). Apart from the possibility of alternative calculations of the Wigner quasiprobability for special states the knowledge of different representation may become important in case that one gains a deeper insight in the nature of the emergence of such functions.
The reconstruction of the density operator from the Wigner quasiprobability can be made by the formula
The Wigner quasiprobability in representation by the variables is restricted in its values by
As another possible basic definition of the (normalized) Wigner quasiprobability in complex conjugate coordinates follows from (2.9) using (5.5)
where the delta “functions” in representation by real coordinates and complex
conjugate coordinates are related by . Using (6.1) and
taking into account that is an arbitrary density operator from (6.5) follows the pure operator relation
It is normalized as follows by partial integration
This is in agreement with
where we used (7.9) but the arising geometric series is only conditionally convergent.
In anti normal ordering of the operator involved in the definition (6.5) one finds
Introducing into (6.9) the completeness relation (5.17) of the coherent states one obtains from (6.5)
One may introduce in this way by the coherent-state quasiprobability also called Husimi-Kano quasiprobability which is connected with the Wigner quasiprobability as follows (“*” is notation for convolution of two functions)
The normalization of follows immediately using partial integration
The coherent-state quasiprobability (correspondingly ) is real-valued and restricted by
due to definition (6.11). It is easy to show that similarly to the Wigner quasiprobability a displacement of the state displaces the function corresponding to according to . In the process of “desmoothing” the singular zero points of form the germ cells for the formation of regions of negative values of the Wigner quasiprobability .
In comparison to the Wigner quasiprobability the quasiprobability
is smoothed by the operator that is not connected
with a loss of information. Inversely from given according to (6.11) one may derive the basic formula (6.11) for the Wigner quasiprobability obtained from . In Appendix C we give a derivation with interesting intermediate formulae for applications.
One may consider more general real-valued quasiprobabilities with a real parameter r by the following convolution of the Wigner quasiprobability with normalized Gaussian functions
For one obtains from the Husimi-Kano quasiprobability and for the Glauber-Sudarshan quasiprobability which may be indirectly introduced by the following formula   (Equation (7.48)) and  (chap. 13) ( means normal ordering of content in braces)
At first place is here set the reconstruction formula of the density operator from the function .
The quasiprobability possesses an interesting property. All classical normalized probability functions which are by definition positively semi-definite but may be not bounded provide inserted into the formula for in (6.15) a possible quantum-optical density operator of one mode. For coherent states one finds and from pure states these are the only states with positively semi-definite (generalized) functions as they are also possible for classical probability functions. All other classical probability functions inserted in (6.15) provide mixed quantum-optical density operators. The corresponding Wigner quasiprobabilities for such states since they are smoothed functions of according to
cannot possess regions of negativity or values equal to zero but they are not identical with all non-negative such functions. For example, from the pure states they do not give the squeezed coherent states. The mentioned states with classical probability functions can be considered at once as the “most classical” quantum-optical states and a measure of nonclassicality of an arbitrary state may be defined as the nearest distance to such a state (in case of pure states the nearest distance to a coherent state   ).
The possible quantum-optical functions form a greater manifold compared with the possible classical probability functions . For the transition to classical optics by the limiting procedure all possible quantum-optical (usually generalized) functions have to make the transition into a classically possible probability function which is then not specific for its quantum-optical origin that may be one key to a deeper understanding of this function which seems to be absent now. It has to be noted that for the transition one has primarily to make the transition from variables to real canonical variables according to (6.3). Planck’s constant is involved in the denominators of and apart from thermodynamic formulae (mixed states) does not fully appear in representations of the quasiprobabilities by the complex variables .
7. Wigner Quasiprobability in Number-State Representation
In this Section we derive the number representation of the displaced number states and formulae for the Wigner quasiprobability of arbitrary states in expansions of the number states. For symmetries in the formulae it is favorable to use the Laguerre 2D polynomials instead of the usual generalized Laguerre or Laguerre-Sonin polynomials for the representations. The Laguerre 2D polynomials are alternatively defined by (e.g.,  )
with the special and limiting cases
For the derivatives of from (7.1) follows
The more general formulae
follow immediately by repeated application of derivatives from (7.3).
The relation of the Laguerre 2D polynomials to the generalized Laguerre (or Laguerre-Sonin) polynomials is 
that shows that the Laguerre-Sonin polynomials may be considered as the radial rudiments of the Laguerre 2D polynomials.
We now consider the displaced number states defined in (5.15). By expansion of the operators as functions of in Taylor series and accomplishment of their actions onto the number states it is easy to obtain the following representation of the displaced number states by the number states  (Equation (5.8))
We see using (7.2) that the special cases and are correctly obtained. From (7.6) follows for the scalar products of displaced number states
Another basic relation for the displaced number states follows from (5.14) inserting there (see Appendix D)
For this means the (over-)completeness of the displaced number states for arbitrary including the coherent states as special case . In (7.8) the numbers are fixed. If one fixes the displacement one finds immediately
This is a completeness relation of the infinite set of displaced number states for arbitrary fixed and together with (7.8) we have for them two different systems of completeness relations.
The Wigner quasiprobability in number-state representation is
Using now the following formula of the form of a generating function for products of Laguerre 2D polynomials  (Equation (8.4))
in the special case we find for the Wigner quasiprobability from (7.10)
In representation by the canonical variables this is equivalent to (see transformations (5.3) and (5.5))
In special case of the Wigner quasiprobabilities for the number states one finds from (7.12) 
and from (7.13)
The Wigner quasiprobability for the first 6 number states is illustrated in Figure 3. In principle, it is known  but we illustrate it for easy comparison with the eigenstates of the Hamiltonian for an impenetrable square well discussed in Section 4.
8. Coherent Phase States with London Phase States as Their Limiting Case and Quasi-Distribution of Phase
In this and in the following Sections we discuss coherent phase states and calculate their Wigner quasiprobability.
As phase states one defines the eigenstates of phase operators. Since the phase is only defined up to multiples of 2π only 2π-periodically functions of the phase are uniquely defined and as classical basic functions one may choose the functions which play a role in Fourier analysis. For the
Figure 3. Wigner quasiprobability for first 6 number (Fock) states to . The maximal possible height is and the maximal possible depth . We set ; (made by “Mathematica 6” in 2011).
quantum-mechanical equivalent operators there exist two concepts. The first is the general Weyl correspondence of symmetrically ordered operators  and its implementation for different categories of functions (see, e.g.,  ). The other possibility is to choose the Susskind-Glogower formalism  of correspondence of the operators and as substitution of the classical functions and    of the phase and which are
where are the number states as normalized eigenstates of the harmonic oscillator. From this follows
We consider now the normalizable right-hand and left-hand eigenstates of the operator to eigenvalues or , respectively, according to
which are well known (e.g.,  ) and are
They are called coherent phase states. The scalar product of two coherent phase states is
It affirms the chosen normalization factor in (8.4) and shows that the coherent phase states are nonorthogonal. They are (over-)complete and are already complete for arbitrary constant with that can be shown by contour integral in connection with the dual states  . That they are (over-)complete as a whole follows from ( )
In the limiting case the coherent phase states become non-normalizable. We define
These are the London phase states (after Fritz London6  ). They are complete according to
Their scalar product is
This is a generalized function, here an analytic functional, which for , does not give a finite value of . To make it a genuine analytic functional of variable we have to determine the way to deal with the singularity at that we will not discuss here.
The London phase states can be used to define for arbitrary (pure or mixed) states with density operator a normalized 2π-periodic phase distribution according to ( )
Since the London phase states are non-orthogonal it is not a genuine probability but only a “one-dimensional” quasiprobability (however, of other kind than the Wigner quasiprobability since it is positively semi-definite) defined over the unit circle .
For coherent phase states one finds for
The centered distribution for coherent phase states (i.e., for ) which we denote by possesses a very simple Fourier decomposition
The used identities are well known from the theory of Chebyshev polynomials and involve the relation and one of the generating functions for the Chebyshev polynomials of first kind (e.g., , chap. 10.11., Equations (2) and (29)). The Fourier series of the more general distribution (8.11) can be obtained from (8.12) by the substitution . The phase distribution (8.12) is illustrated in Figure 4 for a coherent phase state with (and ).
The Fourier amplitudes in (8.12) are strictly decreasing for from on to both sides of n. In the limiting case one derives or affirms from (8.12) the known relation of the theory of generalized functions
Figure 4. Phase distribution for a phase state with and folded and unfolded. The mean value of the number operator N is from which follows ; (“Mathematica 10” in 2018).
(e.g., Vladimirov , chap. 2, § 9, Equation (35))
which is also an example for the Poisson transformation of a sum from ordinary to reciprocal lattice.
9. Coherent Phase States as SU(1,1) Coherent States
Up to now we developed the elementary theory of coherent phase states . Less elementary is that the coherent phase states are coherent states     . The definition (8.4) of coherent phase states can be also written
form a realization of the Lie algebra with the commutation relations (see also (4.9))
and with the Casimir operator (confer also (4.9) with the same value k)
Interestingly, the index of the unitary irreducible representation of
is the same as for the system of a particle in a square well with infinitely high impenetrable walls (see Section 4).
The generation of the coherent phase states from the vacuum state by a unitary operator can now be written
and (9.1) follows from the disentanglement of this unitary operator. The first relation on the right-hand side is the unitary approach that means the application of a unitary operator to the vacuum state which guarantees the normalization . The second relation on the right-hand side is that what we call the non-unitary approach meaning the application of the non-unitary operator to the vacuum state. The normalization factor can be obtained in this case by finding disentanglement relations for general group operations into products of simpler group elements. All formulae for the group given in Appendix A of  among them the disentanglement relations are applicable to the present case.
10. Wave Functions and Further Characteristics of Coherent Phase States
We begin with the Bargmann (-Segal) representation of states (e.g.,  ) which is the scalar product of the considered state with analytic but non-normalized
coherent states concerning complex variable and
which leads in our case of coherent phase states to
where we introduced the very unorthodox entire function of the complex variable z defined by
This function which is difficult to deal with plays a role, practically, in all representations of the coherent phase states   7. Although the Taylor series (10.2) represents an entire function which, therefore, converges in the whole complex plane this convergence is “very slow” and direct computer calculations from the series for high values of (say, for about ) are very difficult to make. About some progress with this function we give a few information in Appendix D.
From the Bargmann representation (10.1) of the coherent phase states one obtains immediately the Husimi-Kano quasiprobability according to
From the normalization of according to (6.12) follows by the substitutions
This means that increases for real exponentially not faster than with arbitrary small . The order of the
entire function is equal to (see  and Appendix D).
For the wave functions one finds
where we applied the generation of the Hermite polynomials from the monomials according to
with a parameter c. With the function defined in (10.2) this and analogously can be written
The operator with applied to a function makes a smoothing of this function and the operator with
, correspondingly, the opposite of this (“desmoothing”).
11. Calculation of the Wigner Quasiprobability of a Coherent Phase State
For the calculation of the Wigner quasiprobability we have different possibilities. The first is that we calculate it from the general relation to the Husimi-Kano quasiprobability and using the special form (10.3) of the last (see Appendix C)
This is in number representation equivalent to the formula (see (7.12) and (7.13))
where and can be taken from (8.4) and which we used for the calculations. To get we have to transform it by relations (5.3). A second possibility is the calculation directly from the wave function or from via definitions (2.4) or (2.6) with density operator . A difficulty in the calculation of the Wigner quasiprobability for coherent phase states is the slow convergence of the involved series and the presence of the unorthodox entire function in the formulae. The convergence is caused only by the powers which for is very slow.
In Figure 5 we illustrate the calculation of the Wigner quasiprobability for a coherent phase state for real , i.e. for , made by formula (C.11) using their number representation (8.4) with “Mathematica 6” in 20118.
The maximal value of the Wigner quasiprobability occurs under increasing parameter for increasing values and does not reach
Figure 5. Wigner quasiprobability for a coherent phase state compared with a squeezed coherent state in units . The parameters of the coherent phase states are: , , , , , and therefore . The parameters of the squeezed coherent state are: , , , , and therefore . We set . The partial picture in second position of second line which is a front sight onto the Wigner quasiprobability shows weakly its negativities. A good graphical representation of the small negativities is only possible by showing a small part of the Wigner quasiprobability with distortion of the natural scales. It shows that the greatest negativities lie in the neighborhood of small values of q corresponding to the zeros of the non-orthodox function ; (made by “Mathematica 6” in 2011).
the maximally possible value for Wigner quasiprobabilities, in contrast to
that for squeezed coherent states, which apart from their displacement from the coordinate origin in comparison the squeezed vacuum states are eigenstates of the parity operator to parity +1. The Wigner quasiprobability for coherent phase states possesses also regions of negativity. They appear in the neighborhood of the (singular) zeros of the Husimi-Kano quasiprobability since the last is some averaging of the Wigner quasiprobability and this is only possible if the Wigner quasiprobability possesses regions of negativity in its neighborhood. According to (10.3) the singular zeros of (or ) are determined by the zeros of the unorthodox function (see also Appendix D). That the Wigner quasiprobability for coherent phase states should possess regions of negativity is also clear from the product (Weierstrass) representation of analytic functions because such functions without zeros are functions of the exponential type and the squeezed coherent states are already the most general set of normalizable functions with wave functions and Wigner quasiprobabilities of Gaussian type.
Let us make a few remarks about the article  . According to its title, in this article the Wigner function of a phase state is calculated and some related functions are represented graphically. As phase state the authors use the London phase states given here in (8.7) which are non-normalizable (see (8.9)). They declare as the density operator but it is and the obtained “Wigner function” is also non-normalizable and, therefore, is not a genuine Wigner quasiprobability. Expectation values cannot be calculated with such a function. One may think that the calculated function plays, at least, an auxiliary role for the calculation of interesting properties of normalizable phase states for defined in (8.4) from which the London phase states (8.7) follow as their limiting case without the normalization factor . However, we did not find a relation where they may play an auxiliary role. The authors  made their London phase states finite by truncating the sum over n in (8.7) with certain but then the scalar product of these
changed states is and is not normalized to the value equal to 1 and the
results for the Wigner function are badly defined. In case of our normalizable coherent phase states the truncation is made “in natural way” by the powers for which is a very slowly convergent procedure and which caused our extremely long calculation times.
We have to underline the following. To get use from the calculated functions in  (or its generalization for ) as an auxiliary quantity for calculations, for example, of expectation values it has to be given, at least, in general analytic form. However, due to the involved unorthodox entire function this is very difficult to achieve. This also concerns our calculations of a genuine example of the Wigner quasiprobability for coherent phase states in Figure 5. This may only serve as a certain illustration but for the calculation of expectation values from the Wigner quasiprobability this is inappropriate. Some expectation values for the coherent phase states can be effectively calculated in other way that we demonstrate in the next Sections.
12. Expectation Values of Functions of the Number Operator N for Coherent Phase States
The calculation of expectation values via the Wigner quasiprobability is the most illustrative one concerning the analogy to classical phase space but it is not in every case the simplest one and in case of coherent phase states the difficulties are almost not to overcome. Therefore, we discuss here and in the next Section some other possibilities of calculation of expectation values for coherent phase states .
The expectation values of powers of the number operator N are
where denotes the Eulerian numbers   and also  with some of the first given in Table 1.
For which by definition is the boundary between sub- and super-Poissonian statistics we find
Table 1. Eulerian numbers .
This shows that among the coherent phase states are not states with sub-Poissonian statistics for which by definition should hold . The measure shows that the coherent phase states possess for a great asymmetry around the mean values . Furthermore from (12.2) follows
We use this result in Section 13 in the calculation of and .
The Hilbert-Schmidt distance to the vacuum state with density operator is
For the Hilbert-Schmidt distance   to the vacuum state goes to the maximally possible ones equal to . The Hilbert-Schmidt distance of to an arbitrary coherent state is
The minimal distance to a coherent state as possible measure of non-classicality of a state  is obtained if and possess the same phase and is then determined by the minimum of (12.6) or the maximum of with respect to variation of . Due to the difficulties with the unorthodox function it is not easy to derive approximations. For the minimal distance (12.6) goes to that means to the maximal possible one for normalized states.
13. Expectation Values for Canonical Operators Q and P and Their Variances for Coherent Phase States
In this Section we discuss the calculation of expectation values of the canonical operators Q and P and of simple functions of these operators for coherent phase states .
We begin with the calculation of the expectation value of the annihilation operator a for which we find from (8.4) the following series
In the same way we find for the expectation value of the operator a2
The right-hand sides of (13.1) and (13.2) are series roughly speaking of the type of a Geometric series , however, with additional slowly increasing coefficients in front of which we more generally denoted and evaluated in  by
The numbers are generalized Eulerian numbers for which we found the general formula
We see that the sum (13.1) is of the type and (13.2) of the type of a Generalized Geometric series where is not an integer and, therefore, where the Generalized Eulerian numbers according to
(13.4) are non-vanishing for arbitrary . For integer k the Eulerian numbers are different from zero only for as shown in preceding Section    . Thus we may write the result of evaluation of (13.1) in the form
and the result of evaluation of (13.2) in the form
The splitting of the functions in our case from and of from is very important for the acceleration of the
convergence of series in the evaluation, in particular, for in the neighborhood of and for approximate solutions it is then necessary to
take into account only a few number of terms of the series or . With a few terms of the expansion we find for the series involved in (13.5)
satisfying for the inequality
and for the series involved in (13.7)
satisfying for the inequality
In the following we need also the square of the expression (13.9) for which we find the first sum terms of the expansion
As an interesting observation we remark that in contrast to the expectation values and in (13.13) in the coefficients in braces we do not have denominators different from 1 in the corresponding expansions for the squared expectation values and . We checked that this is true even up to 25 sum terms in the approximation but do not try to generalize it for arbitrary order and to prove it. Although this seems to be unimportant for numerical calculations it is perhaps interesting for number-theoretic considerations (likely, this is
connected with in the generalized Eulerian numbers that their
squares possess the mentioned property).
For the expectation values of the canonical operators Q and P we find
For their squares we calculate
and for their product
We come now to the expectation values of the operators and for which we find using (13.7)
We inserted here the result (12.4) for the expectation value of . The lastly needed expectation value of quadratic combinations of the canonical operators Q and P is
This expression is vanishing for that means if is a real or an imaginary number. In all other cases of complex numbers it is non-vanishing.
Using the results of this Section we calculate in next Section variances to the expectation values of the canonical operators.
14. Uncertainties, Uncertainty Matrix and Uncertainty Correlations
The uncertainties of the expectation values of the canonical operators emphasizing their transformation and invariance properties are well summarized in the following symmetric uncertainty matrix (e.g.,  )
where and and expectation values are defined by with the density operator of the considered state. In the preceding Section we prepared the calculation of this uncertainty matrix for coherent phase states with density operator by the calculation of partial building stones.
By combination of the results in Equations (13.17) and (13.15) we find for the variances and
and by combination of (13.18) and (13.16) for the uncertainty correlation
We find the following few first terms of expansions of common partial expressions in (14.2) and (14.3)
The corresponding expansion for the expression was
already given in (13.13).
The uncertainty sum is the trace of the uncertainty matrix and we find
It depends only on the modulus of the complex but does not depend on the angle . Thus it is invariant with respect to rotations of the coherent phase states in the phase plane of variables . In the limiting case using (13.13) and in the limiting case we obtain
The second invariant of the two-dimensional matrix with respect to similarity transformations is its determinant which provides here the uncertainty product taking into account the uncertainty correlations for which we find using (13.17) and (13.18) together with (13.15) and (13.16)
In the same way as the uncertainty sum it depends only on the modulus of but not on its angle. If we apply the binomial formula to the two quadratic terms in (14.7) then the last two terms cancel but we do not come in this way to an essentially shorter representation. We come back to this below. In the limiting cases we find
The most expressions which we may calculate from Q and P are not invariant with respect to the phase of the complex parameter . For example, the mean values and depend on the phase of . If we choose
then we find
The vector of the mean values lies on the positive q-axis and continuously increases there with increasing . For the squares we find
and for the expectation values and
For the corresponding variances we now obtain the formally asymmetric expressions for and
The uncertainty correlation (14.3) vanishes in the special case (14.9) that means
and the matrix (14.1) becomes diagonal. The uncertainty sum is invariant with respect to rotations of and therefore we find from (14.13) again the expression (14.5). For the uncertainty product in the special case (14.9) follows from (14.13)
This is the same as (14.7) in an alternative representation but in the considered special case (14.9) the uncertainty correlation vanishes. In general, the uncertainty product is not invariant with respect of rotations of the state in the phase space or here with respect to the phase of . The uncertainties obey the following general chain of inequalities 
As known, in general, the minimal uncertainty product for a given state is only obtained under special positions of the axes of the phase space but the uncertainty product taking into account the uncertainty correlations provide it in every case. The inequality between the square of the uncertainty product taking into account the uncertainty correlation and the half uncertainty sum is the inequality between geometric and arithmetic mean. For imaginary parameter instead of real one the roles of the axes are interchanged.
The uncertainties for coherent phase states are graphically represented in Figure 6.
We derived in this article formulae for the calculation of the Wigner quasiprobability with emphasis of relations which are symmetric in the canonical operators . We applied them to the eigenstates of the one-dimensional square well with infinitely high impenetrable walls and in quantum optics to the coherent phase states. Both considered examples are quantum-mechanical systems with symmetry. For the coherent phase states we discussed in detail the difficulties of calculations due to a very unorthodox entire function involved in such calculations. We tried to bring more light in the calculations of properties of coherent phase states and of their Wigner quasiprobability which did not be dealt with up to now in satisfactory form.
An interesting question without a clear answer is whether nature can realize
Figure 6. Uncertainties for coherent phase states . In the partial figure to the left we have chosen in and, therefore, . The partial figure to the right is invariant concerning the choice of . In the lower curve we see 4 different approximations. They are generated with taking into account different numbers of sum terms in the expressions which were 4, 16, 64, 256. With the upper curve it was made the same but here the differences are not to see; (“Mathematica 10” in 2018).
coherent phase states or not. It is meant by this not only an approximate realization, for example, by superposition of a few coherent states on a line beginning from the center or by squeezed coherent states. The reason as mentioned is that in the description of the genuine coherent phase states a very unorthodox function
plays a main role which by digital calculations makes considerable
trouble. Is nature really capable to realize this by an analog physical process without such a trouble?
The Wigner quasiprobability is an essential mean for the illustration of the connection of quantum theory with classical mechanics but not in all cases the calculation of properties of states from the Wigner quasiprobability is the most suited and easiest one as the considered examples show (remind that it was first developed for corrections to thermodynamic equilibrium). Interesting in a wider sense is, for example, also the interferences seen in the Wigner quasiprobability for superpositions of coherent states in the regions of their overlapping which, however, could not be considered here. A deeper explanation of the origin of the Wigner quasiprobability in possible formulae for expectation values would be desirable. We did not discuss here the transition from quantum to classical mechanics by transition to the infinite number of commutative irreducible representations of the basic operators of the Heisenberg-Weyl algebra for which the operators form appropriate bases in the space  .
Finally, we mention that the Wigner quasiprobability possesses also something which is to feel as a very aesthetical aspect.
S. Skorokhodov sent me a nice e-mail in 2008 (see  with essential passages in the footnote in Appendix D) but in preparation of the present work I was no more able with my present computer to open the appended file. I thank him very much. In the following years after 2008 I was involved in other topics and was confronted with other problems and so I only now genuinely returned to this.
Fourier Transformation of Coherent States and Representation of Displaced Parity Operator
For the following we need some auxiliary integrals. By transition to polar (or real) coordinates one finds
where c is a parameter. From this one calculates the following more general integrals (see (7.1))
With the substitutions one obtains the special
We apply this now.
Using the first formula in (A.3) with one calculates the Fourier transform of a coherent state according to
and in analogous way
and the displaced parity operator becomes
This together with (6.1) leads to the representation (6.2) of the Wigner quasiprobability.
Completeness Relation over Heisenberg-Weyl Group and Application to Displaced Number States
From the two-dimensional integrals over the complex z-plane
which can be calculated by transition to polar coordinates one obtains
Now, using the definitions (7.1) of Laguerre 2D polynomials and their differentiations (7.3) and, furthermore, partial integration we calculate the following integrals (remind )
From this follows using the decomposition (7.6) of displaced number states into number states
For an arbitrary operator A follows from this relation
This is the (over-)completeness integral (5.14) for displaced operators A and follows also from general orthogonality relations of group representations if one knows the invariant measure over the Lie group.
In an analogous way using the number representation of the coherent states one calculates
where we used the evaluation of the integral in (B.1).
Wigner Quasiprobability W(α, α*) from Husimi-Kano Quasiprobability Q(α, α*) via Bargmann Representation
The Wigner quasiprobability can be obtained from the Husimi-Kano quasiprobability by the relation
We introduce now in the Bargmann representation of a pure state by an analytic function of the complex variable
where are the analytic (non-normalized) coherent states.
and the Wigner quasiprobability (C.1) becomes
We will give this formula another principal form using the Lie group . This may be useful for similar considerations to more general quasiprobabilities.
Introducing the operators
one finds that they satisfy the commutation relations (9.3) and, therefore, form a realization of the Lie algebra to the Lie group . The corresponding Casimir operator C is
Using the commutation relations
one may transforms (C.4)
to the form
The entangled operator of can be
disentangled using the relations in  (Appendix A, Equation (A.14)) according to
belonging to the unimodular matrix with (see ,
Equation (A.7)). This leads to the representation of (C.7)
The applied technique is a special case ( ) of the technique applied
in  (Appendix A there) and the operator is the operator of
multiplication of both arguments