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# Special Distribution Functions

In the same way in which the kernel determines the association between operators and scalars, it also defines an operator ordering [Meh77,AM77]. In this context an expansion (A.7) of an operator is called ordered if it is written as a superposition of terms that are of the form . The essential point of this definition is that the multiplication of the exponential with the kernel function yields a characteristic composition of exponential operators, the explicit form of which depends on the particular choice of .

The implications of this definition become clearer when some specific examples are considered:

• For the kernel function
 (A.10)

one obtains from equation (A.13), integrating over and and substituting , the WIGNER distribution function [Wig32]:
 (A.11)

It corresponds to the WEYL ordering of operators [Wey31,DGS72],
 (A.12)

and by equations (A.7, A.8) gives rise to the WEYL transform of the operator :
 (A.13)

The last two terms of expression (A.16) indicate the association of operators and scalars that is defined by , as discussed in section A.1.

Using the WIGNER distribution function, the overlap of two states and described by the WIGNER functions and can be computed as the overlap of the corresponding WIGNER functions in phase space:

 (A.14)

Besides the HUSIMI distribution function (which is introduced in section A.3 below), the WIGNER function is the most commonly used quantum phase space distribution function.

• The kernel function
 (A.15)

defines the standard ordering of operators; it is characterized by all -dependent terms preceding the -dependent terms,
 (A.16)

which leads to the standard-ordered distribution functionA.4:
 (A.17)

Essentially, is the product of the position and momentum representations of the state .

• Setting
 (A.18)

and thus having all -dependent terms precede the -dependent terms, one gets the antistandard-ordered distribution function (also known as the KIRKWOOD distribution function [Kir33] or RIHACZEK distribution function [Rih68]):
 (A.19)

It is obtained from its standard-ordered counterpart by complex conjugation.

• For systems that can be described as a harmonic oscillator with mass and frequency -- in this appendix no scaling as in section 2.1 is employed, such that the parameters and are retained in the formulae -- or as an ensemble of harmonic oscillators, the normal-ordered and the antinormal-ordered distribution functions and are useful.A.5They are defined by requiring operators to be (anti-) standard-ordered not with respect to , , but with respect to the ladder operators, i.e. with respect to the creation operator and the annihilation operator , where is given by
 (A.20)

as usual. (Equation (2.30) is obtained from this definition in the case of the scaling (1.15, 2.4), i.e. by formally setting .)

This (anti-) standard ordering of operators is achieved by using the kernels

as can easily be confirmed by direct computation:A.6

where the complex parameter is defined as

 (A.19)

The corresponding distribution functions are

The normal-ordered distribution is also called GLAUBER-SUDARSHAN distribution function or -function [Gla63b,Gla65,Sud63], while the antinormal-ordered distribution is sometimes referred to as the -function [Gla65].

Defining as a Gaussian wave packet centered at in phase space,A.7

 (A.19)

with , the antinormal-ordered distribution function can be written in a particularly concise way:
 (A.20)

as is easily verified by substituting formula (A.29) into equation (A.30) and comparing the result with the definition (A.13) for .

Therefore, is essentially obtained by computing the convolution of the state with a Gaussian in position space. The physical meaning of this convolution becomes clearer in section A.5. From equation (A.30) it is also clear that is non-negative; this is of importance for the interpretation of as a (quasi-) probability distribution function in section A.5. in the form of equation (A.30) finds its main application in the discussion of the HUSIMI distribution function in section A.3.

Since there are infinitely many different functions that can be chosen as kernel functions, there exist equally many different distribution functions . The important point is that all these different are equivalent: each of them contains the same information about the state , and each can be used to compute the expectation values (A.9a) of any operator that is expanded as in equation (A.7).

The connection between the WIGNER distribution function and the antinormal-ordered distribution function is an important example:

 (A.21)

This unveils the antinormal-ordered distribution function as the convolution of the WIGNER function with a Gaussian in phase space; I discuss this fact in some more detail in section A.5.

Formula (A.31) may be proved in the following way: using similar arguments as those leading to equation (A.13), with the definition (A.6) I first rewrite as

 (A.22)

and obtain for the expectation value in the integrand:
 (A.23)

equation (A.31) then follows by integration over and .

Other conversion formulae between arbitrary different distribution functions and with can be obtained by similar computations. Some formulae of this type are listed in [Lee95].

#### Footnotes

... functionA.4
This slightly inaccurate naming convention is common practice. In order to avoid misunderstandings I want to stress here that in the narrow sense it is only operators that are (or are not) standard-ordered. For distribution functions (standard) ordering is not defined at all. Therefore, the standard-ordered distribution is not standard-ordered. Similar statements hold for all the other orderings of operators and their associated distribution functions.
... useful.A.5
Typical applications may be found in quantum optics; see [Vou94,KH95b] for some examples. Another field where and are utilized frequently is the modelling of a heat bath by an ensemble of harmonic oscillators [Coh94,HB95,HB96]. Cf. also the monographs by Louisell [Lou64,Lou73] and Dineykhan et al. [DEGN95].
... computation:A.6