Phase Space Distribution Functions

One possible motivation for introducing quantum distribution functions is their utility for the comparison of classical and quantum mechanics, as mentioned above. In addition to this there is another, equally important motivation for studying these functions: they can be used to compute expectation values in a comparatively simple way, where the computational simplification is mainly due to the fact that the corresponding formulae depend on scalars only, as opposed to conventional expressions of quantum expectation values which typically involve operators.

Consider a classical observable that depends on the position and
momentum variables and . The expectation value of can be
computed as

Rather than discussing
a general
quantum
observable
, with the
position and the momentum operators and , I begin by
considering a particular operator instead, namely
,
with constants
.
Exponentials of this type are used below in the FOURIER expansion
(A.7) to construct any other operator.
In order to obtain an expression analogous to equation (A.1)
the operator
somehow has to be substituted by a
corresponding scalar expression. For example one could set

as well, with
another
distribution function .
But due to the fact that and do not commute one has

(A.1) |

, respectively.

In order to avoid this ambiguity one first chooses a complex-valued
*kernel function* ; then the scalar
is defined to be associated with the operator
*exclusively*,

The
quantization rule
(A.5) not only defines how to
associate exponential operators with scalars, but is much more general,
as it can be applied to each term of the FOURIER expansion of any
operator ,

which is defined unambiguously.

where equation (A.9b) is the desired expression analogous to equation (A.1).

An *explicit* expression for the distribution function
can be obtained from the *implicit* definition (A.6) by
FOURIER transformation. For a system in the state
at
time , the expectation values are given by
, and one gets

which by insertion of the identity operator can also be written as

The exponential acts as a translation operator in position space (cf. equation (5.23));

such that

Inserting this into equation (A.10) and substituting , I finally obtain a convenient explicit formula for the distribution function :

- ...
formula.
^{A.2} -
A simplified BAKER-CAMPBELL-HAUSDORFF (BCH) formula

holds in the special case when the operators and both commute with their commutator (which is true in the present case, because is a c-number). See [Per93] for a proof, and [Wil67,Ote91] for more on BCH formulae. - ... unambiguously.
^{A.3} -
The question needs to be addressed if the theory,
and in particular the evaluation of integrals like that in equation
(A.8),
could be spoiled by
kernel functions that have zeroes or are singular for some values
of . I do not discuss this issue here, but refer the reader
to [SPM99] where it is shown that the formalism can be
applied smoothly even in such notorious cases.
Note that the kernel function can also be chosen, more generally, as a functional of the quantum state of the system itself: . COHEN gives an example for such a kernel function that, despite being quite complicated, leads to the very simple

*COHEN distribution function*that nicely combines the position and momentum representations of in an intuitive way [Coh66]. However, since choosing a -dependent has a number of unfavourable consequences -- for example, equation (A.8) indicates that in this case the function associated with the operator becomes -dependent, too -- I do not further discuss kernels that are functionals of .