Wigner-Weyl-Moyal quantization procedures and asymptotic morphisms are described as general quantization procedures, beyond first, second or canonical quantization methods employed in quantum theories.
The more general quantization techniques beyond canonical quantization revolve around using operator
kernels in representing asymptotic morphisms. A fundamental example is an asymptotic
morphism
as expressed by the Moyal
`deformation' :
where
and the operators
are of trace
class. In Connes (1994), it is called the `Heisenberg deformation'.
An elegant way of generalizing this construction entails the introduction of the
tangent groupoid,
, of a suitable space
and using asymptotic morphisms. Putting aside
a number of technical details which can be found in either Connes (1994) or Landsman (1998), the
tangent groupoid
is defined as the normal groupoid
of a pair Lie groupoid
which is obtained by `blowing up' the diagonal
in
. More specifically, if
is a (smooth) manifold, then let
and
, from which it can be seen
and
. Then in terms of disjoint unions one has:
In this way
shapes up both as a smooth groupoid
,
as well as a manifold
with boundary.
Quantization relative to
is outlined by Várilly (1997) to which the reader is referred
for further details. The procedure entails characterizing a function
on
in terms of a
pair of functions on
and
respectively, the first of which will be a kernel and the
second will be the inverse Fourier transform
of a function defined on
. It will be
instructive to consider the case
as a suitable example. Thus, one can take a function
on
whose inverse Fourier transform
yields a function on
. Consider next the terms
which on solving leads to
and
. Then,
the following family of operator kernels
This mechanism can be generalized to quantize any function on
when
is a Riemannian
manifold, and produces an asymptotic morphism
.
Furthermore, there is the corresponding K-theory map
, which is the
analytic index map of Atiyah-Singer (see Berline et al., 1991, Connes, 1994). As an example,
suppose
is an even dimensional spin
manifold together with a `prequantum' line bundle
. Then one can define a `twisted Dirac operator',
, and a `virtual' Hilbert space
given by
This subsection defines the important notion of an asymptotic morphism following Connes
(1994). Suppose we have two C*-algebras (see below)
and
, together
with a continuous field
of C*-algebras over
whose fiber at
0
is
,and whose restriction to
is the constant field
with fiber
, for
. This may be called a strong 'deformation'
from
to
.
For any
, it can be shown that there exists a continuous
section
of the above field satisfying
. Choosing such an
for each
, we set
,
for all
.
Given the continuity of norm
for any continuous section
,
consider the following conditions :
Then an asymptotic morphism from
to
is given by a family of
maps
, from
to
satisfying conditions (1) and (2) above.
As of this snapshot date, this entry was owned by bci1.