1. What is AQ?
The simplest way to understand AQ is to derive it from CQ. The classical variables, p & q, lead to self-adjoint quantum operators, P & Q, that cover the real line, i.e.,
, and obey
. Next we introduce several versions of
, specifically
(1)
This equation serves to introduce the “dilation” operator
1 which leads to
. While
are the foundation of CQ,
are the foundation of AQ. Another way to examine this story is to let
, while
.
Observe, for CQ, that while q & Q range over the whole real line, that is not possible for AQ. If
then d covers the real line, but if
then
and p is helpless. To eliminate this possibility we require
. While this may seem to be a problem, it can be very useful to limit such variables, like
, or
, or even both.2
2. A Look at Quantum Field Theory
2.1. Selected Poor and Good Results
Classical field theory normally deals with a field
and a momentum
, where x denotes a spatial point in an underlying space.3
A common model for the Hamiltonian is given by
(2)
where
is the power of the interaction term,
is the dimension of the spatial field, and
, which adds the time dimension. Using CQ, such a model is nonrenormalizable when
, which leads to “free” model results [2]. Such results are similar for
and
, which is a case where
[3] [4] [5]. When using AQ, the same models lead to “non-free” results [2] [6].
Solubility of classical models involves only a single path, while quantization involves a vast number of paths, a fact well illustrated by path-integral quantization. The set of acceptable paths can shrink significantly when a nonrenormalizable term is introduced. Divergent paths of integration are like those for which
when
. A procedure that forbids possibly divergent paths would eliminate nonrenormalizable behavior. As we note below, AQ provides such a procedure.
2.2. The Classical and Quantum Affine Story
Classical affine field variables are
and
. The quantum versions are
and
, with
. The affine quantum version of (2) becomes
(3)
The spacial differential term restricts
to continuous operator functions, maintaining
. In that case, it follows that
which implies that
for all
, a most remarkable feature because it forbids nonrenormalizability!4
Adopting a Schrödinger representation, where
, simplifies
, which also implies that
. This relation suggests that a general wave function is like
, as if
acts as the representation of a family of similar wave functions.
We now take a Fourier transformation of the absolute square of a regularized wave function that looks like5
(4)
Normalization ensures that if all
, then
, which leads to
(5)
Finally, we let
to secure a complete Fourier transformation6
(6)
This particular process side-steps any divergences that may normally arise in
when using more traditional procedures.
3. The Absence of Nonrenormalizablity, and the Next Fourier Transformation
Observe the factor
in (4) which is prepared to insert a zero divergence for each and every
when
. However, the factor
in (4) turns that possibility into a very different story given in (6).
Another Fourier transformation can take us back to a suitable function of the field,
. That task involves pure mathematics, and it deserves a separate analysis of its own.
NOTES
1Even if Q does not cover the whole real line, which means that
, yet
. This leads to
.
2For example, affine quantization of gravity can restrict operator metrics to positivity,i.e.,
, straight away [1].
3In order to avoid problems with spacial infinity we restrict our space to the surface of a large,
-dimensional sphere.
4For Monte Carlo studies, concern for the term
has been resolved by successful usage of
, where
[2] [6].
5The remainder of this article updates and improves a recent article by the author [7].
6Any change of
due to
is left implicit.