7 Statistical Quantization

russell_5.jpg "Wherever possible, substitute constructions out of known entities for inferences to unknown entities"
Bertrand Russell, philosopher and logician (1872 - 1970)

As pointed out already, at the end of section 1, an understanding of the transition from QT to classical physics, as well as the inverse transition, should provide a clue to the understanding (and the proper interpretation) of QT. As mentioned in section 4.3107 , a detailed analysis [85] of the classical limit of QT shows that QT can only be interpreted in the sense of the SI; otherwise, if interpreted in an individualistic sense, the ℏ   →    0 limit of Schrödingers equation had to agree with Newton’s equation. This is not the case.

The inverse transition, from classical physics to QT, is also of great interest. The question ”How to interpret QT?” is closely related to the question ”how to ’derive’ QT?”. We use the term quantization to describe this process of derivation of QT (generalizing slightly the standard meaning). The standard quantization method starts from a classical Hamiltonian whose basic variables are then ’transformed’, by means of well-known correspondence rules,
        ℏ   d                 ℏ  d
p  →    -------, E   →    -   ------, ,
         i dx                 i dt

into operators. The physical origin of this recipe is unclear. Many physicists do not care about this because they think that a mathematically well-defined rule is all that can be achieved, or all that is really needed. Unfortunately, this rule is neither intuitively understandable nor is it mathematically well-defined as shown in papers by Groenewold [59], Shewell [139], and others. The above rule can, of course, be justified by its success, but that is not quite the point here. The problem has not been discussed intensely in recent years. This is due to the increasing trend to axiomatics and mathematical abstraction, which conveniently hides such problems.

On the other hand many scientists have been trying over the years to understand QT, or to analyze its origins, in terms of physically meaningful assumptions. A number of these works used the statistical point of view to start from [see e.g. the quotations in the introduction of this paper]. A complete statistical quantization method which is based on very simple (in the sense of ’fundamental’) concepts has been designed by the present author and is described in the following works:

It seems possible to extend the ideas which form the basis of these papers to systems with an infinite number of degrees of freedom (quantum field theory) . A very interesting question is, of course, whether or not these ideas will survive the transition to relativistic physics.