This paper is a sequel to a previous work [18] of the present author, which will be referred to as I. In I an attempt has been undertaken to construct a new ’statistical’ approach to quantum theory. This approach is based on the idea that quantum mechanics is not a theory about particles but about statistical ensembles. It is well known that the dynamic numerical output of quantum mechanics consists of probabilities. A probability is a ”deterministic” prediction which can be verified in a statistical sense only, i.e. by performing experiments on a large number of identically prepared individual systems [7],[24]. Therefore, quantum mechanics is a theory about statistical ensembles [5] and can only be used to make predictions about individual events if additional intellectual constructs, which are not part of the physical formalism, are introduced. The work reported in I, as well as the present one, can be characterized by complete absence of such constructs.
Adopting this point of view one is immediately led to the idea that a quantization procedure should exist, which is based on statistical concepts and assumptions. It must, of course, lead to the same results as the standard (canonical) quantization method but should be based on physically interpretable assumptions. In this respect, the ’classical’ canonical quantization procedure, which is based on the single-particle picture, is not satisfying because it consists of a number of purely formal rules. From a positivistic point of view all quantization methods leading to the same final result are equivalent. From the present point of view this is not the case and comprehensibility matters.
The quantization method reported in I is essentially based on the validity of the following three assumptions: (i) two differential equations which are similar in structure to the canonical equations of classical mechanics but with observables replaced by expectation values, (ii) a local conservation law of probability with a particular form of the probability current, and (iii) a differential version (minimal Fisher information) of the statistical principle of maximal disorder. As has been shown in I these postulates imply Schrödinger’s equation ”for a single particle” (i.e. for an ensemble of identically prepared single particles) in an external mechanical potential. This derivation provides a statistical explanation for the ad hoc rules of the conventional (single-particle) canonical quantization method. The treatment in I was restricted to a single spatial dimension.
In the present paper the work reported in I is extended to three spatial dimensions, gauge fields and spin. In section 2 the fundamental ideas are reviewed and the basic equations of the three-dimensional theory are listed. In section 3 an integral relation which provides a basis for later calculation is derived. In the central section 4 we pose the following question: Which constraints on admissible forces exist for the present class of statistical theories ? The answer is that only macroscopic forces of the form of the Lorentz force can occur in nature. These forces are statistically represented by potentials, i.e. by the familiar gauge coupling terms in matter field equations. The present statistical approach provides a natural explanation for the long-standing question why potentials play an indispensable role in the field equations of physics. In section 5 it is shown that among all statistical theories only the time-dependent Schrödinger equation follows the logical requirement of maximal disorder or minimal Fisher information. In section 6 the basic equations for a generalized theory, using the double number of dynamic variables, are formulated. The final result obtained in section 7 is Pauli’s equation for a spin ensemble of particles. In section 8 the classical limit of quantum mechanics is studied and the misleading role of the principle of reductionism is pointed out. In Section 9 related questions concerning the role of potentials and the general interpretation of quantum mechanics are discussed. A comparison of electrodynamical and inertial gauge fields is reported in appendix A. The final Section 10 contains concluding remarks.