9 Discussion

In section 4 it as been shown that only the Lorentz force can exist as fundamental (quantum mechanical) force if the statistical assumptions of section 3 are true. It is the only force (see however the remarks on spin below) that can be incorporated in a ’standard’ differential equation for the dynamical variables ρ , S . The corresponding terms in the statistical field equations, representing the Lorentz force, must be given by the familiar gauge (minimal) coupling terms containing the potentials. The important fact that all forces in nature follow this ’principle of minimal coupling’ is commonly explained as a consequence of local gauge invariance. The present treatment offers an alternative statistical explanation.

Let us use the following symbolic notation to represent the relation between the local force and the terms representing its action in a statistical context:
Φ,   A⃗  ⇒    e ⃗E   +     ⃗v  ×   ⃗B.

The fields  ⃗
E and ⃗
B are uniquely defined in terms of the potentials φ and ⃗
A [see (30)] while the inverse is not true. Roughly speaking, the local fields are ’derivatives’ of the potentials - and the potentials are ’integrals’ of the local field; this mathematical relation reflects the physical role of the potentials φ and  ⃗
A as statistical representatives of the the local fields  ⃗
E and  ⃗
B , as well as their non-uniqueness. It might seem that the logical chain displayed in (108) is already realized in the classical treatment of a particle-field system, where potentials have to be introduced in order to construct a Lagrangian [21]. However, in this case, the form of the local force is not derived but postulated. The present treatment ’explains’ the form of the Lagrangian - as a consequence of the basic assumptions listed in section 3.

The generalization of the present theory to spin, reported in sections 6 and 7, leads to a correspondence similar to Eq. (108), namely
⃗μ  ⃗B   →    ⃗μ  ⋅ -----B⃗.
                 ∂  ⃗x

The term linear in  ⃗
B , on the l.h.s. of (109), plays the role of a ’potential’ for the local force on the r.h.s. All points discussed after Eq. (108) apply here as well [As a matter of fact we consider  ⃗
B as a unique physical quantity; it would not be unique if it would be defined in terms of the tensor on the r.h.s. of (109)]. We see that the present approach allows for a certain unification of the usual gauge and spin interaction terms - comparable with a relativistic formulation, where the spin coupling is introduced simultaneously with the other gauge coupling terms. Unfortunately, the derivation of the spin force on the r.h.s. of (109) differs from the derivation of the Lorentz force insofar as additional assumptions had to be made in order to arrive at the final result (see the remarks in section 7).

Our notation for potentials       ⃗
φ,   A , fields  ⃗    ⃗
E  ,  B , and parameters e,  c suggests that these quantities are electrodynamical in nature. However, this is not necessarily true. The constraint (31) yields four equations which are not sufficient to determine the six fields ⃗     ⃗
E  , B ; additional conditions must be imposed. The most familiar possibility is, of course, the second pair of Maxwell’s equations leading to the electrodynamical gauge field. A second possible realization for the fields ⃗     ⃗
E  , B is given by the inertial forces acting on a mass m in an arbitrarily accelerated reference frame [15]. In Appendix A a brief discussion of the inertial gauge field and its interplay with the Maxwell field is given. It is remarkable that the present theory establishes a (admittedly somewhat vague) link between the two extremely separated physical fields of inertia and quantum theory. An interesting point is that the mathematical axioms of U  (1  ) gauge theory imply the Maxwell field but say nothing about the inertial field which is just as real as the former. The inertial field  ⃗
BI [see (114)] may also lead to a spin response of the ensemble. Experiments to verify such inertial effects of spin have been proposed by Mashhoon and Kaiser [25].

It is generally assumed that the electrodynamic potentials have a particular significance in quantum mechanics which they do not have in classical physics. Let us analyze this statement in detail. (we restrict ourselves in the following discussion to the electrodynamical gauge field). The first part of this statement, concerning the significance of the potentials, is of course true. The second part of the statement, asserting that in classical physics all external influences can be described solely in terms of field strengths, is wrong. More precisely, it is true for classical mechanics but not for classical physics in general. A counterexample - a theory belonging to classical physics but with potentials playing an indispensable role - is provided by the classical limit (104),(105) of Schrödinger’s equation. In this field theory the potentials play an indispensable role because (in contrast to particle theories, like the canonical equations) no further derivatives of the Hamiltonian, which could restore the fields, are to be performed. This means that the significance of the potentials is not restricted to quantum theory but rather holds for the whole class of statistical theories discussed above, which contains both quantum theory and its classical limit theory as special cases. This result is in agreement with the present interpretation of the potentials as statistical representatives of the local fields of particle physics.

The precise characterization of the role of the potentials is important for the interpretation of the Aharonov-Bohm effect. The ’typical quantum-mechanical features’ observed in these phase shift experiments should be identified by comparing the quantum mechanical results not with classical mechanics but with the predictions of the classical statistical theory (104),(105). The predictions of two statistical theories, both of which use potentials to describe the influence of the external field, have to be compared.

The interpretation of a physical theory is commonly considered as a matter that cannot be described by mathematical means. This may be generally true, but nevertheless mathematical facts exist which have an immediate bearing on questions of interpretation. The most important of these is probably the limiting behavior of a theory. The fact that the classical limit of quantum mechanics - discussed in section 8 - is not classical mechanics but a special (configuration space) classical statistical theory is widely unknown. This lack of knowledge is one of the main reasons for the widespread erroneous belief that quantum mechanics can be used to describe the dynamics of individual particles. Unfortunately, this erroneous belief is historically grown and firmly established in our thinking as shown by the ubiquitous use of phrases like ’the wave function of the electron’ or ’quantum mechanics’.

It is clear that an erroneous identification of the domain of validity of a physical theory will automatically create all kinds of mysteries, ill-posed questions and unsolvable problems. Above, we have identified one of the more subtle problems of this kind, concerning the role of potentials in quantum mechanics (a paradigmatic example is the ’measurement problem’ which is unsolved since its creation eighty years ago). Generalizing the above argumentation concerning potentials, we claim that characteristic features of quantum theory cannot be identified by comparison with classical mechanics. Instead, quantum theory should be compared with its classical limit, which is in the present 3D -case given by (104), (105). The latter theory is probably much more difficult to solve (numerically) than Schrödinger’s equation because is is no longer linear in ψ . Nevertheless, it would be very interesting to compare the solutions of (104), (105) with those of (58), (59) to find out which ’typical quantum-theoretic features’ are already given by statistical (nonlocal) correlations of the classical limit theory and which features are really quantum-theoretical in nature - related to the nonzero value of ℏ .