12 Concluding remarks

The present derivation was based on the assumption that dynamical predictions are only possible for statistical averages and not for single events - leaving completely open the question why predictions on single events are impossible. This deep question remains unanswered; some speculative remarks on a possible source of the indeterminacy have been given elsewhere [26].

Throughout this work it has been assumed that ρ is not a (in single events) observable quantity but plays the role of a probability density. There are of course other possibilities, besides this ’probabilistic’ (or ’immaterial’) interpretation of ρ  =   |ψ  |2 . According to the original point of view of Schrödinger and de Broglie, |ψ  |2 was an observable field measuring something like the density distribution of an ’extended particle’. More recently, this interpretation has been reconsidered in an interesting paper by Barut [4]. But the results of modern high-precision measurements [47] strongly support the probabilistic interpretation and exclude, in our opinion, the original ’material’ view of Schrödinger and de Broglie.

A third interpretation for ρ is possible. It could play the role of a density of a stream of particles in the framework of an approximate continuum theory. This ρ would also be an observable quantity. An interesting problem for future research is the question if the present derivation can be adapted in such a way that this interpretation of ρ makes sense. The resulting Schrödinger equation (32) would be a classical field equation despite the fact that it contains an adjustable parameter ℏ . However, it would be only approximately true (what is actually observed are particles and not fields) and could, therefore, play a role as a starting point for a procedure known as second quantization.

’Interaction between individual objects’ and the corresponding notion of force are macroscopic concepts. In the microscopic domain, where according to the present point of view only statistical laws are valid, the concept of force looses its meaning. In fact, in the quantum-mechanical formalism ’interaction’ is not described in terms of forces but in terms of potentials (as is well known, this leads to a number of subtle questions concerning the role of the vector potential in quantum mechanics). The relation between these two concepts is still not completely understood; it seems that the present statistical approach offers a new point of view to study this problem [24].

In a previous work [26] of the present author, Schrödinger’s equation has been derived from a different set of assumptions including the postulate that the dynamic equation of state may be formulated by means of a complex-valued state variable ψ . The physical meaning of this assumption is unclear even if it sounds plausible from a mathematical point of view. The present paper may be seen as a continuation and completion of this previous work, insofar as this purely mathematical assumption has been replaced now by other requirements which may be interpreted more easily in physical terms. Finally, we mention that there are, besides the points mentioned already, several other open problems for future research, extending the range of validity of the present approach. These include the consideration of gauge fields [24], a generalization of the present formalism to internal degrees of freedom (spin) and higher-dimensional configuration space (several particles), as well as a relativistic formulation.