Jan Jerzy Slawianowski
Dirac-Hamilton Representation of the Schroedingerer Equation and Geometric Nonlinearities
Let us forget temporarilly the physical content of the Schroedinger equation and consider it just as some differential equation of mathematical physics. To simplify everything as far as possible, we begin with what physicists often call a finite-level quantum system, i.e., one in a finite-dimensional unitary space.From the formal point of view it is a Hamiltonian system. However, if one deals with the usual Schroedinger equation,it is a degenerate Lagrangian-Hamiltonian system in the Dirac sense. The structure of primary and secondary constraints is discussed in some details. Later on we admit in Lagrangian the term quadratic in generalized velocities. There are no Dirac constraints then and in this sense some kind of "regularization" is achieved. There is no continuous transition between both models. And there are still some open problems concerning the statistical interpretation if again interpreting the resulting equation as a quantum-mechanical one.One deals with similar problems in field theory, when one uses the Klein-Gordon-Dirac equation involwing both first-order and second-order time derivatives. In some past we discussed such a kind of equation within the context of conformal invariance in gauge gravitation theory. Many years ago there were also some models by Barut motivated by a similar philosophy and combining the first-order and second-order derivatives. The next step concerns the possible nonlinearity models. We show that there exists some interesting class of nonlinearities which is not introduced by hand, but has a deep geometrical motivation. These nonlinearities appear in a consequence of making the scalar product a dynamical variable, not an absolute quantity (compare this formally with the transition from special to general relativity). Some effective nonlinearity of the total system of equations appears and one can hope that this kind of nonlinearity may help us with discussing decoherence, reduction, and measurement problems. The dynamical scalar product represents in some "averaged" sense the surrounding of the object and may be interpreted as a kind of description of open systems.In this way, starting from a rather formal analysis, we return step by step to quantumm mechanics and its "paradoxes".