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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

A fundamental equation in quantum mechanics?

L Mackinnon

Published Nov 1, 1981• 24 citations
Publisher
DOI
10.1007/bf02745126

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In this paper Quantum Mechanics with Fundamental Length is built as a deformation of Quantum Mechanics. To this aim an approach is used which does not take into account commutator deformation as usually it is done, but density matrix deformation. The corresponding deformed density matrix, which is called density pro-matrix is given explicitly. It properties have been investigated as well as some dynamical aspects of the theory. In particular, the deformation of Liouville equation is analyzed in detail. It was shown that Liouville equation in Quantum Mechanics appears as a low energy limit of deformed Liouville equation in Quantum Mechanics with Fundamental Length. Some implications of obtained results are presented as well as their application to the calculation of black hole entropy.
Abstract: Here within the lacking part of ultimate theory, i.e. the Everlasting Theory, I derived the fundamental equation of the Matrix Quantum Mechanics i.e. the commutator. It follows from the phase transitions of the fundamental spacetime that are based on the halfintegral-spin constancy. The fundamental equation results from the entanglement that leads to the infinitesimal transformations. In reality, the Matrix Quantum Mechanics that describes excited states of fields, i.e. the quantum particles, is timeless and non-local i.e. nondeterministic. But the Matrix Quantum Mechanics leads to the time-dependent, so deterministic, wave functions that are characteristic for the Statistical Quantum Mechanics. It is the reason why the wave functions appear in the equations of motion. The Statistical Quantum Mechanics or the Quantum Theory of Fields, are the semiclassical/semi-quantum theories. The presented here extended Matrix Quantum Mechanics leads to the methods applied in the Quantum Theory of Fields and points the limitations. 1.
The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero). A "functional" formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers.