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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

An attempt to construct quantum mechanics from Newton equations

R. Hojman, Sergio A. Hojman

Published Dec 1, 1985• 5 citations
Publisher
DOI
10.1007/bf02722902

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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.
Born was convinced that the ‘indeterministic foundations’ of quantum mechanics were here to stay. He also left a calling, that has been largely ignored, to show that the ‘apparently deterministic’ laws of classical physics were derivable from the indeterminacy of quantum mechanics. Here, we attempt to answer the calling left to us by Born and we do so first by showing that the beautiful laws of classical optics are directly derivable from the N -slit quantum probability equation. Secondly, we use the quantum probability dependent intensity to arrive, via Heisenberg's uncertainty principle, first at Newton's second law and then to Newton's law of universal gravitational. What this shows is that there is an explicit cohesiveness in physics with quantum mechanics at the foundations. The only ‘price to pay’ is that the laws of classical physics become approximate rather than ‘exact.’ No more determinism. This is entirely consistent with the fields of experimental physics, quantum or classical, where no measurement is exact.
Intuitionistic real numbers are constructed as sheaves on the state space of the Schrodinger representation of a CCRalgebra with a finite number of degrees of freedom. These numbers are used as the values of position and momentum variables that obey Newton's equations of motion. Heisenberg's operator equations of motion are shown to give rise to numerical equations that, on a family of open subsets of state space, are local approximations to Newton's equations of motion for the intuitionistically valued variables.