A classical realization of quantum mechanics

Vladimir V. Kisil

One can introduce so-called {\em Plain Mechanics} having an {\bf operator realization}. Then the set of one-dimension representations of this operator realization may be identified with the Classical Mechanics. Different irreducible infinite-dimension representations may be recognized as Quantum Mechanics for different $\hbar$ (the Planck constant). It can be done in the such manner that the following diagram will be commutative. Plain Mechanics / \ / \ / \ Quantum Mechanics --> Classical Mechanics h->0 Here the horizontal arrow is well known correspondence between Quantum and Classical Mechanics if Planck constant tensing to zero. A {\em realization} of this scheme for a particle in $n$-dimensional space by two-sided convolutions on the Heisenberg group is constructed. We also introduce the {\em motion equations} for observables in this realization. The left arrow of the given diagram carries this equation to the Heisenberg one and the right arrow maps it to the Hamilton equation.

I. S. Arzhanykh

An attempt was made to develop a universal algorithm which in one realization describes the equations of classical mechanics and in another realization the equations of quantum mechanics. Hence, a synthesis can be derived as the difference and the relation between the first and sccond realizations. (R.V.J.)

Ghenadie N. Mardari

L. D. Faddeev

Abstract In this article Prof. Ludwig Faddeev sketches a rather formal framework into which Quantum Mechanics satisfactorily fits. He shows that both Classical and Quantum Mechanics can be realized in terms of the same objects. Quantum Mechanics can be understood as a deformation of Classical Mechanics, the mathematical deformation parameter being Planck's constant. (Red.)