Mathematics – Operator Algebras
Scientific paper
1994-05-02
J. of Natural Geometry, v. 9 (1996), no. 1, pp. 1-14
Mathematics
Operator Algebras
15 pages, LaTeX2e; on 05/02/2004 files were updated to produce PS
Scientific paper
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.
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