Axiomatics of general Hamiltonian theories including classical and quantum ones as particular cases

Abstract

The axioms of Hamilton theory are formulated in a form in which they are suitable for classical or quantum theories. These axioms define an algebra with two operations of multiplication; the algebra is called the free Hamiltonian algebra. Classical and quantum mechanics are obtained as factor algebras with respect to different equivalence relations. The axiomatics allows essentially new Hamiltonian theories with nonassociative multiplication of observables. 1. In their standard form, the bases of classical and quantum mechanics appear completely different, and they do not reflect the deep connections between these theories. The aim of the present paper is to distinguish completely the identical and the different aspects of these theories. The aspects that are common to both distinguish the set of Hamiltonian physical theories, of which the classical and the quantum are particular cases. We shall assume the physical systems under consideration have finitely many degrees of freedom. This makes it possible to concentrate attention on the algebraic aspects of the problem, since here, in contrast to quantum field theory, questions of topology are not decisive.