Hilbert space structure in classical mechanics. I
Abstract
In this paper we study the Hilbert space structure underlying the Koopmanvon Neumann (KvN) operatorial formulation of classical mechanics. KvN limited themselves to study the Hilbert space of zeroforms that are the square integrable functions on phase space. They proved that in this Hilbert space the evolution is unitary for every system. In this paper we extend the KvN Hilbert space to higher forms which are basically functions of the phase space points and the differentials on phase space. We prove that if we equip this space with a positive definite scalar product the evolution can turn out to be nonunitary for some systems. Vice versa, if we insist in having a unitary evolution for every system then the scalar product cannot be positive definite. Identifying the oneforms with the Jacobi fields we provide a physical explanation of these phenomena. We also prove that the unitary/nonunitary character of the evolution is invariant under canonical transformations.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 December 2003
 DOI:
 10.1063/1.1623333
 arXiv:
 arXiv:quantph/0208046
 Bibcode:
 2003JMP....44.5902D
 Keywords:

 45.20.Dd;
 45.05.+x;
 02.30.Uu;
 02.10.Ud;
 General theory of classical mechanics of discrete systems;
 Integral transforms;
 Linear algebra;
 Quantum Physics;
 High Energy Physics  Theory
 EPrint:
 74 pages, 2 figures, RevTex