Some remarks on hamiltonian systems and quantum mechanics

Chernoff, P. and J. E Marsden

Foundations of probability theory, statistical inference and statistical theories of science (Proc. Internat. Res. Colloq., Univ. Western Ontario, London, Ont., 1973), III, (1977), 35-53; see also Univ. Western Ontario Ser. Philos. Sci., 6, Reidel, Dordrecht,674-679

Abstract:

These notes contain some remarks on the general structure of a class of physical systems called Hamiltonian, and on quantum mechanical systems in particular. Our goal here is to point out some unifying structures and special properties that may be of interest to this conference. Thus, many of our remarks are deliberately brief and sometimes vague. Most of the results are known in the literature (cf. [18, 23]), although perhaps from a different point of view.

We shall begin in §2 with the general features that a physical system admitting a probabilistic interpretation should have. The distinguishing features of classical and quantum mechanical systems are pointed out. in §3 the C ^* -algebra approach to quantum mechanics as delineated by Segal, is reviewed. Then in §4 we study the dynamics of classical and quantum mechanics--we endeavor to show that both systems are Hamiltonian, when the latter condition is interpreted from the modern point of view of symplectic manifolds (see [2]). In §5 we briefly describe some other classes of Hamiltonian systems: specifically hydrodynamics and general relativity. Finally in §6,7, we mention a few problems connected with hidden variables and the theory of measurement.