Classical pure states: Information and symmetry in statistical mechanics

The aim of this paper is to find classical counterparts of pure quantum states. It is shown that these are singular probability distributions concentrated on the so-called maximal null manifolds in a phase space. They are equivalent to densities studied by Van Vleck and Schiller and to WKB solutions (cf. Van Vleck, 1928; Schiller, 1962). Properties of such distributions and their relativistic generalisations have been studied in previous papers (Sawianowski, 1971; Slawianowski, 1972). However, it has not been shown there that such distributions arise actually in the limith0. When working with the standard apparatus of differential geometry we mostly use the language of Kobayashi & Nomizu (1963).