Abstract. We study the problem of correspondence between classical and quantum statistical models. We show that (contrary to a rather common opinion) it is possible to construct a natural pre-quantum classical statistical model. The crucial point is that such a pre-quantum classical statistical model is not the conventional classical statistical mechanics on the phase space R²ⁿ, but its infinite-dimensional analogue. Here the phase space Ω = H × H, where H is the (real separable) Hilbert space. The classical → quantum correspondence is based on the Taylor expansion of classical physical variables—maps f:Ω → R. The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and dispersion h. The quantum statistical model is obtained as the lim_h→0 of the classical one. All quantum states including so-called 'pure states' (wavefunctions) are simply Gaussian fluctuations of the 'vacuum field', ω = 0 Ω, having dispersions of the Planck magnitude.

PACS number: 03.65.Ta

Print publication: Issue 41 (14 October 2005)
Received 1 June 2005, in final form 31 August 2005
Published 28 September 2005