The Canonical Structure of a Classical Theory, Quantization Procedures and Non-Equilibrium Quantum Statistical Mechanics

Abstract

The aim of this lecture is to show that the harmonic oscillator model of the non-equilibrium quantum statistical mechanics which was presented by Emch [1] we can obtain by the natural quantization of the two interesting transformations in the phase-space of the one-dimensional classical harmonic oscillator:

$$ \Pi _{\beta /2} :\left[ {\begin{array}{*{20}c} z \hfill \\ {z^* } \hfill \\ \end{array} } \right] \to \left[ {\begin{array}{*{20}c} {e^{ - \frac{{\beta \omega }} {2}} \;\;z} \hfill \\ {e^{\frac{{\beta \omega }} {2}} \;\;z*} \hfill \\ \end{array} } \right]$$

(1)

$$ \gamma (s):\left[ {\begin{array}{*{20}c} z \hfill \\ {z*} \hfill \\ \end{array} } \right]\,\, \to \,\,\left[ {_{e^{ - \lambda s} \,\,\,z*}^{e^{ - \lambda s} \,\,\,z} } \right]\,\,\,\,,\,s \geqslant 0$$

(2)

where z= ω^1/2q + iω ^−1/2p, the physical constants n = 1, k = 1, and β is the inverse temperature.

Seminar given at XV. Internationale Universitätswochen für Kernphysik,Schladming,Austria,February 16–27,1976.

Supported in part by N.S.F. grant No. GF-41958.