Phase-Space Formulation of the Dynamics of Canonical Variables

Statistical reformulation of quantum mechanics in terms of phase‐space distribution functions as given by Moyal using Weyl's correspondence rule between classical functions and operators has been extended to various different correspondence rules. The dynamical bracket in the Weyl correspondence (the ``Moyal'' or the ``sine'' bracket) is shown to be a Lie bracket. It is further shown that if the theory is restricted to Lie brackets of the form $$\mathrm{[u(p,\hspace{0.17em}q),\upsilon (p,\hspace{0.17em}q)]=}\left\{f\left(\frac{{\partial }^2}{{\mathrm{\partial q}}_1{\mathrm{\hspace{0.17em}\partial p}}_2}-\frac{{\partial }^2}{{\mathrm{\partial q}}_2{\mathrm{\hspace{0.17em}\partial p}}_1}\right){\mathrm{u(p}}_1{\mathrm{,\hspace{0.17em}q}}_1{\mathrm{)\upsilon (p}}_2{\mathrm{,\hspace{0.17em}q}}_2)\right\}$$ , evaluated for p₁ = p₂ = p; q₁ = q₂ = q after differentiation, then the only admissible functional form of f is f(x) = β[(sin αx)/α], where α and β are constants. A law of multiplication which is associative and distributive with respect to addition is also introduced in each case. It gives a correct correspondence between operator multiplication and the multiplication of classical functions. The dynamical brackets obtained in each case are also found to be Lie brackets. Conditions on the phase‐space distribution functions to describe pure states are also given.