Group-theoretical formalism of quantum mechanics based on quantum generalization of characteristic functions

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Abstract
Fourier transforms of the von Neumann density operators could be regarded as functions on a Lie group whose infinitesimal generators correspond to dynamical variables characterizing a given quantum system. Called ‘‘characteristic functions’’ in this paper, they are discussed with a view to reformulating quantum statistical mechanics in a group-theoretical formalism. It is shown that these characteristic functions allow a universal differentiating procedure for calculating averages of arbitrary ordered noncommuting observables and satisfy the same criterion of non-negative definiteness as the Fourier transforms of the everywhere non-negative probability distribution (the characteristic functions in classical probability theory). The group-theoretical characteristic-function formalism is especially useful in providing easy reductions of the dynamics of complicated quantum systems to the evolutions of a set of quantum variables that are of particular interest. As examples of applications, the following results are derived: (a) a hierarchy of equations of motion for subsystems of a given N-body quantum system (a quantum analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy), (b) an equation of motion for centers of mass of two interacting quantum systems, and (c) the behavior of spin fluctuations of an ideal spin gas in the thermodynamic limit.