Exact formulation of centroid dynamics

EXACT FORMULATION OF CENTROID DYNAMICS III.1 QUASI-DENSITY OPERATOR 

For an arbitrary canonical density operator, the phase space centroid distribution function is uniquely defined. However, this function does not directly contain any dynamical information from the quantum ensemble because such information has been lost in the course of the trace operation. The lost information may be recovered by associating to each value of the centroid distribution function the following normalized operator  

Integration of the operator of Eq. (10) over xc and pc results in the following important identity  

This expression suggests that the canonical ensemble can be considered to be an incoherent mixture of the QDO s, each with different position and momentum centroids, and the latter having a probability density given by pc (xc, pc) / Z. Each QDO can then be interpreted as a representation of a thermally mixed state localized around (xc,pc), with its width being defined by the temperature and the system Hamiltonian. 

For any physical observable corresponding to the operator A, one can define a Corresponding centroid variable as 

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