Extension of the Glauber-Sudarshan Mapping For Classical and Quantum Energy Spectra

This paper begins by reviewing the general form of the Glauber-Sudarshan P mapping, a cornerstone of coherence theory in quantum optics, which defines a two-way mapping between ensembles of states S of any classical Hamiltonian field theory and a subset of the allowed density matrices {\rho} in the corresponding canonical bosonic quantum field theory (QFT). It has been proven that Tr(H{\rho})=E(S), where E is the classical energy and H is the normal form Hamiltonian. The new result of this paper is that ensembles of S of definite energy E map into density matrices which are mixes of eigenstates of eigenvalue zero of N(H-E,H-E) where N represents the normal product. This raises interesting questions and opportunities for future research, including questions about the relation between canonical QFT and QFT in the Feynman path formulation. In the Appendix, a corresponding Boltzmann distribution is shown, using a dual version of the P mapping, for the case of statistically compressible dynamical systems.