Thermo-Algebras in Phase Space: Quantum and Classical Systems

E. Celeghini, DE MARTINO S., DE MARTINO S. +3 more

H. Matsumoto, Hiroomi Umezawa

John Stephen. Hebron

H. Matsumoto

Masashi Ban

An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time-evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.

R. Cartas-Fuentevilla

Abstract Using a hyperbolic ring based formalism, it is shown that special unitary groups correspond to thermo-algebras for closed quantum systems, while for open systems, the full unitary algebra is the suitable underlying structure. Specifically the groups U (2) and U (1, 1) are realized as the dynamical symmetry groups for open quantum systems through continuous tensorial products; these unitary groups are obtained by mean of abelian extensions through U (1)-factors, from the standard thermo-algebras SU (2), and SU (1, 1).