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arXiv:nucl-th/0102014 · Progress of Theoretical Physics

Utility of su(1, 1)-Algebra in a Schematic Nuclear su(2)-Model

H. Akaike, Y. Tsue, Constança Providência, João da Providência, A. Kuriyama, M. Yamamura

Published May 1, 2001• 1 citation
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Abstract

The su(2)-algebraic model interacting with an environment is investigated from the viewpoint of treating a dissipative system. By using a time-dependent variational approach with a coherent state and with the help of the canonicity condition, the time-evolution of this quantum many-body system is described in terms of the canonical equations of motion in the classical mechanics. Then, it is shown that the su(1, 1)-algebra plays an essential role in treating this model. An exact solution with appropriate initial conditions is obtained in terms of Jacobi's elliptic function. The implications for the dissipative process are discussed.

DOI
10.1143/ptp.105.717

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The su(2)-algebraic model interacting with an environment is investigated from the viewpoint of treating a dissipative system. By using a time-dependent variational approach with a coherent state and with the help of the canonicity condition, the time-evolution of this quantum many-body system is described in terms of the canonical equations of motion in the classical mechanics. Then, it is shown that the su(1, 1)-algebra plays an essential role in treating this model. An exact solution with appropriate initial conditions is obtained in terms of Jacobi's elliptic function. The implications for the dissipative process are discussed.
The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.
The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.
A time-dependent variational approach with a mixed-mode coherent state is applied to su(1, 1)-algebraic models derived from nuclear su(2)-models, which are the single orbit shell model with pairing correlation and the Lipkin model. The equations of motion are derived in a canonical form similar to that in the time-dependent Hartree-Fock theory, and the solutions are derived by the use of conserved action variables together with an appropriate canonical transformation. It is shown that thermal effects appear in the form of the bose distribution function. Also, it is found that the energy concentrated in a pure state at initial time dissipates to the other states in a manner similar to the case of the damped-and-amplified harmonic oscillator model investigated in our previous work.
A time-dependent variational approach with a mixed-mode coherent state is applied to su(1, 1)-algebraic models derived from nuclear su(2)-models, which are the single orbit shell model with pairing correlation and the Lipkin model. The equations of motion are derived in a canonical form similar to that in the time-dependent Hartree-Fock theory, and the solutions are derived by the use of conserved action variables together with an appropriate canonical transformation. It is shown that thermal effects appear in the form of the bose distribution function. Also, it is found that the energy concentrated in a pure state at initial time dissipates to the other states in a manner similar to the case of the damped-and-amplified harmonic oscillator model investigated in our previous work.
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.
We discuss dissipative systems in Quantum Field Theory by studying the canonical quantization of the damped harmonic oscillator (dho). We show that the set of states of the system splits into unitarily inequivalent representations of the canonical commutation relations. The irreversibility of time evolution is expressed as tunneling among the unitarily inequivalent representations. Canonical quantization is shown to lead to time dependent SU(1,1) coherent states. We derive the exact action for the dho from the path integral formulation of the quantum Brownian motion developed by Schwinger and by Feynman and Vernon. The doubling of the phase-space degrees of freedom for dissipative systems is related to quantum noise effects. Finally, we express the time evolution generator of the dho in terms of operators of the qqq-deformation of the Weyl-Heisenberg algebra. The qqq-parameter acts as a label for the unitarily inequivalent representations.