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arXiv:quant-ph/0308053

Thermofield Dynamics of Time-Dependent Boson and Fermion Systems

Sang Pyo Kim, F. C. Khanna

Published Aug 9, 2003• 1 citation
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Abstract

We formulate the thermofield dynamics for time-dependent systems by combining the Liouville-von Neumann equation, its invariant operators, and the basic notions of thermofield dynamics. The new formulation is applied to time-dependent bosons and fermions by using the time-dependent annihilation and creation operators that satisfy the Liouville-von Neumann equation. It is shown that the thermal state is the time- and temperature-dependent vacuum state and a general formula is derived to calculate the thermal expectation value of operators.

DOI
10.48550/arxiv.quant-ph/0308053

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We formulate the thermofield dynamics for time-dependent systems by combining the Liouville-von Neumann equation, its invariant operators, and the basic notions of thermofield dynamics. The new formulation is applied to time-dependent bosons and fermions by using the time-dependent annihilation and creation operators that satisfy the Liouville-von Neumann equation. It is shown that the thermal state is the time- and temperature-dependent vacuum state and a general formula is derived to calculate the thermal expectation value of operators.
Perturbation theory in thermofield dynamics (a real time finite temperature quantum field theory) is formulated in the interaction representation and a diagrammatic expansion for the expectation value of time ordered products is derived solely on the basis of the thermal state condition and the algebraic properties of the operators. This new formulation has a much wider domain of applicability than those presented previously. The method is applicable even in the case of purely quantum mechanical systems. When certain bosonic operators, which commute with the unperturbed Hamiltonian, contribute to the renormalization of the thermal state vacuum the vacuum diagrams do not cancel in the usual way. It will be shown how these diagrams arise and how they may be computed.
Out of thermal equilibrium state, the vacuum is unstable and evolves in time.Consequently, the annihilation operators associated with the unstable vacuum depend on time.This dissipative time-evolution of quantum systems can be systematically treated, within the canonical operator formalism referred to as non-equilibrium thermo-field dynamics.Given is an alternative route to derive the time-dependent annihilation operators within the formalism.As an example, time-dependent annihilation operators for the systems of bosonic and fermionic semi-free fields are derived.
A free boson system out of equilibrium is studied with a time-dependent Hamiltonian. The density operator is determined by an operator hatI(t)\\hat{I} (t)hatI(t), satisfying the quantum Liouville-von Neumann equation. When the system evolves from an initial equilibrium, the temperature is obtained as T(t)=Ti(overline<hatH(t)>Psi/overline<hatI(t)>Psi)T(t) = T_i (\\overline{< \\hat{H} (t) >_{\\Psi}}/ \\overline{< \\hat{I} (t) >_{\\Psi}})T(t)=Ti​(overline<hatH(t)>Psi​/overline<hatI(t)>Psi​), where the expectation value is taken with respect to the exact quantum state of Schr\\"{o}dinger equation and the time average also is taken over the period of system. It recovers the result for the adiabatic (quasi-equilibrium) and the nonadiabatic (out of equilibrium) evolution by taking into account the factor for particle production. The temperature also is obtained for fermion system.
We show how a generalized time‐dependent harmonic oscillator (GTHO) is extended to a finite temperature case by using thermo field dynamics (TFD). We derive the general time‐dependent annihilation and creation operators for the system, and obtain the time‐dependent quasiparticle annihilation and creation operators for the GTHO by using the temperature‐dependent Bogoliubov transformation of TFD. We also obtain the thermal state as a two‐mode squeezed vacuum state in the time‐dependent case as well as in the time‐independent case. The general formula is derived to calculate the thermal expectation value of operators.
The effective theory of an open boson or fermion system is studied, which evolves out of equilibrium with time-dependent Hamiltonian H^(t)\hat{H}(t)H^(t). A measure of nonequilibrium temperature for the open system evolving from an equilibrium is proposed as the time-averaged energy expectation value T(t)=Ti(<H^(t)>Ψˉ/<I^(t)>Ψˉ)T(t) = T_i(\bar{< \hat{H} (t) >_Ψ}/ \bar{< \hat{I} (t) >_Ψ})T(t)=Ti​(<H^(t)>Ψ​ˉ​/<I^(t)>Ψ​ˉ​), where I^(t)\hat{I} (t)I^(t), the action operator, satisfies the quantum Liouville-von Neumann equation and determines the true density operator. It recovers the result for the adiabatic (quasi-equilibrium) and nonadiabatic (out of equilibrium) evolution from one static Hamiltonian to another and takes into account the particle production due to the intermediate processes.
We analyze the many-particle Schrodinger equation for fermions in a thermal ensemble by introducing an exponential operator expansion, defined in the context of thermofield dynamics. The expansion is optimized variationally at each time step through changes in the basis of excitations, which leads to a method of generating approximate differential equations to solve the time dependent problem, and can also be used to cool the system in imaginary time. The method is applied for a specific set of basis transformations and truncation scheme, leading to an explicit set of differential equations that reduce to the Hartree Fock solution in the low temperature limit. This procedure can also be generalized to include quantum correlation, which will be pursued in a future publication.