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Annihilation operators associated with unstable vacua in non-equilibrium thermo-eld dynamics

Kazuhiro Yoshida, Toshihico Arimitsu

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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-eld 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 elds are derived.

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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-eld 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 elds are derived.
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.
Most of the mathematical approaches for quantum Langevin equation are based on the non-commutativity of the random force operators. Non-commutative random force operators are introduced in order to guarantee that the equal-time commutation relation for the stochastic annihilation and creation operators preserves in time. If it is true, it means that the origin of dissipation is of quantum mechanical. However, physically, it is hard to believe it. By making use of the unified canonical operator formalism for the system of the quantum stochastic differential equations within Non-Equilibrium Thermo Field Dynamics, it is shown that it is not true in general.
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.
A fundamental framework for the construction of a quantum field theory for open systems, which will be called non-equilibrium thermo field dynamics (NETFD), is built upon two concepts. One is the thermal state in the thermal-Liouville space, by which the quasi-particle superoperator is introduced generally from the thermal state condition. The other is the coarse graining which is realized by projecting out some partial space from the complete thermal-Liouville space, and by which the dissipation is introduced. Most properties of the usual quantum field theory (e.g., the operator formalism, the time-ordered formulation of the Green’s functions, the Feynman diagram method in real time, etc.) are preserved in NETFD. The entropy for the nonequilibrium state can be also introduced in NETFD.
In their constructions of system of quantum stochastic differential equations, mathematicians and/or several physicists interpret that the function of random force operator is to preserve the canonical commutation relation in time, i.e., to secure the unitarity of time evolution generator even for dissipative systems. If this is the case, it means physically that the origin of dissipation is attributed to quantum non-commutativity ({\it quantumness}). The mechanism that the mathematician's approaches rest on will be investigated from the unified view point of Non-Equilibrium Thermo Field Dynamics (NETFD) which is a canonical operator formalism of quantum systems in far-from-equilibrium state including the system of quantum {\it stochastic} equations.