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arXiv:quant-ph/9511015

Spontaneous dissipation from generalized radiative corrections

Masahiro Morikawa

Published Nov 14, 1995• 0 citations
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Abstract

We derive dissipative effective Hamiltonian for the unstable Lee model without any ad hoc coarse graining procedure. Generalized radiative corrections, utilizing the in-in formalism of quantum field theory, automatically yield irreversibility as well as the decay of quantum coherence. Especially we do not need to extend the ordinary Hilbert space for describing the intrinsically dissipative system if we use the generalized in-in formalism of quantum field theory.

DOI
10.48550/arxiv.quant-ph/9511015

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We derive dissipative effective Hamiltonian for the unstable Lee model without any ad hoc coarse graining procedure. Generalized radiative corrections, utilizing the in-in formalism of quantum field theory, automatically yield irreversibility as well as the decay of quantum coherence. Especially we do not need to extend the ordinary Hilbert space for describing the intrinsically dissipative system if we use the generalized in-in formalism of quantum field theory.
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In this paper we shall study whether dissipation in a λϕ4λϕ^{4}λϕ4 may be described, in the long wavelength, low frequency limit, with a simple Ohmic term κϕ˙κ\dotϕκϕ˙​, as it is usually done, for example, in studies of defect formation in nonequilibrium phase transitions. We shall obtain an effective theory for the long wavelength modes through the coarse graining of shorter wavelengths. We shall implement this coarse graining by iterating a Wilsonian renormalization group transformation, where infinitesimal momentum shells are coarse-grained one at a time, on the influence action describing the dissipative dynamics of the long wavelength modes. To the best of our knowledge, this is the first application of the nonequilibrium renormalization group to the calculation of a damping coefficient in quantum field theory.
It is demonstrated that, making minimal changes in ordinary quantum mechanics, a reasonable irreversible quantum mechanics can be obtained. This theory has a more general spectral decomposition, with eigenvectors corresponding to unstable states that vanish when $t\ensuremath{\rightarrow}\ensuremath{\infty}.$ These Gamov vectors have zero norm, in such a way that the norm and the energy of the physical states remain constant. The evolution operator has no inverse, showing that we are really dealing with a time-asymmetric theory. Using the Friedrichs model, reasonable physical results are obtained, e.g., the remnant of an unstable decaying state reappears, in the continuous spectrum of the model, with its primitive energy.
Previous work of the authors on a three-level quantum system is extended to allow the radiation field in interaction with the system to have a continuous spectrum of possible frequencies. The limiting procedure involved in the passage from discrete to continuous spectra is complicated by the need to express sums over different discrete spectra as integrals with well-behaved limits. Exact expressions are found, for a spontaneous emission problem, for the time evolution of the probabilities that the system be in (each) one of its three states and representative calculations of these probabilities are presented. The emergence of irreversible behavior upon constructing the thermodynamic limit of the problem is plainly seen and this demonstration permits for the first time a discussion of the relative effectiveness of competing decay channels in three-level quantum systems without imposing any ad hoc assumptions (such as exponential decay). Rather, the actual form of the decay emerges as a consequence of the structure and parameters of the Hamiltonian defining the model, and hence one can examine the variety of circumstances in which the evolution can reasonably be described as exponential. The results obtained should be of great use in clarifying certain outstanding conceptual problems in radiation physics, particularly those which deal with the universality of exponential decay in three- (and two-) level quantum systems in interaction with a radiation field and the conditions under which nonexponential or even nonergodic behavior can emerge in such dissipative quantum systems.
A compact description of the evolution of a many-body quantum system, e.g., a dilute gas, is provided by the generalization of the usual reaction U matrix to a system U matrix. Using this tool, it is demonstrated that in such systems (i) time-reversal-invariant states are unstable and cannot exist in nature or be ``constructed in the laboratory;'' (ii) the time-reversal invariance of the Hamiltonian leads inexorably to relaxation towards thermal equilibrium of arbitrary nonequilibrium states, i.e., to dissipation, which thus allows the definition of a quantum arrow of time; (iii) the second law of thermodynamics, and hence the complete field of thermodynamics, is a consequence of quantum physics. These results have at their basis the hermiticity of the Hamiltonian and the Heisenberg uncertainty relations.