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arXiv:gr-qc/0007044 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

How large are dissipative effects in noncritical Liouville string theory?

John R. Ellis, Nick E. Mavromatos, D.V. Nanopoulos

Published Dec 29, 2000• 41 citations
arXivHTMLPDF

Abstract

In the context of non-critical Liouville strings, we clarify why we expect non-quantum-mechanical dissipative effects to be $\mathcal{O}{(E}^{2}{/M}_{P}),$ where E is a typical energy scale of the probe, and ${M}_{P}$ is the Planck scale. In Liouville strings, energy is conserved at best only as a statistical average, as distinct from Lindblad systems, where it is strictly conserved at an operator level, and the magnitude of dissipative effects could only be much smaller. We also emphasize the importance of nonlinear terms in the evolution equation for the density matrix, which are important for any analysis of complete positivity.

DOI
10.1103/physrevd.63.024024

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In the context of non-critical Liouville strings, we clarify why we expect non-quantum-mechanical dissipative effects to be O(E2/MP),\mathcal{O}{(E}^{2}{/M}_{P}),O(E2/MP​), where E is a typical energy scale of the probe, and MP{M}_{P}MP​ is the Planck scale. In Liouville strings, energy is conserved at best only as a statistical average, as distinct from Lindblad systems, where it is strictly conserved at an operator level, and the magnitude of dissipative effects could only be much smaller. We also emphasize the importance of nonlinear terms in the evolution equation for the density matrix, which are important for any analysis of complete positivity.
In the context of non-critical Liouville strings, we clarify why we expect non-quantummechanical dissipative eects to be O E 2 =MP ,w hereE is a typical energy scale of the probe, and MP is the Planck scale. In Liouville strings, energy is conserved at best only as a statistical average, as distinct from Lindblad systems, where it is strictly conserved at an operator level, and the magnitude of dissipative eects could only be much smaller. We also emphasize the importance of nonlinear terms in the evolution equation for the density matrix, which are important for any analysis of complete positivity.
The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.
We discuss differential-- versus integral--equation based methods describing out--of thermal equilibrium systems and emphasize the importance of a well defined reduction to statistical observables. Applying the projection operator approach, we investigate on the time evolution of expectation values of linear and quadratic polynomials in position and momentum for a statistical anharmonic oscillator with quartic potential. Based on the exact integro-differential equations of motion, we study the first and naive second order approximation which breaks down at secular time-scales. A method is proposed to improve the expansion by a non--perturbative resummation of all quadratic operator correlators consistent with energy conservation for all times. Motion cannot be described by an effective Hamiltonian local in time reflecting non-unitarity of the dissipative entropy generating evolution. We numerically integrate the consistently improved equations of motion for large times. We relate entropy to the uncertainty product, both being expressible in terms of the observables under consideration.
The possible role of decoherence due to space-time foam is discussed within the context of two models, one based on string/brane theory. and the other based on properties of black hole horizons in general relativity. It is argued that the density matrix satisfies a dissipative master equation, primarily from the study of renormalization group flows in non-critical string theory.This interpretation of the zero mode of the Liouville field as time leads necessarily to the CPT operator being ill defined. One striking consequence is that the quantum mechanical correlations of pair states of neutral mesons produced in meson factories are changed from the usual EPR state. The magnitude of this departure from EPR correlations is characterised by a parameter ωωω. The predicted value of ωωω is very small or zero. However it is shown explicitly that the the non-vanishing of ωωω is only a feature of the model based on string/brane theory.
Based on a recently developed quantum dissipation formulation [R. X. Xu and Y. J. Yan, J. Chem. Phys. 116, 9196 (2002)], we present a reduced Liouville-space approach to evaluate the response and correlation functions of dissipative systems. The weak system-bath interaction is treated properly for its effects on the initial state, the evolution, and the correlation between coherent driving and non-Markovian dissipation. Numerical demonstration shows this correlated effect cannot be neglected even in the calculation of linear response quantities that do not explicitly depend on external fields. Highlighted in this paper is also the proper choice of theory among various formulations in the weak system-bath interaction regime.
Green's functions and transition self-energies of dissipative quantum-mechanical many-body systems are formulated in terms of operator-basis expansions of Heisenberg equations of motion in Liouville space. Matrix elements of the Liouvillian with respect to arbitrary nonorthogonal operator bases play a primary role in the analysis. The self-energies are expressed as functionals of initial statistical ensemble averages which are presumed known and not restricted to systems at or near equilibrium; this provides a first-principles approach to use of spectral line shapes as a diagnostic of the nonequilibrium state of a dense gas, plasma, and/or radiation field. A perturbation expansion of the self-energy is derived and the general term is interpreted in terms of a subtraction of reducible parts. A separation into resonant and nonresonant contributions plays a crucial role in the analysis. The first-order resonant contribution yields a generalized statistical Hartree-Fock approximation, whereas the second-order contribution generalizes the Weisskopf-Wigner theory of decaying states to include thermal and, more generally, nonequilibrium environmental effects on transition-energy shifts and widths. Off-diagonal contributions to the self-energy are shown to be essential for inclusion of environmental contributions.