The Caldeira–Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

We quantize the topological σ-model. The quantum master equation of the Batalin-Vilkovisky formalism ΔρΨ=0 appears as a condition which eliminates the exact states from the BRST invariant states Ψ defined by QΨ=0. The phase space of the BV formalism is a supermanifold with a specific symplectic structure, called the fermionic Kähler manifold.

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FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

International Journal of Modern Physics B ◽

10.1142/s0217979207045086 ◽

2007 ◽

Vol 21 (23n24) ◽

pp. 3993-3999

Author(s):

SUMIYOSHI ABE

Keyword(s):

Brownian Motion ◽

Quantum Theory ◽

Fractional Calculus ◽

Diffusion Equation ◽

Master Equation ◽

Anomalous Diffusion ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Quantum Master Equation ◽

Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

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Master equation in phase space applied to the quantum Brownian motion in a tilted periodic potential

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/45/10/105002 ◽

2012 ◽

Vol 45 (10) ◽

pp. 105002 ◽

Cited By ~ 3

Author(s):

William T Coffey ◽

Yuri P Kalmykov ◽

Serguey V Titov ◽

Liam Cleary ◽

William J Dowling

Keyword(s):

Brownian Motion ◽

Phase Space ◽

Master Equation ◽

Periodic Potential ◽

Quantum Brownian Motion

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Erratum: Master equation in phase space applied to the quantum Brownian motion in a tilted periodic potential

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/45/17/179601 ◽

2012 ◽

Vol 45 (17) ◽

pp. 179601

Author(s):

William T Coffey ◽

Yuri P Kalmykov ◽

Serguey V Titov ◽

Liam Cleary ◽

William J Dowling

Keyword(s):

Brownian Motion ◽

Phase Space ◽

Master Equation ◽

Periodic Potential ◽

Quantum Brownian Motion

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Semiclassical master equation in Wigners phase space applied to Brownian motion in a periodic potential

Physical Review E ◽

10.1103/physreve.75.041117 ◽

2007 ◽

Vol 75 (4) ◽

Cited By ~ 24

Author(s):

W. T. Coffey ◽

Yu. P. Kalmykov ◽

S. V. Titov ◽

B. P. Mulligan

Keyword(s):

Brownian Motion ◽

Phase Space ◽

Master Equation ◽

Periodic Potential

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ON POSSIBLE GENERALIZATIONS OF FIELD-ANTIFIELD FORMALISM

International Journal of Modern Physics A ◽

10.1142/s0217751x93000928 ◽

1993 ◽

Vol 08 (13) ◽

pp. 2333-2350 ◽

Cited By ~ 47

Author(s):

I. A. BATALIN ◽

I. V. TYUTIN

Keyword(s):

Phase Space ◽

Master Equation ◽

Jacobian Matrix ◽

Functional Integral ◽

Quantum Master Equation ◽

Arbitrary Field ◽

Constraint Surface

A generalized version is proposed for the field–antifield formalism. The antibracket operation is defined in arbitrary field–antifield coordinates. The antisymmetric definitions are given for first- and second-class constraints. In the case of second-class constraints the Dirac's antibracket operation is defined. The quantum master equation as well as the hypergauge fixing procedure are formulated in a coordinate–invariant way. The general hypergauge functions are shown to be antisymmetric first–class constraints whose Jacobian matrix determinant is constant on the constraint surface. The BRST-type generalized transformations are defined and the functional integral is shown to be independent of the hypergauge variations admitted. In the case of reduced phase space the Dirac's antibrackets are used instead of the ordinary ones.

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