Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Translation invariant quantum master equation

Dmitri Botvich, Malshev, V.A., Anatoly Dmitrievich Manita

Published Sep 20, 1991• 1 citation
Publisher

Abstract

We give the first rigorous derivation of a kinetic equation in a nontrivial translation invariant situation. We consider the heavy particle (mass scaled as me −δ , δ≥0) in the ideal fermi gas with translation invariant interaction eV between them. In the Heisenberg representation we prove convergence to the strongly continuous completely-positive semigroup T s for any value of scaled time s, t=s/e 2 and for δ>2. When δ=2 T s is not a semigroup. For 0≤δ<2 we announce some more weak results

DOI
10.5169/seals-116331

Related papers

Rigorous derivation of Lindblad equations from quantum jumps processes in 1D

Christophe Gomez, Maxime Hauray

We are interested by the behaviour of a 1D single heavy particle, interacting with an environment made of very fast particles in a thermal state. Assuming that the interactions are instantaneous, we construct an appropriate quantum jump process for the density operator of the heavy particle. In a weak-coupling limit (many interactions with few effect), we show that the solutions of jump process converge in law in the appropriate space towards the solution of a Lindbald master equation. To the best of our knowledge, it seems to be the first rigorous derivation of a dissipative quantum evolution equation.

arXiv:1603.07969•Mar 25, 2016•1 citation

A STOCHASTIC GOLDEN RULE AND QUANTUM LANGEVIN EQUATION FOR THE LOW DENSITY LIMIT

Luigi Accardi, Alexander Nikolaevich Pechen, Igor Vasil'evich Volovich

A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.

arXiv:math-ph/0206032•Infinite Dimensional Analysis Quantum Probability and Related Topics•Sep 1, 2003•18 citations

Master-equations for the study of decoherence

Bassano Vacchini

Different structures of master-equation used for the description of decoherence of a microsystem interacting through collisions with a surrounding environment are considered and compared. These results are connected to the general expression of the generator of a quantum dynamical semigroup in presence of translation invariance recently found by Holevo. 1

•Jan 1, 2003•16 citations

Quantum kinetic field theory

Esteban Calzetta, B. L. Hu

The Kadanoff–Baym equations Quantum kinetic field theory is the theme of this chapter. In this section we get right to the heart of it by showing a derivation of the celebrated Kadanoff-Baym (KB) equations [KadBay62]. The basic idea is that close to equilibrium, propagators are nearly translation invariant. It is possible to define a partial Fourier transform with respect to the relative position of the arguments. The Kadanoff – Baym equations then determine how the partial Fourier transform depends on the average (or “center of mass”) of the arguments in the original propagators. Besides the presentation in Kadanoff and Baym's textbook, there are several derivations of these equations in the literature [Dub67, Dan84a, MroDan90, MroHei94, ZhuHei98]. We shall follow [CalHu88, CaHuRa00]. See also [Hen95, IvKnVo00, KnIvVo01, Nie02, Koi02]. References [BoVeWa00, WBVS00] follow a different path towards quantum kinetic theory, based on the so-called dynamical renormalization group. The model . To better appreciate the main points in this derivation, we shall consider a simple model, namely, the KB equations for the theory of a single real self-interacting λΦ 4 scalar field, in the absence of background fields. Actually, the key ideas are not sensitive to the particular models, but for concreteness it will be helpful to have a model in mind. The classical action is given by equation (6.106). A translation-invariant propagator G ab depends on its arguments x and x ′ only through the so-called “relative” variable u = x − x ′.

Cambridge University Press eBooks•Jul 24, 2008•0 citations

Structures of master-equation for the study of decoherence

Bassano Vacchini

Different structures of master-equation used for the description of decoherence of a microsystem interacting through collisions with a surrounding environment are considered and compared. These results are connected to the general expression of the generator of a quantum dynamical semigroup in presence of translation invariance recently found by Holevo. 1

•Jan 10, 2003•0 citations

ON THE WEAK COUPLING LIMIT FOR A FERMI GAS IN A RANDOM POTENTIAL

Tin-Lun Ho, Lawrence J. Landau, A. J. WILKINS

General conditions are derived for the essential self-adjointness of –∆ + V, where V is a translation-invariant random potential, and for the existence of the perturbation expansion. A sequence of graphs is exhibited violating Dell'Antonio's bound for skeleton graphs. For a translation-invariant and clustering Gaussian random potential V, and a translation-invariant and clustering initial state S of the Fermi gas, uncorrelated with the random potential, the weak coupling limit (Van Hove limit) yields increase of entropy, propagation of chaos, convergence of the state for sufficiently small values of the parameter τ to a gauge-invariant and quasi-free asymptotic state, and the semigroup describing the evolution of the two-point function. The asymptotic system is Bernoulli. Results are obtained not only for the average over the random potential but also with probability one. If the random potential V′ is absolutely continuous with respect to V, and if the state S′ is given by a density matrix in the GNS representation for S, then the weak coupling limit is the same as for V and S.

Reviews in Mathematical Physics•Jun 1, 1993•33 citations
We give the first rigorous derivation of a kinetic equation in a nontrivial translation invariant situation. We consider the heavy particle (mass scaled as me −δ , δ≥0) in the ideal fermi gas with translation invariant interaction eV between them. In the Heisenberg representation we prove convergence to the strongly continuous completely-positive semigroup T s for any value of scaled time s, t=s/e 2 and for δ>2. When δ=2 T s is not a semigroup. For 0≤δ<2 we announce some more weak results
We are interested by the behaviour of a 1D single heavy particle, interacting with an environment made of very fast particles in a thermal state. Assuming that the interactions are instantaneous, we construct an appropriate quantum jump process for the density operator of the heavy particle. In a weak-coupling limit (many interactions with few effect), we show that the solutions of jump process converge in law in the appropriate space towards the solution of a Lindbald master equation. To the best of our knowledge, it seems to be the first rigorous derivation of a dissipative quantum evolution equation.
A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.
The Kadanoff–Baym equations Quantum kinetic field theory is the theme of this chapter. In this section we get right to the heart of it by showing a derivation of the celebrated Kadanoff-Baym (KB) equations [KadBay62]. The basic idea is that close to equilibrium, propagators are nearly translation invariant. It is possible to define a partial Fourier transform with respect to the relative position of the arguments. The Kadanoff – Baym equations then determine how the partial Fourier transform depends on the average (or “center of mass”) of the arguments in the original propagators. Besides the presentation in Kadanoff and Baym's textbook, there are several derivations of these equations in the literature [Dub67, Dan84a, MroDan90, MroHei94, ZhuHei98]. We shall follow [CalHu88, CaHuRa00]. See also [Hen95, IvKnVo00, KnIvVo01, Nie02, Koi02]. References [BoVeWa00, WBVS00] follow a different path towards quantum kinetic theory, based on the so-called dynamical renormalization group. The model . To better appreciate the main points in this derivation, we shall consider a simple model, namely, the KB equations for the theory of a single real self-interacting λΦ 4 scalar field, in the absence of background fields. Actually, the key ideas are not sensitive to the particular models, but for concreteness it will be helpful to have a model in mind. The classical action is given by equation (6.106). A translation-invariant propagator G ab depends on its arguments x and x ′ only through the so-called “relative” variable u = x − x ′.
General conditions are derived for the essential self-adjointness of –∆ + V, where V is a translation-invariant random potential, and for the existence of the perturbation expansion. A sequence of graphs is exhibited violating Dell'Antonio's bound for skeleton graphs. For a translation-invariant and clustering Gaussian random potential V, and a translation-invariant and clustering initial state S of the Fermi gas, uncorrelated with the random potential, the weak coupling limit (Van Hove limit) yields increase of entropy, propagation of chaos, convergence of the state for sufficiently small values of the parameter τ to a gauge-invariant and quasi-free asymptotic state, and the semigroup describing the evolution of the two-point function. The asymptotic system is Bernoulli. Results are obtained not only for the average over the random potential but also with probability one. If the random potential V′ is absolutely continuous with respect to V, and if the state S′ is given by a density matrix in the GNS representation for S, then the weak coupling limit is the same as for V and S.