26 research outputs found
On the solutions of the nonlinear Liouville hierarchy
We investigate the initial-value problem of the non-linear Liouville
hierarchy. For the general form of the interaction potential we construct an
explicit solution in terms of an expansion over particle clusters whose
evolution is described by the corresponding-order cumulant of evolution
operators of a system of finitely many particles. For the initial data from the
space of integrable functions the existence of a strong solution of the Cauchy
problem is proved.Comment: 9 page
New concept of relativistic invariance in NC space-time: twisted Poincar\'e symmetry and its implications
We present a systematic framework for noncommutative (NC) QFT within the new
concept of relativistic invariance based on the notion of twisted Poincar\'e
symmetry (with all 10 generators), as proposed in ref. [7]. This allows to
formulate and investigate all fundamental issues of relativistic QFT and offers
a firm frame for the classification of particles according to the
representation theory of the twisted Poincar\'e symmetry and as a result for
the NC versions of CPT and spin-statistics theorems, among others, discussed
earlier in the literature. As a further application of this new concept of
relativism we prove the NC analog of Haag's theorem.Comment: 15 page
Transport Equations from Liouville Equations for Fractional Systems
We consider dynamical systems that are described by fractional power of
coordinates and momenta. The fractional powers can be considered as a
convenient way to describe systems in the fractional dimension space. For the
usual space the fractional systems are non-Hamiltonian. Generalized transport
equation is derived from Liouville and Bogoliubov equations for fractional
systems. Fractional generalization of average values and reduced distribution
functions are defined. Hydrodynamic equations for fractional systems are
derived from the generalized transport equation.Comment: 11 pages, LaTe
Deterministic constant-temperature dynamics for dissipative quantum systems
A novel method is introduced in order to treat the dissipative dynamics of
quantum systems interacting with a bath of classical degrees of freedom. The
method is based upon an extension of the Nos\`e-Hoover chain (constant
temperature) dynamics to quantum-classical systems. Both adiabatic and
nonadiabatic numerical calculations on the relaxation dynamics of the
spin-boson model show that the quantum-classical Nos\`e-Hoover chain dynamics
represents the thermal noise of the bath in an accurate and simple way.
Numerical comparisons, both with the constant energy calculation and with the
quantum-classical Brownian motion treatment of the bath, show that the
quantum-classical Nos\`e-Hoover Chain dynamics can be used to introduce
dissipation in the evolution of a quantum subsystem even with just one degree
of freedom for the bath. The algorithm can be computationally advantageous in
modeling, within computer simulation, the dynamics of a quantum subsystem
interacting with complex molecular environments.Comment: Revised versio
On "full" twisted Poincare' symmetry and QFT on Moyal-Weyl spaces
We explore some general consequences of a proper, full enforcement of the
"twisted Poincare'" covariance of Chaichian et al. [14], Wess [50], Koch et al.
[34], Oeckl [41] upon many-particle quantum mechanics and field quantization on
a Moyal-Weyl noncommutative space(time). This entails the associated braided
tensor product with an involutive braiding (or -tensor product in the
parlance of Aschieri et al. [3,4]) prescription for any coordinates pair of
generating two different copies of the space(time); the associated
nontrivial commutation relations between them imply that is central and
its Poincar\'e transformation properties remain undeformed. As a consequence,
in QFT (even with space-time noncommutativity) one can reproduce notions (like
space-like separation, time- and normal-ordering, Wightman or Green's
functions, etc), impose constraints (Wightman axioms), and construct free or
interacting theories which essentially coincide with the undeformed ones, since
the only observable quantities involve coordinate differences. In other words,
one may thus well realize QM and QFT's where the effect of space(time)
noncommutativity amounts to a practically unobservable common noncommutative
translation of all reference frames.Comment: Latex file, 24 pages. Final version to appear in PR
Quantum Kinetic Evolution of Marginal Observables
We develop a rigorous formalism for the description of the evolution of
observables of quantum systems of particles in the mean-field scaling limit.
The corresponding asymptotics of a solution of the initial-value problem of the
dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution
of marginal observables and the evolution of quantum states described in terms
of a one-particle marginal density operator are established. Such approach
gives the alternative description of the kinetic evolution of quantum
many-particle systems to generally accepted approach on basis of kinetic
equations.Comment: 18 page
The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems
The Cauchy problem for the von Neumann hierarchy of nonlinear equations is
investigated. One describes the evolution of all possible states of quantum
many-particle systems by the correlation operators. A solution of such
nonlinear equations is constructed in the form of an expansion over particle
clusters whose evolution is described by the corresponding order cumulant
(semi-invariant) of evolution operators for the von Neumann equations. For the
initial data from the space of sequences of trace class operators the existence
of a strong and a weak solution of the Cauchy problem is proved. We discuss the
relationships of this solution both with the -particle statistical
operators, which are solutions of the BBGKY hierarchy, and with the
-particle correlation operators of quantum systems.Comment: 26 page
Fractional Liouville and BBGKI Equations
We consider the fractional generalizations of Liouville equation. The
normalization condition, phase volume, and average values are generalized for
fractional case.The interpretation of fractional analog of phase space as a
space with fractal dimension and as a space with fractional measure are
discussed. The fractional analogs of the Hamiltonian systems are considered as
a special class of non-Hamiltonian systems. The fractional generalization of
the reduced distribution functions are suggested. The fractional analogs of the
BBGKI equations are derived from the fractional Liouville equation.Comment: 20 page
Fractional Systems and Fractional Bogoliubov Hierarchy Equations
We consider the fractional generalizations of the phase volume, volume
element and Poisson brackets. These generalizations lead us to the fractional
analog of the phase space. We consider systems on this fractional phase space
and fractional analogs of the Hamilton equations. The fractional generalization
of the average value is suggested. The fractional analogs of the Bogoliubov
hierarchy equations are derived from the fractional Liouville equation. We
define the fractional reduced distribution functions. The fractional analog of
the Vlasov equation and the Debye radius are considered.Comment: 12 page
Towards Rigorous Derivation of Quantum Kinetic Equations
We develop a rigorous formalism for the description of the evolution of
states of quantum many-particle systems in terms of a one-particle density
operator. For initial states which are specified in terms of a one-particle
density operator the equivalence of the description of the evolution of quantum
many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and
by the Cauchy problem of the generalized quantum kinetic equation together with
a sequence of explicitly defined functionals of a solution of stated kinetic
equation is established in the space of trace class operators. The links of the
specific quantum kinetic equations with the generalized quantum kinetic
equation are discussed.Comment: 25 page