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Reduced and extended weak coupling limit

Jan Dereziński, Wojciech De Roeck (2007)

Banach Center Publications

The main aim of our lectures is to give a pedagogical introduction to various mathematical formalisms used to describe open quantum systems: completely positive semigroups, dilations of semigroups, quantum Langevin dynamics and the so-called Pauli-Fierz Hamiltonians. We explain two kinds of the weak coupling limit. Both of them show that Hamiltonian dynamics of a small quantum system interacting with a large resevoir can be approximated by simpler dynamics. The better known reduced weak coupling...

Refined Algebraic Quantization: Systems with a single constraint

Donald Marolf (1997)

Banach Center Publications

This paper explores in some detail a recent proposal (the Rieffel induction/refined algebraic quantization scheme) for the quantization of constrained gauge systems. Below, the focus is on systems with a single constraint and, in this context, on the uniqueness of the construction. While in general the results depend heavily on the choices made for certain auxiliary structures, an additional physical argument leads to a unique result for typical cases. We also discuss the 'superselection laws' that...

Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics

Othmar Koch, Christian Lubich (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We discuss the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of the time-dependent Schrödinger equation in quantum molecular dynamics. This method approximates the high-dimensional nuclear wave function by a linear combination of products of functions depending only on a single degree of freedom. The equations of motion, obtained via the Dirac-Frenkel time-dependent variational principle, consist of a coupled system of low-dimensional nonlinear partial differential...

Relating quantum and braided Lie algebras

X. Gomez, S. Majid (2003)

Banach Center Publications

We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if Γ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space k ⊕ Γ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra U ( Γ ) is a bialgebra in the category of A-comodules.

Relative conditional expectations on a logic

Oľga Nánásiová, Sylvia Pulmannová (1985)

Aplikace matematiky

In this paper, the authors introduce the notion of conditional expectation of an observable x on a logic with respect to a sublogic, in a state m , relative to an element a of the logic. This conditional expectation is an analogue of the expectation of an integrable function on a probability space.

Relatively additive states on quantum logics

Pavel Pták, Hans Weber (2005)

Commentationes Mathematicae Universitatis Carolinae

In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous...

Relativistic stability of matter (I).

Charles L. Fefferman, Rafael de la Llave (1986)

Revista Matemática Iberoamericana

In this article, we study the quantum mechanics of N electrons and M nuclei interacting by Coulomb forces. Motivated by an important idea of Chandrasekhar and following Herbst [H], we modify the usual kinetic energy -∆ to take into account an effect from special relativity. As a result, the system can implode for unfavorable values of the nuclear charge Z and the fine structure constant α. This is analogous to the gravitational collapse of a heavy star. Our goal here is to find those values of α...

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