Some basic concepts of algebraic quantum theory

Kasia Rejzner

Ellen B. Stechel

A quantum ergodic theory for finite systems (such as isolated molecules) is developed by introducing the concept of a quantum measure algebra. The basic concept in classical ergodic theory is that of a measure space. A measure space is a set M, together with a specified sigma algebra of subsets in M and a measure defined on that algebra. A sigma algebra is closed under the formation of intersections and symmetric differences. A measure is a nonnegative and countably additive set function. For this to be further classified as a dynamical system, a measurable transformation is introduced. A measurable transformation is a mapping from a measure space into a measure space, such that the inverse image of every measurable set is measurable. In conservative dynamical systems, a measurable transformation is measure preserving, which is to say that the inverse image of every measurable set has the same measure as the original set. Once the measure space and the measurable transformation are defined, ergodic theory can be investigated on three levels—describable as analytic, geometric and algebraic. The analytic level studies linear operators induced by a transformation. The geometric level is concerned directly with transformations on a measure space and the algebraic treatments substitute a measure algebra for the measure space and basically equate sets that differ only by sets of measure zero. It is this latter approach that is most directly paralleled here. A measure algebra for a quantum dynamical system is defined within which stochastic concepts in quantum mechanics can be investigated. The quantum measure algebra differs from a normal measure algebra only in that multiplication is noncommutative and addition is nonassociative. Nonetheless, the quantum measure algebra preserves the essence of a normal measure algebra.

Valter Moretti

Anjan Kundu

An overview of the quantum integrable systems (QIS) is presented. Basic concepts of the theory are highlighted stressing on the unifying algebraic properties, which not only helps to generate systematically the representative Lax operators of different models, but also solves the related eigenvalue problem in an almost model independent way. Difference between the approaches in the integrable ultralocal and nonultralocal quantum models are explained and the interrelation between the QIS and other subjects are focussed on.

Valter Moretti

Andrés Fernando Reyes-Lega

Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics. Aspects of the representation theory of C*-algebras will be motivated and illustrated in physical terms. Particular emphasis will be given to explicit examples from the theory of quantum phase transitions, where concepts coming from strands as diverse as quantum information theory, algebraic quantum physics and statistical mechanics agreeably converge, providing a more complete picture of the physical phenomena involved.