Quantum and Non-Commutative Analysis Past, Present and Future Perspe
In the past decade, there has been a sudden and vigorous development in a number of research areas in mathematics and mathematical physics, such as theory of operator algebras, knot theory, theory of manifolds, infinite dimensional Lie algebras and quantu
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MATHEMATICAL PHYSICS STUDIES
Series Editor:
M. FLATO, Universite de Bourgogne, Dijon, France
VOLUME 16
Quantum and Non-Commutative Analysis Past, Present and Future Perspectives
edited by
Huzihiro Araki Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
Keiichi R. Ito Department of Mathematics and Information, College of Liberal Arts, Kyoto Unversity, Kyoto, Japan
Akitaka Kishimoto Department of Mathematics, Hokkaido University, Sapporo, Japan
and
Izumi Ojima Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data Quantum and non-commutative analysis : past, presant, and future perspectives 1 edited by Huzihiro Araki ... [et al.]. p. cm. -- R ~ r. A state which can trigger some n-fold coincidence at a time t but no (n+ 1)-fold coincidence is called n-fold localized at the time. A single particle state is a state which is singly localized at all times. This gives an operational definition of the concept of particle. The asymptotic localization number for t -+ oo becomes the number of outgoing particles. This approach, suggested in [1] and [2] has been developed in the last decade especially by Buchholz [3]-[5] to a remarkable degree so that the treatment of collision theory in the presence of zero mass particles and the discussion of completeness of the particle picture becomes possible. One very important conclusion from this discussion is that the theory must have some "compactness" or "nuclearity" property. One version of this is the following. Let SE denote the set of states with energy below E and S E( CJ) their restriction to the region CJ (as positive linear forms over 2l( CJ).) Then, to characterize a state in SE( CJ) up to precision c; in the norm topology we need only a finite pumber N(e) of parameters. We expect that the exploitation of this property can provide a key to the problem of classifying theories within the frame described. The third class of physically interesting results concerns statistical mechanics. The formulation of the theory in terms of abstract local algebras gives us the benefit of treating states with non-vanishing matter density extending to infinity and if we wish we can do so in the relativistic setting. These are states in Rep H and among them are the equilibrium states in the
LOCAL QUANTUM PHYSICS AND BEYOND
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thermodynamics limit. Time is too short now to engage in this vast area. So I shall only say that the description of Gibbs ensembles given by Kubo and by Martin and Schwinger assumes a very natural form in the algebraic setting. It establishes contact with the powerful mathematical theory of modular automorphisms by Tomita and Takesaki, allows an operational characterization of equilibrium, and thermodynamic phases, relates the chemical potentials to the superselection structure of RepEp· Let us finally look at the shortcomings of the frame developed thus far and at open problems. The outstanding questions are of course:
