Encyclopedia of Mathematical Physics, Jean-Pierre Françoise

Encyclopedia of Mathematical Physics, Jean-Pierre Françoise.
 
   The Encyclopedia is intended primarily for experienced researchers but should be of use also to beginning graduate students. For the latter category of readers,we have included eight elementary introductory articles for easy reference, with those on mathematics aimed at physics graduates and those on physics aimed at mathematics graduates, so that these articles can serve as their first port of call to enable them to embark on any of the main articles without the need to consult other material beforehand. In fact, we think these articles may even form the foundation of advanced undergraduate courses, as we know that some authors have already made such use of them. In addition to the printed version, an on-line version of the Encyclopedia is planned, which will allow both the contents and the articles themselves to be updated if and when the occasion arises. This is probably a necessary provision in such a rapidly advancing field.




Geometric Modular Action.
The fact that the modular objects in quantum field theory associated with wedge-shaped regions and the vacuum state in Minkowski space have geometric significance (“geometric modular action”) was originally discovered in the framework of the Wightman axioms. As an algebraic quantum field theory (AQFT) does not rely on the concept of Wightman fields, it was natural to ask (i) when does geometric modular action hold in AQFT and (ii) which physically relevant consequences follow from this feature?

There are two approaches to the study of geometric modular action. In the first, attention is focused on modular covariance, expressed in terms of the modular groups associated with wedge algebras and the vacuum state in Minkowski space. Modular covariance has been proven to obtain in conformally invariant AQFT, in any massive theory satisfying asymptotic completeness, and also in the presence of other, physically natural assumptions. To mention only three of its consequences, both the spin-statistics theorem and the PCT theorem, as well as the existence of a continuous unitary representation of the Poincare group acting covariantly upon the observable algebras and satisfying the spectrum condition follow from modular covariance.

CONTENTS.
INTRODUCTORY ARTICLES.
PHYSICS SUBJECTS.
Classical Mechanics.
Classical, Conformal and Topological Field Theory.
Condensed Matter and Optics.
Disordered Systems.
Dynamical Systems.
Equilibrium Statistical Mechanics.
Fluid Dynamics.
Gauge Theory.
General Relativity.
Integrable Systems.
M-Theory see String Theory and M-Theory.
Nonequilibrium Statistical Mechanics.
Quantum Field Theory.
Quantum Gravity.
Quantum Information and Computation.
Quantum Mechanics.
String Theory and M-Theory.
Yang–Mills Theory see Gauge Theory.
RELATED MATHEMATICS SUBJECTS.
Algebraic Techniques.
Algebraic Topology.
Complex Geometry.
Differential Geometry.
Discrete Mathematic.
Functional Analysis and Operator Algebras.
Lie Groups and Lie Algebras.
Low Dimensional Geometry.
Noncommutative Geometry.
Ordinary and Partial Differential Equations.
Quantization Methods and Path Integration.
Quantum Groups.
Stochastic Methods.
Symplectic Geometry and Topology.
Variational Techniques.



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Теги: энциклопедия по математике :: математика :: Jean-Pierre Françoise