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Statistical symmetries in physics. (English) Zbl 0839.17019

Every law of physics is invariant under some group of transformations and is therefore the expression of some type of symmetry. Symmetries are classified as geometrical, dynamical or statistical. At the most fundamental level, statistical symmetries are expressed in the field theories of the elementary particles. This paper traces some of the developments from the discovery of Bose statistics, one of the two fundamental symmetries of physics. A series of generalizations of Bose statistics is described. A supersymmetric generalization accomodates fermions as well as bosons, and further generalizations, including parastatistics, modular statistics and graded statistics, accomodate particles with properties such as ‘colour’. A factorization of elements of \(ggl(n_b, n_f)\) can be used to define truncated boson operators. A general construction is given for \(q\)-deformed boson operators, and explicit constructions of the same type are given for various ‘deformed’ algebras; these include a rather simple \(Q\)-deformed variety as well as the well known \(q\)-deformed variety. A summary is given of some of the applications and potential applications.

MSC:

17B81 Applications of Lie (super)algebras to physics, etc.
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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