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Relativistic quantification. (Quantification relativiste.) (French) Zbl 0745.35057
In this memoir, the author provides a thorough introduction to the Klein- Gordon symbolic calculus of operators, and discusses an application to the Mathieu oscillator. Various old and new results are presented in a unified and systematic fashion, thus greatly enhancing their accessibility. The central theme is the analogy between the (relativistic) Klein-Gordon calculus and the (classical) Weyl calculus: while the former arises naturally from the Klein-Gordon equation, the latter can be derived analogously from Schrödinger’s equation. Moreover, the Weyl calculus can be obtained as the non-relativistic limit of the Klein-Gordon calculus.
Preceded by a very instructive introduction, the material is arranged in the following sixteen sections: Space-time, and the space of observers. The Klein-Gordon symbolic calculus. A digression on the Weyl calculus. Coherent states, and the relativistic Wick calculus. Invariant operators on the mass hyperboloid. The connection between Klein-Gordon symbols and relativistic Wick symbols. Klein-Gordon symbols, standard symbols, and others. Geometric inequalities, and classes of symbols. Continuity of the Klein-Gordon operators. Characterization of operators and composition. Composition of symbols. Symbols of the infinitesimal generators of the Bargmann-Wigner representation. The envelope algebra. Asymptotic expansions. Relativistic Euler operator, de Sitter group, and Mathieu oscillator. The Weyl calculus as a non-relativisitic limit of the Klein- Gordon calculus.

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
47G30 Pseudodifferential operators
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
81S99 General quantum mechanics and problems of quantization
35Q40 PDEs in connection with quantum mechanics
47N50 Applications of operator theory in the physical sciences
58J40 Pseudodifferential and Fourier integral operators on manifolds
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