Pseudodifferential analysis, quantum mechanics and relativity. (English) Zbl 0659.35114

The author develops a calculus of pseudo differential type, connected with quantum mechanics and relativity. More precisely, starting from symbols a(t,x,v) defined on the product of Minkowski’s spacetime \(M_ 4\) by the unit ball \(B_ 3\) of admissible speeds \((| v| <c)\), operators A are defined acting on a suitable Hilbert space H. The construction is the following, in short: the author begins by describing a Fuchs type calculus on a hyperboloid, then transfers it to a Fuchs calculus on a ball, and then defines A by taking a conjugation under the Fourier transformation. When the speed of light c goes to infinity, the preceding calculus contracts to the standard pseudo differential Weyl calculus in \({\mathbb{R}}^ 3\).
Reviewer: L.Rodino


35S05 Pseudodifferential operators as generalizations of partial differential operators
47A60 Functional calculus for linear operators
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