×

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

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
47A60 Functional calculus for linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Goulaouic C., Arch. Rational Mech. Anal 34 pp 361– (1970)
[2] Bourdaud G., Sur les operateurs pseudo–differentiels à coefficients peu réguliers (1983)
[3] Feynman R.P., Quantum Electrodynamics (1962) · Zbl 0112.45703
[4] Guillemin V., Symplectic techniques in physics (1984) · Zbl 0576.58012
[5] Landau, L. et Lifschitz, E. Théorie de champ, Editions Mir (1966), Moscou
[6] Lax P.D., C.P.A.M 32 pp 617– (1979)
[7] Magnus W., Formulas and theorems for the special functions of mathematical physics 3 (1966) · Zbl 0143.08502
[8] DOI: 10.1007/BF02392873 · Zbl 0492.58023
[9] Reed M., Methods of modern mathematical physics 2 (1975)
[10] Simon B., The P()2 euclidean (quantum) field theory (1974) · Zbl 1175.81146
[11] Stein E.M., Singular integrals and differentiability properties of functions (1970) · Zbl 0207.13501
[12] Unterberger, A. Pseudodifferential operators and applications : an introduction, Aarhus Universitet Lecture Notes · Zbl 0342.47028
[13] DOI: 10.1080/03605308408820361 · Zbl 0561.35081
[14] Unterberger, A. L’analyse harmonique et 1’analyse pseudo-differentielle du cone ou d’un domaine mixte, to appear
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.