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Ali, S. Twareque; Prugovečki, Eduard

Mathematical problems of stochastic quantum mechanics: Harmonic analysis on phase space and quantum geometry. (English) Zbl 0603.22010

Acta Appl. Math. 6, 1-18 (1986).
The authors review in this paper the main problems and results of functional analysis and representation theory of locally compact groups which are relevant to their program of stochastic quantum mechanics and quantum spacetime [the second author, Stochastic quantum mechanics and quantum spacetime (1984; Zbl 0536.60100); the first author, Riv. Nuovo Cim. 8, No.11, 1-128 (1985)].
They consider positive-operator-valued measures and stress the role of these measures in providing, via systems of covariance, generalizations of Mackey’s imprimitivity theorem [the reviewer, Comment. Math. Helv. 54, 629-641 (1979; Zbl 0425.22010); the first author, Can. Math. Bull. 27, 390-397 (1984; Zbl 0596.22003)] as well as their connection with reproducing kernel Hilbert spaces [the reviewer, J. Math. Phys. 23, 659- 664 (1982)]. Moreover, they show how systems of covariance are central in the fibre bundle formulation of quantum geometries [the second author, Nuovo Cim. A 89, 105-125 (1985)].
Reviewer: U.Cattaneo

Cited in 2 Reviews
Cited in 9 Documents

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
83A05 Special relativity
81T60 Supersymmetric field theories in quantum mechanics

Keywords:

kinematical groups; harmonic analysis; POV measure; phase space representation; review; systems of covariance; reproducing kernel Hilbert spaces; quantum geometries

Citations:

Zbl 0603.22011; Zbl 0603.22012; Zbl 0536.60100; Zbl 0425.22010; Zbl 0596.22003
Cite Review PDF
Full Text: DOI

References:

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