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Classical representations of quantum mechanics related to statistically complete observables. (English) Zbl 0898.46069

Berlin: Wissenschaft & Technik Verlag (Habil.). 117 p. DM 48.00 (1997).
The problem of reformulating quantum mechanics in terms of probability densities and functions on phase space originated with the famous paper of Eugene P. Wigner from 1932 and has been a subject of much research ever since. The interest on that problem ranges from practical calculational needs to fundamental interpretational questions. The accumulation of knowledge on the relations between irreducible projective group representations, continuous resolutions of identity, and statistically complete observables has opened a systematic way to the phase space representations of quantum mechanics. The book of Werner Stulpe offers a compact, rigorous treatment of such representations.
In the introductory chapter of the book, the author gives a simple direct proof of the well-known fact that there is no phase space representation of quantum mechanics which could reproduce the usual probability densities for the position and momentum observables. Chapter 2 introduces the fundamental notion of the statistical completeness of a family of observables and gives a basic characterization of such families. A construction of single statistically complete observables is given and it is shown that statistically complete observables are necessarily noncommutative. This means that such observables can only be given as positive operator measures and not as projection measures.
In Chapter 3 the definition of a classical representation of quantum mechanics on a sample space is formulated and it is shown that each statistically complete observable defines such a representation and that, conversely, any classical representation of quantum mechanics determines a unique statistically complete observable. A reformulation of quantum dynamics in terms of measures evolving in time on the classical sample space is also worked out.
Chapter 4 investigates continuous resolutions of identity, showing, in particular, how they are obtained from irreducible projective unitary representations of locally compact second countable topological groups. The remaining three chapters specify the abstract setting by dealing with the concrete \(2N\)-dimensional real phase space \(\mathbb{R}^{2N}\). Chapter 5 introduces joint position-momentum observables via continuous resolutions of identity obtained by an irreducible projective unitary representation of the additive group \(\mathbb{R}^{2N}\). Their marginal observables are determined to be approximate position and approximate momentum observables, respectively.
Chapter 6 investigates Hilbert spaces of bounded infinitely differentiable wave functions on phase space and it determines the action of position and momentum operators in a phase space representation. In the final chapter of the book, a class of statistically complete joint position-momentum observables is worked out and the related classical representations of quantum mechanics on phase space are investigated in great detail.
This is a concise rigorous treatise on the phase space representations of quantum mechanics. Advanced students and researchers in quantum physics as well as mathematicians working on operator theory should find this book interesting and useful. The price and the size of the book are also favourable. Readers of it might also enjoy reading the recent book of F. E. Schroeck, jun. [Quantum mechanics on phase space (Fundamental Theories of Physics, Vol. 74, Kluwer Academic Publishers, Dordrecht) (1996; Zbl 0876.00025)].

MSC:

46N50 Applications of functional analysis in quantum physics
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81P05 General and philosophical questions in quantum theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
81R30 Coherent states
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47N50 Applications of operator theory in the physical sciences
60A05 Axioms; other general questions in probability

Citations:

Zbl 0876.00025
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