An axiomatic basis for quantum mechanics. Vol. 1: Derivation of Hilbert space structure.

*(English)*Zbl 0582.46065
Berlin etc.: Springer-Verlag. X, 243 p. DM 118.00 (1985).

This book is the first volume of a two-volume work on the fundamental concepts of quantum mechanics. It is the author’s aim to deduce the description of microsystems solely from their production and detection by macroscopic devices.

First of all, it is pointed out that experiments with microsystems must be describable by ”pretheories” for quantum mechanics the essential feature of which is the ”objectivating manner of description”. The axiomatic scheme presented here bases on macrosystems being composed of two parts with ”directed interaction” where later on microsystems are discovered as ”action carriers”. Statistically equivalent ”preparing” and ”registration devices” for the action carriers define ”ensembles” and ”effects”.

In the central chapter of the book it is shown that the set K of ensembles and the set L of effects can be embedded as convex sets in a dual pair of a base norm and an order-unit norm space, the duality given by the probability function \(\mu\) defined on \(K\times L\). This development was initiated by the author himself in the sixties and includes classical statistical as well as quantum theories. In the following chapter ”observables” and ”preparators” are introduced in the context of the general formulation.

In order to derive the Hilbert space structure of quantum mechanics, empirically founded axioms are formulated. By means of these so-called ”main laws of preparation and registration” it is possible to introduce ”decision effects” (which correspond to the usual propositions of quantum logic) and to prove the structure of the partially ordered set \(G\subset L\) of decision effects as an atomic orthomodular complete orthocomplemented lattice. The Hilbert space enters by representation theory of modular and orthomodular orthocomplemented lattices. Presuming G to be irreducible, the author finally shows by use of Gleason’s theorem that the statistical dual pair is given by the space of self-adjoint trace-class operators in Hilbert space, the space of bounded self-adjoint operators, and the trace.

Further investigations about the relation between quantum mechanics and macrophysics, quantum statistical thermodynamics as well as the measuring problem will be presented in a second volume. The entire book and especially the first volume should be of greatest interest to all researchers and advanced students concerned with the fundamental concepts of quantum theory or the general mathematical structure of statistical physical theories. The book supplements another two-volume work of the author, namely, ”Foundations of quantum mechanics I, II”, New York etc.: Springer-Verlag (1983; Zbl 0509.46057), (1985; Zbl 0574.46057) to which the reader is sometimes referred.

First of all, it is pointed out that experiments with microsystems must be describable by ”pretheories” for quantum mechanics the essential feature of which is the ”objectivating manner of description”. The axiomatic scheme presented here bases on macrosystems being composed of two parts with ”directed interaction” where later on microsystems are discovered as ”action carriers”. Statistically equivalent ”preparing” and ”registration devices” for the action carriers define ”ensembles” and ”effects”.

In the central chapter of the book it is shown that the set K of ensembles and the set L of effects can be embedded as convex sets in a dual pair of a base norm and an order-unit norm space, the duality given by the probability function \(\mu\) defined on \(K\times L\). This development was initiated by the author himself in the sixties and includes classical statistical as well as quantum theories. In the following chapter ”observables” and ”preparators” are introduced in the context of the general formulation.

In order to derive the Hilbert space structure of quantum mechanics, empirically founded axioms are formulated. By means of these so-called ”main laws of preparation and registration” it is possible to introduce ”decision effects” (which correspond to the usual propositions of quantum logic) and to prove the structure of the partially ordered set \(G\subset L\) of decision effects as an atomic orthomodular complete orthocomplemented lattice. The Hilbert space enters by representation theory of modular and orthomodular orthocomplemented lattices. Presuming G to be irreducible, the author finally shows by use of Gleason’s theorem that the statistical dual pair is given by the space of self-adjoint trace-class operators in Hilbert space, the space of bounded self-adjoint operators, and the trace.

Further investigations about the relation between quantum mechanics and macrophysics, quantum statistical thermodynamics as well as the measuring problem will be presented in a second volume. The entire book and especially the first volume should be of greatest interest to all researchers and advanced students concerned with the fundamental concepts of quantum theory or the general mathematical structure of statistical physical theories. The book supplements another two-volume work of the author, namely, ”Foundations of quantum mechanics I, II”, New York etc.: Springer-Verlag (1983; Zbl 0509.46057), (1985; Zbl 0574.46057) to which the reader is sometimes referred.

Reviewer: W.Stulpe

##### MSC:

46N99 | Miscellaneous applications of functional analysis |

46A40 | Ordered topological linear spaces, vector lattices |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

46B42 | Banach lattices |

47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |