Book review

The book under review is devoted to the proof of the famous Gleason's theorem on the states on a Hilbert space and its applications to quantum logic. A quantum logic, or in short a logic, is a set of quantum mechanical events which is closed under countable disjunction of countably many mutually exclusive events.

The Kolmogorov probabilistic model of a physical system is a very important one, but not satisfactory in describing physical systems in which, for microlevel measurements, nonnegligible indeterminacies must be respected in a physical model.

J. von Neumann in his book "Mathematische Grundlagen der Quantummechanik", Springer-Verlag, Berlin 1932, proposed a mathematical apparatus and an interpretational logic, where the states of quantum system are unit vectors of a complex separable Hilbert space H and the observables are self-adjoint operators on H.

In G.W. Mackey's kook "Mathematical foundations of Quantum Mechanics", Benjamin, New-York 1963, the quantum logic of a Hilbert space is the set L(H) of all closed subspaces of a separable Hilbert space H and a state on L(H) is a probability measure m: L(H) -> [0,1]. Obviously that, for a unit vector <formula> is a probability measure on L(H), i.e. a state on L(H). The same is true for any sigma-convex combination <formula>. The remarkable result of A.M. Gleason J. Math. Mech. 6 (1957), 885 - 893, asserts that all the states on L(H) are of this form, if H is a separable Hilbert space with dim H != 2. There are examples of states on a two-dimensional Hilbert space which are not of this form.

The heart of the proof of Gleason's theorem lies on its real three dimensional version. The original proof is very complicated and have been made many attempts to simplify it. The most elementary proof known until now belongs to R. Cooke, M. Keane and W. Moran, Proc. Cambridge Phil. Soc. 98 (1985), 117 - 128, and it is presented in the third Chapter of the book. Beside this proof, this chapter contains also extensions of Gleason's theorem to signed measures, to finitely additive measures and to non-separable Hilbert spaces, as well as various applications, some of them unexpected as for example the characterization, obtained by the author, of non-measurable cardinals: Any finite sigma-additive measure on L(H) is a Gleason measure if and only if the dimension of H is a non-measurable cardinal != 2 (Theorem 3.5.1).

Chapter 1, Hilbert Space Theory, has an introductory character, presenting the basic facts from Hilbert space theory and operator theory on Hilbert spaces.

In Chapter 2, Theory of Quantum Logic, in addition to basic facts it is presented the complete solution, obtained by the author, of the existence of joint distribution of observables in general quantum logics.

Other chapters of the book are Chapter 4, Gleason's Theorem and Completeness Criteria, giving characterizations of the completeness of inner product spaces in terms of completely additive signed measures, defined on families of closed subspaces of these spaces, and Chapter 5, Applications of Gleason's Theorem. In this last chapter are presented the non-classical Hilbert spaces of H.A. Keller, Math. Z. 172 (1980), 41 - 49, where the states can be also described by a Gleason type theorem.

Presenting an interdisciplinary theory - the theory of L(H)-quantum logics, with many still unsolved problems, the book will be of great interest, first of all for physicists and mathematicians, but also for researchers in social sciences, psychology, the activity of human brain, where Heisenberg type uncertainity principles arise in a natural way. For instance, in psychology, the patient's psychological stress Delta-p and the accuracy of his (or her) answers Delta-x are connected by a relation <formula>.