The branch of theoretical physics which omits to make heuristic assumptions but investigates the rigor of their mathematical formulations and conclusions based on them as well as develops suitable mathematical structures is called mathematical physics. Mathematical physics has yielded many fields of activity, partially known under specific names, such that the encyclopedia describing these fields in short essays of about ten pages is very useful and will be highly appreciated by the scientific community.
In his foreword Gerard ’t Hooft highlights the symbiotic relationship of mathematics and physics. The preface by the editors addresses the encyclopedia to experienced researchers as well as to beginning graduate students. Since the alphabetic order of the articles according to their titles is, clearly, at random according to the subjects, there is a contents list by subjects, where the articles are grouped together according to the subject to which their topics belong. The classification scheme contains 35 subjects. There are eight introductory articles to the main fields. An index is put at the end in Vol. 5, but one does not find each concept mentioned in the articles, e.g the Kochen-Specker theorem is mentioned in an article by Roger Penrose or quantum operations are used in articles by Michael Keyl, both concepts do not appear in the index. It seems that only those concepts which are explicitly introduced or explained are contained in the index. Eight articles are classified as introductory ones and placed at the beginning.
Two of the introductory articles, ‘Classical Mechanics’ and ‘Equilibrium Statistical Mechanics’ by G. Galavotti, are of around 35 pages, the other ones have the ‘usual’ length of about ten pages. This seems sufficient for ‘Electromagnetism’ by N. M. Woodhouse, and ‘Minkowski Space and Special Relativity’ by G. L. Naber, but it seems too short for ‘Differential Geometry’ and ‘Functional Analysis’ by S. Psycha, ‘Quantum Mechanics by G. F. dell’Antonio, and ‘Topology’ by Tso Sheung Tsun. The article on mechanics is restricted to point mechanics and conservative forces which is warranted by the fact, that nonconservative forces arise in open systems. One asks why d’Alembert’s principle and non-holonomic constraints are not considered at all in the encyclopedia. The introductory article ‘Quantum Mechanics’ as well as the article ‘Quantum Mechanics: Foundations’ by Roger Penrose both confine themselves to the orthodox (Copenhagen) interpretation with all its narrowness and difficulties about quantum measurement and quantum reality as it is known from the first half of the last century. The research on foundations of quantum mechanics in mathematical physics concerning quantum logic, effect algebras and, particularly, the operational approach [see for a comprehensive description e.g. S. Gudder, Stochastic methods in quantum mechanics. North Holland Series in Probability and Applied Mathematics. New York, Oxford: North Holland (1979; Zbl 0439.46047); and see for collections of recent papers the Pecca Lahti Festschrift, part 1: [J] Found. Phys. 39, No. 6 (2009), part 2: [J] Found. Phys. 39, No. 7 (2009)] of the second half of the last century and contemporary is almost completely ignored. Concepts developed in the operational approach are central in quantum information, e.g. quantum operations as completely positive maps, so they are used in articles on quantum information. The research not considered, moreover, sheds new light on the difficulties with the Copenhagen interpretation. Here, only articles on Bohmian mechanics and hidden variables are present as alternatives.
Clearly, most of the working activities in mathematical physics are well described and brought to an order such that quick information can be achieved. Many of the articles contain at the end a section on open problems. Each article contains a list of references for further reading. This encyclopedia is a very valuable and useful work which can be best recommended.