Shapely figures to contemplate

A Course in Modern Mathematical Physics
December 2, 2005

This is a beautifully crafted book. Mathematical physics is not any longer so much about how to juggle partial differential equations and intriguing integrals. Although analytical skills are still desirable, the modern scope of the field is more focused on mathematical structure. Peter Szekeres presents in the most elegant and compelling manner a magnificent overview of how classic areas such as algebra, topology, vector spaces and differential geometry form a consistent and unified language that has enabled us to develop a description of the physical world reaching a truly profound level of comprehension.

The book is very self-contained, and a dedicated student with a good A-level background will be able to use it for self-study. The book's pedagogical impact is enhanced significantly by a large selection of well-composed problems and exercises. As a textbook for a lecture course, it can be used for undergraduate as well as graduate students in physics and mathematics. Even curious engineering students would find some interesting reading here. Lecturers can obtain a PDF with worked-out solutions to all the problems (though not the exercises, which are supposed to be somewhat easier); a wonderful opportunity for a busy academic.

Szekeres's style is clear, thorough and immensely readable. His selection of topics concentrates on areas where a fully developed rigorous mathematical exposition is possible. Often the classic theorem-lemma-corollary exposition is used, but not in a dry and excluding way. Szekeres includes introductions, explanations and comments in rich measure to ensure that the reader always understands the purpose of the discussion.

This is not yet another "Mathematical methods for scientists" book. The intention is not to teach analytical skills - plenty of good books already exist for that purpose. Szekeres's ambition is to get the reader to understand the mathematics at a level that goes far beyond a superficial algorithmic ability. To understand the essentials of mathematics one needs to grasp the proofs; only that way does one appreciate the deeper content and meaning of mathematical concepts. This then facilitates a more profound understanding of the physics. Remember that in physics many discoveries have been made by a careful analysis of the mathematical formalism. An example is Pauli's discovery of the relation between spin and statistics (a theorem mentioned in the quantum mechanics chapter, but rightly judged to be beyond the scope of the book).

Szekeres shows the reader how blocks such as group theory, vector spaces, algebras, topology, measure theory and differential geometry make a magnificent edifice that enables us to establish marvellous conceptual achievements such as special relativity, thermodynamics, quantum mechanics and general relativity. On the way, we encounter fascinating topics such as the continuum hypothesis and category theory, and we learn in depth about exciting constructs such as Grassmann algebras, Lebesgue measures, quaternions and Lie algebras. I believe many mathematicians would find this enlightening. One cannot help but be slightly awed by the beauty and the capability with which seemingly abstract concepts, often developed in the realms of pure mathematics, turn out to be applicable - "the unreasonable effectiveness of mathematics in the natural sciences", as Eugene Wigner memorably wrote in the title of one of his papers.

The book does not contain material on quantum fields or path integrals, probably because a rigorous treatment is not possible yet in this area.

Stochastic processes are left out, probably for reasons of space. But what is covered is wonderful. I recommend that you get hold of this book for yourself or your library.

Henrik Jeldtoft Jensen is professor of mathematical physics, Imperial College London.

A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry. First edition

Author - Peter Szekeres
Publisher - Cambridge University Press
Pages - 600
Price - £40.00
ISBN - 0 521 82960 7

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