American Scientist

Constitutions of Matter: Mathematically Modeling the Most Everyday of Physical Phenomena. Martin H. Krieger. 343 pp. University of Chicago Press, 1996. $65.

Philosophical books on mathematical physics tend to be fairly rare these days. This is especially true if they deal with the usual psycho-existential questions such as "What induced the big-bang?" or "What is the absolute meaning of time?" In Martin Krieger's Constitutions of Matter we are dealing with the real, hard stuff: the mathematics within. The whole book advocates the following audacious theory: Mathematical tricks and techniques and reasoning are the true driving forces behind most physical theories, the primary goal of which is to model the all-too-complex world we live in. The more accurate our explanations tend to be, the more predictive they are as well, and the greater is the level of mathematical rigor they require. Deep physics is hidden everywhere in the meanders of the apparently only decorative—but indeed crucial—technical details of most mathematical-physical theories.

To illustrate his point, the author has chosen to concentrate on the questions of constitution and stability of matter from a statistical mechanical point of view. The various chapters of the book discuss famous fundamental papers of statistical physics: Darwin-Fowler's derivation of the probability distribution of states of given energy, Lieb and Lebowitz's mathematical construction of the "constitution of matter,'' Onsager's solution of the two-dimensional Ising model, Yang and Lee's evidence of phase transitions, further mathematical reformulation of the Ising model in terms of fermions by Schultz, Mattis and Lieb, and, finally, Baxter's solutions of integrable lattice models.

With great confidence, Krieger takes us through the labyrinths of these rather technical papers to expose the somewhat hidden ideas and constructions, only to show their deep physical implications. The global outcome is a sort of dictionary that translates each mathematical step into physical ideas. For example, the mathematical tool known as the transfer matrix for a two-dimensional statistical system with short-range interactions is nothing but a means of explicitly building up matter out of local bits and pieces, the short-range interaction being justified by the screening of long-range forces within each piece.

The main difficulty in reading this book is inherent to the field it describes: Most arguments are impossible to follow superficially, and even to get a sense of what is going on requires more than a good knowledge of the general principles. It is unrealistic to think that a reader can directly benefit from reading this book without ever having tried to read the underlying papers or the related textbooks. Rather, the book can be seen as an extremely useful reading companion that establishes connections and clarifies ideas by putting them into a broader perspective.

This book is neither a physics textbook nor a philosophical essay. Its whole point is to advocate mathematical tricks, and it is full of them, including two reprints of long original papers by Lars Onsager. To a nonexpert eye this will at best fill a gap: Finally all this mathematical ugliness will have a sense of utility. To physicists, this should definitely shed some light on the hard work of a few of their mathematically oriented colleagues. To the latter, this book is a comforting reassurance, if they nurtured any doubts, that this hard work is going to pay off someday.

At a time when dramatic budget cuts affect fundamental research in physics, and all the more so poor relatives such as mathematical physics, Krieger's book gives us a good opportunity to stop and look at what has been accomplished so far. We may realize how modern mathematics is slowly reshaping physics and vice versa, and how crucial the interplay between these two fields is going to be in the years to come.

Who's afraid of the Big Bad Math? Whoever is, missed the whole point.—Philippe R. Di Francesco, Mathematics, University of North Carolina at Chapel Hill