Constitutions of Matter: Mathematically Modelling the Most Everyday of Physical Phenomena

Most people accept that scientific knowledge occupies a domain that is somewhat distinct from that of everyday knowledge. This applies particularly to physics, where most concepts are expressed in esoteric mathematical formulae, such as Newton's equations of motion or Boyle's Law. What they may not realise is that mathematical knowledge as such is deemed to occupy a separate domain, and that physicists themselves make a further distinction between "theoretical" and "mathematical" physics, almost as if these were two different scientific disciplines. In reality, these disciplines overlap. Martin Krieger's concern is with the relationship between them.

The basic proposition of this book is that the "mathematical" entities and operations that arise in the derivation or formal proof of certain formulae always have a "physical" significance. Thus, for example, every student of physics learns that the force acting on a particle in a gravitational field can be described mathematically as the gradient of a "potential function" which seems to be defined by an arbitrary formula, but which turns out to be none other than that ubiquitous physical quantity - energy. At a much more sophisticated level, the apparently meaningless "negative energy" solutions of Dirac's quantum equations for electrons correspond physically to positron states - and so on.

Krieger's proposition is thus supported by some positive examples - but is it valid in general? Theoretical physicists have often found that it holds the other way round. One of the standard strategies for discovering new mathematical relations in the physical sciences is to designate mathematical symbols with formal properties defined to accord precisely with the known behaviour of certain physical entities, and then see what comes out of the algebra. This, in effect, is how Einstein got into general relativity, and Heisenberg into quantum mechanics. Indeed, one could argue on quite general grounds - though Krieger does not actually do so - that any well-founded "physical" analysis of the properties of a properly defined model system must have a counterpart in more systematic "mathematical" language. In fact, much of the effort in "mathematical physics" is devoted to developing formalisms that illustrate and delimit this type of relationship. It turns out, for example, that some very subtle principles of functional analysis are required to formulate the famous laws of thermodynamics rigorously.

But the general notion that exact mathematical proofs must be deducible from sound physical thinking does not always work in practice, and the converse does not follow. To see this, one has to ask - which Krieger again fails to do - what we mean by a "physical" idea. As near as I can make it, this is a mental entity or operation that can be manipulated in the mind in connection with the subject matter of physics, and communicated in words, symbols or diagrams to others who are similarly accustomed to thinking in that way. That is obviously an almost empty definition that can be filled only by listening to and reading what physicists say. That in turn needs some definition of physics as a scientific discipline, which is surely something more than what happens to be taught under that heading in the best universities.

For my money, physics is the science of the empirically quantitative, excluding all other. Not surprisingly, therefore, although physics is not the same as mathematics, physical thinking is completely permeated with mathematical concepts. Typically, even in referring to such "physical" entities as pressure and volume, a professional physicist such as Krieger would use the mathematical symbols p and v, as if experiencing them mentally as reciprocal variables in Boyle's mathematical formula.

It follows, then, that most widely used physical ideas can be traced back to, or are standardised by reference to, a term or a symbol in a mathematical relationship. It does not follow, though, that all the terms in all the mathematical relationships relevant to physics are sufficiently useful to deserve wide currency amongst physicists. Here is a famous example. Richard Feynman's diagrammatic algorithm for renormalising quantum field theory has proved such a wonderful guide to thought about the interactions of particles that it now seems part of physical reality. On the other hand, nobody now uses Julian Schwinger's algebraic formalism, although it was simultaneous with, precisely equivalent to and exactly as rigorous as Feynman's solution to the same problem.

What is more, theoretical research in physics is often focused on models that do not purport to represent the real world but are deliberately simplified to be amenable to mathematical formulation and analysis. It is hard to see any metaphysical reason why mathematical entities devised to obtain exact formulae for the properties of such models should have any physical significance. Of course, they may turn out to be very valuable By analogy, just as fictional characters like Mr Micawber and Winnie the Pooh play a metaphorical role in our discourse about social life, but that is a matter of artistic choice rather than of philosophical necessity.

Unfortunately, Krieger's lengthy and densely technical book deals only with just such implausible cases. In particular, he is fascinated by one of the most brilliant displays of mathematical virtuosity in theoretical physics: Lars Onsager's derivation, in 1943, of an exact formula for the critical temperature of the two-dimensional Ising model for a magnetic system. For the uninitiated - ie, except for a few thousand specialists - let me say that this model is well understood to be unrealistic, in that it lacks both quantum-mechanical veracity and the third dimension of "everyday matter". Moreover, as the author of one of the few books where an attempt was made to explicate this mathematical tour de force, and to review its physical implications, I am not persuaded by Krieger's rather impressionistic arguments and do not intend to defect from the expert consensus that Onsager's methods cannot be usefully generalised "mathematically" or "physically". Krieger is just nowhere near making his case, empirically or theoretically.

The same objection applies to much else in this book. The proposition motivating it is not uninteresting in principle, but is too narrowly and weakly conceived here to be justified in this way. An occasional specialist in this field of research might be stimulated to an original thought By skimming through it, but I could not, alas, honestly recommend it even to a departmental librarian awash with funds.

John Ziman is emeritus professor of physics, University of Bristol.