Joint review of

On p. 70 of *Mathematical Foundations of Quantum Mechanics*, by none other than George W. Mackey, one of the grand-masters of unitary representation theory (and much else besides), there occurs the following sequence. "Whenever A + B is always defined we can define 'symmetrized products' of bounded observables by setting

A º B = [(A + B)^{2} – A^{2} – B^{2}]/2.
In the model actually in use for quantum mechanics A º B is distributive with respect to addition and the set of all bounded observables forms a commutative, non-associative algebra of a sort known to algebraists as a *Jordan algebra*. [Pasqual] Jordan [was] a physicist who began the study of these algebras in a quantum-mechanical context. Unfortunately there are no known physical reasons for assuming the multiplication A º B to be distributive in general, and Lowdenslager and Sherman have given examples showing that distributivity does not follow from Segal’s axioms."

The reference is to I. E. Segal, of course, whose work, much like Mackey's, occupies a major place in that mysterious place where so many beautiful parts of mathematics and physics meet, with (unitary) representation theory constituting a unifying theme in it all: operators playing in Hilbert space. Mackey and Segal are in fact both tied to a major topic in both quantum mechanics and number theory, namely, the oscillator representation. Closely associated to the Schrödinger representation of a (suitable) Heisenberg group, and therefore rooted in classical quantum mechanics as done by the pioneers themselves, this projective representation is known by number theorists as the Weil representation. The reference here is to André Weil's devastatingly beautiful treatment of the analytic theory of quadratic forms *à la* Carl Ludwig Siegel, going back to the early and middle 1960s.

The point to be taken is that we have here a uniquely beautiful part of mathematics which is also remarkably difficult to characterize: is it physics, is it algebra, is it number theory, is it all of the above? The answer, clearly, is yes.

Mackey’s book (beautiful stuff, of course, and very clear!) is ultimately unabashedly concerned with physics, even as it is "designed for students with a reasonably high degree of facility in dealing with abstract mathematical concepts and little or no knowledge of physics. The reader is assumed to be familiar with the basic concepts of abstract algebra, point set topology, and measure theory, and is given a rapid introduction to coordinate-free tensor analysis on [infinitely differentiable] manifolds and to the theory of self-adjoint operators in Hilbert space." Mackey goes on to say: "The aim of the course [MA 263 at Harvard in the spring of 1960] was to explain quantum mechanics and certain parts of classical physics from a point of view more congenial to pure mathematicians than that commonly encountered in physics texts." And this truly describes the book whose three chapters are, in sequence, "Classical Mechanics," "Quantum Mechanics," and "Group Theory and the Quantum Mechanics of the Atom."

And then there is the other end of the spectrum. [It's so very easy to pun in this area, and nigh-on impossible to resist the temptation! But I plead that indulging this vice is appropriate in light of what’s next: read on!]

The other book under review is Kevin McCrimmon's *A Taste of Jordan Algebras*, whose nature is, if I may put it so, violently algebraic (his Chapter 1, "Jordan Algebras in Physical Antiquity," notwithstanding). Its appeal, which is very considerable, is thus of a very different sort than that of Mackey's book: to quote a line from Weil's famous review of Chevalley’s book on Lie groups, "This is algebra with a vengeance!"

McCrimmon is a pioneer in the subject of Jordan algebras (*qua* algebras, so to speak), having "introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars" (from the back cover). He also is an extremely engaging (if idiosyncratic) expositor: "The reader should see isomorphisms as cloning maps, isotopes as subtle rearrangements of an algebra’s DNA, radicals as pathogens to be isolated by radical surgery, annihilators as biological agents for killing off elements, Peircers as mathematical enzymes." And: "Like Charlie Brown’s kite-eating trees, Jordan theory has Zel'manov's tetrad-eating ideals (though we shall stay clear of these carnivores in our book)." The book is written in this marvellous style, even as it is encyclopaedic in scope, very thorough, and very strong on (the right kind of) pedagogy. The reader will learn a lot of wonderful algebra well, if he takes care to follow McCrimmon's plan: read carefully, do the problems, meditate on what’s going on, follow and absorb the analogies (even as you enjoy the humor and plays on words).

As regards the specific focus of the book, it is largely about the work of Efim Zel'manov. Again, here are McCrimmon's own words: "This book tells the story of one aspect of Jordan structure theory: the origin of the theory in an attempt by quantum physicists to find algebraic systems more general than hermitian matrices, and ending with the surprising proof by Efim Zel'manov that there is really only one such system, the 27-dimensional Albert algebra, much too small to accommodate quantum mechanics [So there!]"

So, yes, it's hard-core algebra, and there is a lot to be learned here, both as regards beautiful material and a way of thinking and doing some hard and intricate mathematics.

Finally, I can't resist a somewhat personal allusion. My four sons (aged 1 to 9) have for some months now been altogether enthralled with the score of the musical *Oklahoma* and I'm apt to be assaulted at any time of the day with a lyric like "Everything’s up to date in Kansas City..." Well, for better or worse (verse? Whose pun is that?), McCrimmon welds the following perverted stanzas into his narrative:

Everything's up to date in structure theory;

They've gone about as fer as they can go;

They went and proved a structure theorem for rings with d.c.c.,

About as fer as a theorem ought to go (Yes, sir!)"

[p. 94]

Everything's up to date in Jordan structure theory;

They've gone about as fer as they can go;

They went and proved a structure theorem for rings with capacity,

About as fer as a theorem ought to go (Yes, sir!)"

[p. 106]

And then the author goes on to perpetrate the following observation: "At the very time this tune was reverberating in nonassociative circles throughout the West, a whole new song without idempotents was being composed in far-off Novosibirsk." The next chapter is titled "The Russian Revolution." Wow!

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University.