Quantum mechanics is one of the wonders and mysteries of the early twentieth century. Richard Feynman is quoted as saying, “I can safely say that nobody understands quantum mechanics,” and this is in itself an epistemological challenge, raising the question of what it means to understand something. Werner Heisenberg, the very first one to travel to this brave new world with his introduction of non-commuting observables that precipitate the centrally important uncertainty relation, famously argued against his friend, Paul Dirac, that theorizing in this realm should be done essentially free of preconceptions from classical physics. Dirac argued for more methodological continuity, so to speak, in regard to corresponding themes in classical physics, but this apparent foray into conservatism notwithstanding, he was of course in truth as much of a revolutionary as his friend. Heisenberg’s view was that to grasp the according workings of nature, one should simply lay out the mathematics working all but in vacuo. Dirac looked for analogy, or at least paid more attention to it, as his formulation of quantum mechanics featuring something like the Poisson bracket suggests. Nonetheless, in both of these formulations, there is a rupture with classical physics, manifest in the uncertainty principle, and this state of affairs ultimately precipitated the Copenhagen interpretation of quantum mechanics, replete with its non-classical statistics, incurring Albert Einstein’s unrelenting ire.

In any case, Heisenberg’s matrix mechanics entered physics as an entirely unexpected motif --- once his thesis advisor, Max Born, identified what Heisenberg had discovered as belonging to matrix algebra. As the story has it (cf. Constance Reid’s wonderful biography, *Hilbert*), Born and Heisenberg took their new mathematics literally next door to the Mathematical Institute at Göttingen and to David Hilbert, who proceeded to say something about a differential equation that would have Heisenberg’s matrices canonically associated with it: they should look for such a beast. Evidently, the two physicists decided the old mathematician didn’t know what he was talking about --- Heisenberg’s approach seemed, to them, altogether incompatible with what Hilbert was suggesting. Of course, soon after this, Erwin Schrödinger came along with his PDE and Hilbert had the last laugh. He didn’t shy from rubbing the physicists’ noses in it: he was known to say that physics was much too difficult for physicists.

Hilbert’s assistant around that time (or slightly later) was none other than the formidable John von Neumann, and it is to him that the focus of the book under review is due. Von Neumann, author of the classic *Mathematical Foundations of Quantum Mechanics*, and the originator of the kind of operator theory that is prevalent in the subject, took the subject to a different level. Starting with the theory of densely defined unbounded operators on a Hilbert space (in which rays, or half-lines, are the states of the quantum mechanical system), i.e. some very substantial functional analysis, von Neumann turned to broader algebras of operators, soon to yield the theory of operator algebras such as C*-algebras, von Neumann algebras, and so on. These are special cases (of huge importance) of the objects the present book is concerned with, namely *-algebras. By definition, a *-algebra is an algebra (a vector space with multiplication) equipped with an involution, i.e. a mapping satisfying a natural linearity condition (if we’re working over the complex numbers conjugate the scalars), commutation with the ring multiplication, and the predictable familiar condition that the mapping is its own inverse. So it is a lot of algebraic fun from the very start. But as the earlier historical comments suggest, there is a lot hiding behind all this.

Well, the cat is already peeking out of the bag. Schmüdgen’s Prologue tells us that if the mathematical formulation of quantum mechanics is along the lines sketched above, with observables being, again by definition, self-adjoint operators on the given Hilbert space, and with (pure) states being (unit) rays (just normalize) in that space, then the explicitly algebraic approach to the subject goes to a higher level of abstraction. Specifically, “the observable algebra is the central object of the theory … a complex unital *-algebra … [with] key postulates … [to the effect that] (A1) Each observable is a Hermitian element … of the *-algebra [and] (A2) Each state is a linear functional on [the *-algebra, satisfying a natural positivity condition].” The connection with C*-algebras arises since the algebra of bounded operators on a Hilbert space is one of these.

This really is beautiful stuff. In his excellent Preface, Schmüdgen goes on to observe that if a group acts as a symmetry group on the observable algebra, meaning that there’s a homomorphism from the group to the *-algebra’s *-automorphisms, then there is a notion of Jordan automorphism of the algebra coming about via the insistence that the Jordan product be conserved. Here the Jordan product of two elements, \( a \) and \( b \), is the average \( (1/2)(ab+ba) \), and the required conservation criterion is pretty deep: “In the case of pure states on a Hilbert space, the transition probability of states was chosen as the relevant concept [sound familiar?]. In the algebraic approach, it is natural to require that symmetries preserve the Jordan product.” Can anyone resist noting in this connection that the Jordan product relates intimately to the uncertainty principle? (With no uncertainty, it reduces to the ordinary product, in either --- irrelevant --- order.)

And as far as mathematics goes, symmetry is of course one of the most profound guiding themes of all, responsible for group theory and, in the present physical context, group representation theory and its applications and evolutions. Schmüdgen’s book’s fourth chapter is in fact devoted to nothing less than *-representations, with his chapters 6 and 8 devoted to, respectively, “Representations of Tensor Algebras” and “Integrable Representations of Commutative *-Algebras.” To drive home the point of parallels with more familiar quantum mechanics (i.e. a more functional analytic approach), § 7.4 deals with “spectral measures of integrable representations.” This clearly resonates with what takes place in the standard (Copenhagen) interpretation of quantum mechanics with probabilities of ending up in certain states being the end-all. To boot, Chapter 9 deals with “Integrable Representations of Enveloping Algebras,” and we accordingly start to encounter Lie theory. Finally, Chapters 12, 13, 14 deal with representation theory with a certain vengeance: “Induced *-Representations,” “Well-behaved Representations,” and “Representations on Rigged spaces and Hilbert C*-modules.”

Finally, harking back to familiar material from, for lack of a better word, quantum mechanics as functional analysis, Chapter 8 bears mentioning. Here we encounter discussions of the Weyl algebra (and we recall the ground-breaking The Theory of Groups and Quantum Mechanics by Hermann Weyl), the Schrödinger representation of this algebra, the Stone-von Neumann theorem, and, indeed, in section 8.7, a return to the start of the whole story, Heisenberg’s uncertainly principle. The more things change, the more they stay the same.

This is a fantastic book. The material it covers is wonderful and exciting. The book is well-written, and in fact pleasant to read. It takes a familiar subject --- that is unfortunately not as popular among mathematicians as it should be --- and presents it from a very evocative perspective. Kudos to Schmüdgen.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.