What is an integrable system? Here’s an oblique answer in the style of V. I. Arnold (cf. p. 1 of the book under review): “In fact, the theorem of Liouville … covers all the problems of dynamics which have been integrated to the present day.” So, we are talking about physics, be it in the classical or quantum version, as the book’s title indicates. What then does the theorem that Arnold refers to say? Here’s what we find on p. 11, under the heading “Arnold-Liouville theorem”: on a \( 2N \)-dimensional symplectic manifold, \( P \), suppose we have \( N \) functions \( f_{i} \) such that we get \( \left\{ f_{i},f_{j} \right\} = 0, \forall i,j \), (we’re playing with the Poisson bracket here) and consider what is called a common level set for these functions, \( P_{c}=\left\{ x \in P | f_{i}(x)=c_{i}, \forall i\right\} \), with constants ci. Suppose, too, that the 1-forms \( df_{i} \) are linearly independent at each \( x \in P_{c} \). Then \( P_{c} \) is a smooth manifold invariant under a Hamiltonian flow induced by the \( f_{i}\). If, additionally, \( P_{c} \) is compact and connected, then it is diffeomorphic to an \( N \)-torus on which the motion under the Hamiltonian is conditionally periodic. Finally, the according equations of motion can be integrated by quadrature. The latter term is perhaps somewhat arcane these days: a quadrature is a process that, by definition, involves “solving a finite number of algebraic equations and computing a finite number of definite integrals.” We are manifestly dealing with very classical material in mathematical physics, seeing that Hamiltonian dynamics is a quintessentially early 19th-century subject.

Regarding the purpose and goals of the book under review, the author states in his Preface that while the subject is certainly equipped with a classical pedigree, as per e.g. the work of Liouville represented by an 1853 paper which, presumably, deals with what Arnold discusses in the quote above, there are remarkable 21st-century developments to take note of. These include connections with the gauge-string duality conjecture, and with quantum field theory on the 2-dimensional string world sheet leading to “spectacular results in … 4-dimensional gauge theory avoiding the evaluation of zillions of Feynman diagrams.” Arutyunov goes on to judge that “perhaps this is … the first time that string theory reveals, through its integrable structure, its wonderful elegance and simplicity in comparison to the standard field-theoretic approach based on perturbation theory.” Given the ubiquity of the latter approach in quantum physics, for example in quantum electrodynamics, and its notoriety due both to daunting computational complexity and the physicists’ penchant for committing mathematical offenses (renormalization, the absence of a well-defined measure for Feynman integrals), this alternative offered by string theory is indeed tantalizing. In connection herewith there is a technical observation to be made: Arutyunov restricts himself to finite-dimensional integrability “leaving out the theory of classical and quantum field theories,” but he notes that “the Bethe Ansatz approach to the spectrum of quantum-mechanical models … [allows one, with] little effort to adapt [our discussion] for the field-theoretic case.” This allusion to the Bethe Ansatz is fascinating in its own right, in the (different) context of the present book, and it is presented in the fifth chapter; again, however, no quantum field theory (or classical field theory), but, as the author stipulates, that is no prohibitive restriction at all.

Thus, it is about mechanics, classical as well as quantum, but it is fair to say that Arutyunov uses the former to get to the latter. The foundational first chapter sets the stage for everything and is clearly non-negotiable; it contains a thorough discussion of Liouville integrability and goes pretty deep, including coverage of Noether’s (physics) theorem (i.e. the equivalence of symmetries and conservation laws) and (Peter) Lax pairs. With symmetries introduced into the story, the author’s second chapter gets to integrability as a consequence of such symmetries, and now the dividends are dramatic. Lie groups appear with gusto, and we even get the notion of the Heisenberg double, which is “a symplectic manifold that can be regarded as a deformation of the cotangent bundle [to a Lie group] … [and] admits a variety of symmetry transformations which makes it a phase space for obtaining non-trivial integrable models …” (cf. p. 118). The context is that of Hamiltonians and dynamical systems of phase spaces, of course.

The book continues in this vein, with sophisticated and physically motivated material front-and-center, what with the last three chapters dealing with, respectively, scattering theory, the Bethe Ansatz, and, finally, integrable thermodynamics. There are five appendices, covering, among other things, variations on themes from Lie theory and additional material from physics including more scattering theory, i.e. material relating to Heisenberg’s S-matrix.

Clearly, this well-written and well thought out book deals with some marvelous material, which, while meant for physicists, and written by a physicist, is of great relevance to mathematics. This book should be very interesting and useful to many mathematicians.

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