Translated by: C.H. Cushman-de Vries
Edited by: R.H. Cushman and G.M. Tuynman
Published by Birkhäuser, Boston, 1997,
This book has three surprises in store for the unsuspecting bookshop browser. First, it is not a book about `dynamical systems', at least not in the sense that most readers of UK Nonlinear News will perceive the subject. As the subtitle suggests, its central theme is that symplectic geometry provides a natural language for describing some of the most fundamental concepts of physics. The second surprise is the ambitious scope of the book. Geometrical frameworks are constructed for classical, statistical and quantum mechanics and used to derive consequences as diverse as a classification of elementary particles, the Stefan-Boltzmann law for black body radiation and the Schrödinger equation.
The first step in the construction of Souriau's big picture is a generalisation of the Lagrangean and Hamiltonian formulations of classical mechanics. He replaces the usual Lagrangean functions with Lagrange two-forms on `evolution space' and proposes as a physical principle (Maxwell's Principle) that these forms should be closed. The motions of the system define a foliation that is tangent to the kernel of the Lagrange form. The next step is central to the whole development. Instead of working on the evolution space, attention is shifted to the `space of motions', ie the quotient of the evolution space by the foliation. If Maxwell's principle is satisfied the Lagrange form descends to a symplectic form on this space. At this stage the original evolution space can be forgotten and a mechanical, or dynamical, system can simply be defined to be a symplectic manifold. Note that this is fundamentally different from the Hamiltonian definition which consists of a Hamiltonian function defined on a symplectic phase space (rather than a space of motions). In particular time and space coordinates are no longer distinguished and non-relativistic and relativistic theories can be developed on an equal footing.
The other main ingredients of the theory are symmetry principles, and in particular that the Galilei group (in the non-relativistic case) or Poincaré group (in the relativistic case) should act on the space of motions U by diffeomorphisms which preserve the symplectic form. This corresponds to invariance under change of reference frame. A moment mapping from U to the dual of the Lie algebra of the symmetry group then associates a number of conserved quantities to each motion. For the 10-dimensional Galilei group these are energy, linear momentum, angular momentum and centre of mass.
In this framework `elementary particles' are defined to be symplectic manifolds on which the appropriate symmetry group acts transitively, so that all motions can be transformed into each other by a change of reference frame. Souriau and other authors have shown that, under appropriate conditions, every such symplectic manifold is a covering space of an orbit of a natural action of the group on the dual of its Lie algebra. Thus the classification of elementary particles is reduced to an exercise in Lie group theory. In the book this is described in detail for both the Galilei group and the Poincaré group.
Souriau's approach to statistical mechanics is to define the statistical states of a system to be probability measures on their spaces of motions. In a non-relativistic context, the `equilibrium' states are those which are invariant under the time-translation subgroup of the Galilei group, and among these the `natural' equilibrium states are those that have the largest entropy. These form the `Gibbs canonical ensemble' of the time-translation symmetry group. These definitions can be generalised to arbitrary subgroups of the Galilei group, and also to subgroups of the Poincaré group. Mathematically the resulting Gibbs canonical ensembles are families of measures indexed by open subsets of the Lie algebras of the symmetry groups. This index determines the temperature in the time-translation case and more generally defines a reference frame. Many of the concepts and principles of classical thermodynamics can be extended to the more general framework where they yield new results on, for example, relativistic ideal gases and the equilibrium states of rotating systems.
The final section of the book is devoted to an account of Souriau's theory of quantization. Along with his work on co-adjoint orbits this is his best known contribution to mechanics. The aim is to produce a theory which assigns to any symplectic space of motions U a Hilbert space H of quantum states, and a mapping from an appropriate space of smooth function on U (the classical observables) to a corresponding space of operators on H (the quantum observables). This correspondence should satisfy a number of compatibility conditions, and of course the resulting theory should agree with the available experimental observations. This set of criteria is notoriously difficult to satisfy!
Although he does not use bundle language explicitly, Souriau's recipe for quantization associates to U a Hermitian line bundle with a connection with curvature given by the symplectic form, and takes H to be a space of sections of this bundle. The connection enables classical observables on U to be lifted to operators on H. This is now known as prequantization, a term that is used in the English translation. If an appropriate line bundle and connection exist then U is said to be prequantizable. Sometimes prequantization can be carried out in more than one way because there is more than one isomorphism class of line bundle on U. A nice example is provided by systems of identical particles for which the appropriate spaces of motions have two inequivalent line bundles. Both of these turn out to be physically relevant, giving `Bose-Einstein' and `Fermi-Dirac' statistics, respectively.
Souriau discusses the problem of lifting the action of a symmetry group on U to an action on H and describes in some detail the prequantization of elementary particles. He also shows how various wave equations, including the Schröodinger equation, can be derived within the prequantum framework, and how it also leads to the creation and annihilation operator formalism of `second quantization'.
It will be apparent that the theories developed in this book use a formidable array of mathematical ideas and techniques. The first 100 pages is devoted to a rapid review of background material on manifolds, vector fields, differential forms, foliations, Lie groups and the calculus of variations. Later in the book are another 40 pages on measure theory. For the most part proof are omitted. I suspect that to make much headway readers will already need to be reasonably familiar with at least the basic ideas from these areas and most will probably skim this material and then refer back to it as necessary.
The third surprise awaiting the bookshop browser is that the original French edition of Souriau's book was published in 1970. The field of `symplectic mechanics', and in particular geometric quantization theory, has seen considerable activity since then, and yet the book still remains highly relevant. I particularly like the way it develops theoretical ideas using sophisticated mathematical machinery without losing sight of the physical concepts and experiments that the theory is supposed to explain. The approach used and ideas developed in the section on classical mechanics is one I would like to follow next time I give an MSc course on Hamiltonian mechanics. I don't think I would use this book as the main course text book, but it would certainly serve as my main inspiration.
The translator and editors have made an excellent job of this edition, my only gripe being that I would have liked a rather more extensive discussion of how the ideas described in the book have fared over the intervening quarter of a century. That apart I think this is an exciting and stimulating book that will sit happily on the shelves of anybody who is interested in the geometric foundations of physics, or at least on the shelves of their University library.
UK Nonlinear News thanks Birkhäuser for providing a copy of this book for review.