×

Geometry of classical fields. (English) Zbl 0675.53065

North-Holland Mathematics Studies, 154; Notas de Matemática, 123. Amsterdam etc.: North-Holland. xviii, 450 p. $ 105.25; Dfl. 200.00 (1988).
This book is addressed to mathematicians who are interested in applications to theoretical physics and to physicists who use differential geometric methods in their modelling. The latter also include methods from Lie group theory, calculus of variations, algebraic topology and nonlinear functional analysis. The book splits into two parts. The first one is devoted to the introduction of the mathematical tools. Their application to the description of physical models is the aim of the second part.
The presentation of the mathematical background contains most of the proofs. The later chapters lack this completeness. There much of the recent research elaborated by the present authors and other scientists working in that area of mathematical physics is included. Hence in many cases only a survey on the main ideas and most important results can be given in the final chapters of the book. The details have to be looked up in the references presented at the end of each single chapter. Nevertheless the book represents an excellent combination of a useful introduction to important tools of modern mathematics and a survey on a comparatively wide area of modern physics to which these methods can be applied. It has to be considered as a very valuable contribution to the existing literature in that area.
Only a brief sketch of the contents can be given here: Cleary at the beginning an exhibition of standard manifold theory is given, including the fundamentals of the theory of Lie groups. Special attention is paid to a discussion of the structure of the Lie groups which are most important in physics. Then the theory of vector bundles is developed, independently from the theory of principal bundles. In this context k- jets and canonical one-forms are treated because of their importance for the calculus of variations. This is followed by the presentation of the basic notions from differential geometry: linear connections, geodesic sprays, exponential map, Riemannian metrics, structures on submanifolds and integrability conditions. The discussion also is extended to the case where the underlying manifold is a Lie group. Some of these notions reappear in the next chapter where principal bundles and their associated fibre bundles are studied. Special attention is given to bundles over homogeneous spaces.
Chapter 5 deals with function spaces and manifolds of maps. Its purpose is to collect the basic material to provide a framework in which fields depending on parameters varying in infinite-dimensional manifolds can be studied. In contrast to the preceding chapters many proofs are omitted. The spaces of maps are equipped with \(C^ k\)-topologies and Sobolev type topologies as well. The local constructions are extended to spaces of maps between manifolds. This is followed by the discussion of these constructions in special cases: embeddings of a compact manifold into some ambient manifold, diffeomorphism groups and automorphism groups of principal bundles.
The second part of the book presents a systematic development of a covariant Hamiltonian formulation of field theory starting from the principle of stationary action. This is exhibited in chapter 6 for systems with a finite number of degrees of freedom, at first in the case of a regular time independent Lagrangian. Techniques from symplectic geometry are introduced in this context. Then degenerate Lagrangians and the corresponding constraints are studied. Here for a covariant formulation the corresponding Cartan form has to be used. Then in the next chapter, to extend these considerations to a covariant Hamiltonian formulation of field theory, the a priori splitting into space and time has to be given up. In the action principle the Lagrangian has to be replaced by the deDonder form. From this an analogue of the Cartan form is derived leading to a Hamiltonian formalism like in the preceding finite-dimensional case. Some attention is paid to a discussion of appropriate boundary conditions. These constructions are illustrated by a discussion of the Klein-Gordon dynamics of a scalar field.
Chapters 8 and 9 are devoted to more examples. The formalism of the preceeding chapter is applied to the study of Yang-Mills fields and gauge transformations. A detailed discussion of electrodynamics prepares the general considerations. Then the extension of this material to non- abelian structure groups is investigated. For a corresponding description of general relativity Lorentzian metrics and torsion free connections are considered as independent variables. Then with the corresponding boundary conditions for Einstein’s equations general relativity admits a covariant Hamiltonian formulation of dynamics. Symmetries and constraints are discussed in this framework. Finally it is shown that asymptotically flat space-times can be described using the methods introduced above, after an embedding into suitable manifolds with boundary has been performed.
Reviewer: Bernd Wegner

MSC:

53C80 Applications of global differential geometry to the sciences
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
70Gxx General models, approaches, and methods in mechanics of particles and systems
22Exx Lie groups
58Exx Variational problems in infinite-dimensional spaces
58Dxx Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
PDFBibTeX XMLCite