Review on our book "Advanced Classical Field Theory" in Mathematical Reviews

MR2527556 (2010h:70028)
Giachetta, Giovanni; Mangiarotti, Luigi; Sardanashvily, Gennadi
Advanced classical field theory. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. x+382 pp. ISBN: 978-981-283-895-7; 981-283-895-3

Unlike quantum field theory, classical field theory is a theory that can be dealt with in a purely mathematical way. This book aims at providing a complete mathematical basis of classical Lagrangian field theory and its BRST extension as a preliminary step towards quantum field theory. Lagrangian field theory is thereby treated in a very general framework relying, among other things, on a geometric approach to the theory of nonlinear differential operators.
  

The first chapter is devoted to an overview of the basic facts about the geometry of fibre bundles, the Frölicher-Nijenhuis calculus of vector-valued forms, jet manifolds (of finite and infinite order), connections on fibre bundles and differential operators. Chapter 2 deals with Lagrangian field theory on fibre bundles, with a discussion of the variational bicomplex and of Lagrangian symmetries and gauge symmetries. Special attention is thereby paid to the case of first-order Lagrangian field theories. Chapter 3 passes to the Lagrangian theory of even and odd fields by means of the Grassmann-graded variational bicomplex. First, an introduction to Grassmann-graded algebraic and differential calculus and to the geometry of graded manifolds is given.
  

Chapter 4 deals with Lagrangian BRST theory. Degenerate Lagrangians are characterised by a family of nontrivial Noether identities. These form a hierarchy which, under certain conditions, can be described by the exact Koszul-Tate complex. By means of a formulation of the inverse second Noether theorem in homology terms, this complex is associated to a cochain sequence of ghosts with an ascent operator, called a gauge operator. The components of this operator represent nontrivial gauge and higher-stage gauge symmetries. Whereas the gauge operator itself in general is not nilpotent, in some cases it may admit a nilpotent extension, which is called the BRST operator and which turns the cochain sequence of ghosts into the BRST complex.
  

Gauge theory on principal bundles is the topic of Chapter 5 with, among others, a study of Yang-Mills gauge theory and supergauge theory, and a discussion of matter fields and Higgs fields. Chapters 6, 7 and 8 are devoted to a complete treatment of gravitation theory on natural bundles, the theory of spinor fields (Dirac spinor and universal spinor structure) and topological field theories (Chern-Simons topological field theory), respectively. Finally, in Chapter 9, some aspects of covariant Hamiltonian field theory are described.
  

To make the exposition as self-contained as possible, the book ends with ten appendices devoted to several mathematical topics, such as differential operators on modules, homology and cohomology theory, sheaf cohomology, local-ringed spaces, leafwise and fibrewise cohomology. In addition, almost every chapter ends with an appendix in which some specific mathematical concept, relevant to the chapter under consideration, is further elucidated.
  

In conclusion, this is a very interesting book which contains a wealth of information regarding the mathematics underlying classical field theories. It is primarily oriented towards a mathematical audience: although the treatment is fairly self-contained, the reader is nevertheless supposed to have a solid background in differential geometry. In the beginning one gets a bit overwhelmed by the rapid succession of definitions, properties and notational conventions, but the effort of struggling through it is definitely rewarding.

Reference:

G.Giachetta, L.Mangiarotti, G.Sardanashvily Advanced Classical Field Theory (2009,WS)