Gauge theory in jet manifolds. (English) Zbl 0811.58004

Hadronic Press Monographs in Mathematics. Palm Harbor, FL: Hadronic Press, Inc. vii, 159 p. (1993).
This book is devoted to the application of the jet manifold formalism to the following aspects of modern field theory: (i) the fiber bundle formulation of gauge theory, (ii) the Lagrangian formalism in jet manifolds, and (iii) the multimomentum generalization of the conventional Hamiltonian formalism when momenta correspond to derivations of fields with respect to all world coordinates, not only time.
The goal is the covariant multimomentum Hamiltonian formulation of constraint field theories. A large family of multimomentum Hamiltonian equations, constraints and gauge conditions for major contemporary field models, including gauge theory and gravitation, are presented.
Contents: Chapters: 1. Fibered manifolds (manifold mappings, fibered manifolds, tangent bundles of fibered manifolds, bundles with an algebraic structure, vertical splitting, tangent-valued forms, differential forms on fibered manifolds),
2. Jet manifolds and connections (jet manifolds, canonical horizontal splitting, first order connections, connections on composite manifolds, geometry of the Legendre manifolds, second order jet manifolds, higher order jet manifolds, differential operators),
3. Lagrangian formalism in jet manifolds,
4. Multimomentum Hamiltonian formalism (multisymplectic structure, multimomentum Hamiltonian forms, Hamiltonian equations, the case of mechanics),
5. Constraint systems (Hamiltonian and Lagrangian systems, hyperregular Lagrangian densities, semiregular Lagrangian densities, almost regular Lagrangian densities, quadratic Lagrangian densities),
6. Field theory (gauge theory of principal connections, scalar matter fields, gravity in multimomentum variables, energy-momentum conservation law, systems on composite manifolds, spontaneous symmetry breaking, gauge gravitation theory, Dirac fermion fields).
Reviewer: R.Schmid (Atlanta)


58-02 Research exposition (monographs, survey articles) pertaining to global analysis
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81T13 Yang-Mills and other gauge theories in quantum field theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H05 Hamilton’s equations
58A20 Jets in global analysis