Luigi Mangiarotti

New Lagrangian and Hamiltonian methods in field theory

This work incorporates three modern aspects of mathematical physics: the jet methods in differntial geometry, the Lagrangian formalism on jet manifolds and the multimomentum approach to the Hamiltonian formalism. Several contemporary field models are investigated in detail. This is not a book on differential geometry, although modern concepts of differential geometry are used throughout the book. Quadratic Lagrangians and Hamiltonians are studied on the general level including the treatment of Hamiltonian formalism on composite fibre manifolds. The book presents new geometric methods and results in field theory.

Connections in Classical and Quantum Field Theory

Elementary gauge theory geometry of fiber bundles geometric gauge theory gravitation topological invariants in field theory jet bundle formalism Hamiltonian formalism in field theory infinite-dimensional bundles.

Advanced Classical Field Theory

Differential Calculus on Fiber Bundles Lagrangian Theory on Fiber Bundles Covariant Hamiltonian Field Theory Grassmann-Graded Lagrangian Theory Lagrangian BRST Theory Gauge Theory on Principal Fiber Bundles Gravitation Theory on Natural Bundles Spinor Fields, Topological Field Theories.

COVARIANT HAMILTON EQUATIONS FOR FIELD THEORY

We study the relationship between the equations of first order Lagrangian field theory on fiber bundles and the covariant Hamilton equations on the finite-dimensional polysymplectic phase space of covariant Hamiltonian field theory. The main peculiarity of these Hamilton equations lies in the fact that, for degenerate systems, they contain additional gauge fixing conditions. We develop the BRST extension of the covariant Hamiltonian formalism, characterized by a Lie superalgebra of BRST and anti-BRST symmetries.We study the relations between the equations of first-order Lagrangian field theory on fibre bundles and the covariant Hamilton equations on the finite-dimensional polysymplectic phase space of covariant Hamiltonian field theory. If a Lagrangian is hyperregular, these equations are equivalent. A degenerate Lagrangian requires a set of associated Hamiltonian forms in order to exhaust all solutions of the Euler-Lagrange equations. The case of quadratic degenerate Lagrangians is studied in detail.

Lagrangian Supersymmetries Depending on Derivatives. Global Analysis and Cohomology

Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational formula and conservation laws for Lagrangian systems on graded manifolds using contact supersymmetries are obtained.

Cohomology of the infinite-order jet space and the inverse problem

Cohomology of the bicomplex of exterior forms on the infinite-order jet space of a smooth fiber bundle of a field model is computed. This provides a solution of the global inverse problem of the calculus of variations in Lagrangian field theory and time-dependent mechanics. In the case of an affine fiber bundle, the outcomes to BRST theory are discussed. We show that there is no topological obstruction to constructing global descent equations in the even field sector of BRST theory.

Noether’s second theorem for BRST symmetries

We present Noethers second theorem for graded Lagrangian systems of even and odd variables on an arbitrary body manifold X in a general case of BRST symmetries depending on derivatives of dynamic variables and ghosts of any finite order. As a preliminary step, Noethers second theorem for Lagrangian systems on fiber bundles over X possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved.We present Noether’s second theorem for graded Lagrangian systems of even and odd variables on an arbitrary body manifold X in a general case of BRST symmetries depending on derivatives of dynamic variables and ghosts of any finite order. As a preliminary step, Noether’s second theorem for Lagrangian systems on fiber bundles Y→X possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved.

Noether's second theorem in a general setting: reducible gauge theories

We prove Noethers direct and inverse second theorems for Lagrangian systems on fiber bundles in the case of gauge symmetries depending on derivatives of dynamic variables of an arbitrary order. The appropriate notions of reducible gauge symmetries and Noethers identities are formulated, and their equivalence by means of certain intertwining operator is proved.We prove Noethers direct and inverse second theorems for Lagrangian systems on fibre bundles in the case of gauge symmetries depending on derivatives of dynamic variables and parameters of an arbitrary order. The appropriate notions of a reducible gauge symmetry and Noether identity are formulated, and their equivalence by means of a certain intertwining operator is proved.

Nonholonomic constraints in time-dependent mechanics

The constraint reaction force of ideal nonholonomic constraints in time-dependent mechanics on a configuration bundle

Bi-Hamiltonian partially integrable systems

Q\to R