G. Sardanashvily

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.

The gauge treatment of gravity

Abstract The gauge gravitation theory, in spite of twenty five years of its history, still remains the single gap in the excellent gauge picture of fundamental interactions. The main disputable point is the gauge status of Einsteins gravitational field, which is a metric or tetrad field, while gauge fields represent connections on fiber bundles. The corner-stones of Einsteins gravitation theory are the Relativity and Equivalence Principles. Having been reformulated in the fiber bundle terms, the gravitation theory turns out to be built from these principles directly as a gauge theory of space-time symmetries, which, however, are spontaneously broken down to the Lorentz symmetries. Metric gravitational fields appear in such a theory as the consequence of this spontaneous symmetry breaking and have the nature of Goldstone type fields. The Lorentz, GL(4, R) and Poincare gauge theories of gravitation are analyzed from these points of view, and some outlooks of the gauge treatment of gravitation, e.g., as the affine-metric theory, are discussed.

The Liouville-Arnold-Nekhoroshev theorem for non-compact invariant manifolds

A method and apparatus for enhancing the distribution of water from an irrigation sprinkler is provided and includes a nozzle disposed at the end of a discharge tube on the sprinkler, which nozzle includes a frusto-pyramidal passage leading to a polygonal nozzle outlet, thereby generating a substantial secondary flow within the nozzle which provides the desired controlled fall-out of water over the entire range of the stream ejected by the sprinkler.Under ceratin conditions, generalized action-angle coordinates can be introduced near non-compact invariant manifolds of completely and partially integrable Hamiltonian systems.

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.

Hamiltonian time-dependent mechanics

The usual formulation of time-dependent mechanics implies a given splitting Y=R×M of an event space Y. This splitting, however, is broken by any time-dependent transformation, including transformations between inertial frames. The goal is the frame-covariant formulation of time-dependent mechanics on a bundle Y→R, whose fibration Y→M is not fixed. Its phase space is the vertical cotangent bundle V*Y, provided with the canonical 3-form and the corresponding canonical Poisson structure. An event space of relativistic mechanics is a manifold Z whose fibration Z→R is not fixed.

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.

On the geometry of spontaneous symmetry breaking

Given a principal bundle P→X with a structure group G and an associated Higgs bundle Σ with a standard fiber G/H, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined. In the presence of a fixed Higgs field σ: X→Σ, matter fields are represented by sections of a matter bundle Eh associated with the corresponding reduced subbundle Ph of P. The totality of matter fields and Higgs fields is described by sections of the bundle E which is the composite bundle EH→Σ→X where EH→Σ is the bundle associated with the principal H‐bundle P→Σ. The bundle E fails to be associated with a principal bundle. To construct a connection Γ: E→J1E on E, the canonical jet bundle morphism J1EH×J1Σ→J1E is used.

Classical Gauge Theory of Gravity

The classical theory of gravity is formulated as a gauge theory on a frame bundle with spontaneous symmetry breaking caused by the existence of Dirac fermionic fields. The pseudo-Reimannian metric (tetrad field) is the corresponding Higgs field. We consider two variants of this theory. In the first variant, gravity is represented by the pseudo-Reimannian metric as in general relativity theory; in the second variant, it is represented by the effective metric as in Logunovs relativistic theory of gravity. The configuration space, Dirac operator, and Lagrangians are constructed for both variants.