Nonrelativistic Geodesic Motion

We show that any second-order dynamic equationon a configuration space X R ofnonrelativistic time-dependent mechanics can be seen asa geodesic equation with respect to some (nonlinear)connection on the tangent bundle TX X of relativisticvelocities. We compare relativistic and nonrelativisticgeodesic equations, and study the Jacobi vector fieldsalong nonrelativistic geodesics.     

Nonequivalent Representations of Nuclear Algebras of Canonical Commutation Relations: Quantum Fields

Non-Fock representations of the canonical commutation relations modeled over an infinite-dimensional nuclear space are constructed in an explicit form. The example of the nuclear space of smooth real functions of rapid decrease results in nonequivalent quantizations of scalar fields.     

Translation Gauge Fields and Space-Time Dislocations

Gauge fields of Poincaré translations define particular nongravitation structure on a space-time, which can be treated as sui generis space-time dislocations. Their source is the canonical energy-momentum tensor of matter, and these dislocations can introduce some corrections to standard values of gravitation effects, e. g. the Yukawa type corrections to the Newtonian gravitation potential.     

Gravity as a goldstone field in the Lorentz gauge theory

A bundle formalism is applied to interpret the Einstein gravitational field in gauge theory; its topological invariants are discussed.     

On the Higgs feature of gravity

The gauge gravitation theory, based on the equivalence principle besides the gauge principle, is formulated in the fibre bundle terms. The correlation between gauge geometry on spinor bundles describing Dirac fermion fields and space-time geometry on a tangent bundle is investigated. We show that field functions of fermion fields in presence of different gravitational fields are always written with respect to different reference frames. Therefore, the conventional quantization procedure is applicable to fermion fields only if gravitational field is fixed. Quantum gravitational fields violate the above mentioned correlation between two geometries.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

Iterated BRST Cohomology

The iterated BRST cohomology is studied by computing cohomology of the variational complex on the infinite order jet space of a smooth fibre bundle. This computation also provides a solution of the global inverse problem of the calculus of variations in Lagrangian field theory.     

Lagrangian Supersymmetries Depending on Derivatives. Global Analysis and Cohomology

Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in a 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.