On the obstruction to the exactness of the variational bicomplex
We study the obstruction to the exactness of the variational complex for a field theory on an affine bundle.
math-ph↗
arXiv subjects
G. Sardanashvily
Publications and source records attributed to G. Sardanashvily.
On quantum evolution as a parallel transport
Time-dependent quantum evolution is described by an algebraic connection on a $C^\infty(R)$-module of sections of a $C^*$-algebra (or Hilbert) fibre bundle.
quant-ph↗
Quantum mechanics with time-dependent parameters
Smooth composite bundles provide the adequate geometric description of classical mechanics with time-dependent parameters. We show that the Berry's phase phenomenon is described in terms of connections on composite Hilbert space bundles.
quant-ph↗
Jets and connections in commutative and noncommutative geometry
It is emphasized that equivalent definitions of connections on modules over commutative rings are not so in noncommutative geometry.
math-ph↗
SUSY-Extended Field Theory
A field model on fibre bundles can be extended in a standard way to the BRS-invariant SUSY field model which possesses the Lie supergroup ISp(2) of symmetries.
hep-th↗
The Koszul-Tate Cohomology in Covariant Hamiltonian Formalism
We show that, in the framework of covariant Hamiltonian field theory, a degenerate almost regular quadratic Lagrangian $L$ admits a complete set of non-degenerate Hamiltonian forms such that solutions of the corresponding Hamilton equations, which live in the Lagrangian constraint space, exhaust solutions of the Euler--Lagrange equations for $L$. We obtain the characteristic splittings of the configuration and momentum phase bundles. Due to the corresponding projection operators, the Koszul-Tate resolution of the Lagrangian constraints for a generic almost regular quadratic Lagrangian is constructed in an explicit form.
hep-th↗
On the Geodesic Form of Non-Relativistic Dynamic Equations
It is shown that any second order dynamic equation on a configuration bundle $Q\to R$ of non-relativistic mechanics is equivalent to a geodesic equation with respect to a (non-linear) connection on the tangent bundle $TQ\to Q$. The case of quadratic dynamic equations is analyzed in details. The equation for Jacobi vector fields is constructed and investigated by the geometric methods.
math-ph↗
Nonrelativistic Geodesic Motion
We show that any second order dynamic equation on a configuration space $X\to R$ of nonrelativistic mechanics can be seen as a geodesic equation with respect to some (nonlinear) connection on the tangent bundle $TX\to X$ of relativistic velocities. We compare relativistic and nonrelativistic geodesic equations, and study the Jacobi vector fields along nonrelativistic geodesics.
gr-qc↗
Constraints in Hamiltonian time-dependent mechanics
In Hamiltonian time-dependent mechanics, the Poisson bracket does not define dynamic equations, that implies the corresponding peculiarities of describing time-dependent holonomic constraints. As in conservative mechanics, one can consider the Poisson bracket of constraints, separate them in first and second class constraints, construct the Koszul-Tate resolution and a BRST complex. However, the Poisson bracket of constraints and a Hamiltonian makes no sense. Hamiltonian vector fields for first class constraints are not generators of gauge transformations. In the case of Lagrangian constraints, we state the comprehensive relations between solutions of the Lagrange equations for an almost regular Lagrangian and solutions of the Hamilton equations for associated Hamiltonian forms, which live in the Lagrangian constraint space. Degenerate quadratic Lagrangian systems are studied in details. We construct the Koszul-Tate resolution for Lagrangian constraints of these systems in an explicit form.
math-ph↗
Covariant Hamiltonian 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.
hep-th↗
On the bracket problem in covariant Hamiltonian field theory
The polysymplectic phase space of covariant Hamiltonian field theory can be provided with the current algebra bracket.
hep-th↗
On the geometric arena of supermechanics
In the case of simple graded manifolds utilized in supermechanics, supervector fields and exterior superforms are represented by global sections of smooth vector bundles.
math-ph↗
Energy-Momentum and Gauge Conservation Laws
We treat energy-momentum conservation laws as particular gauge conservation laws when generators of gauge transformations are horizontal vector fields on fibre bundles. In particular, the generators of general covariant transformations are the canonical horizontal prolongations of vector fields on a world manifold. This is the case of the energy-momentum conservation laws in gravitation theories. We find that, in main gravitational models, the corresponding energy-momentum flows reduce to the generalized Komar superpotential. We show that the superpotential form of a conserved flow is the common property of gauge conservation laws if generators of gauge transformations depend on derivatives of gauge parameters. At the same time, dependence of conserved flows on gauge parameters make gauge conservation laws form-invariant under gauge transformations.
gr-qc↗
Nonholonomic Constraints in Time-Dependent Mechanics
The constraint reaction force of ideal nonholonomic constraints in time-dependent mechanics on a configuration bundle $Q\to R$ is obtained. Using the vertical extension of Hamiltonian formalism to the vertical tangent bundle $VQ$ of $Q\to R$, the Hamiltonian of a nonholonomic constrained system is constructed.
math-ph↗
Dynamic Connections in Analytical Mechanics
It is shown that any dynamic equation on a configuration bundle $Q\to R$ of non-relativistic time-dependent mechanics is associated with connections on the affine jet bundle $J^1Q\to Q$ and on the tangent bundle $TQ\to Q$. As a consequence, any non-relativistic dynamic equation can be seen as a geodesic equation with respect to a (non-linear) connection on the tangent bundle $TQ\to Q$. Using this fact, the relationship between relativistic and non-relativistic equations of motion is studied. The geometric notions of reference frames and relative accelerations in non-relativistic mechanics are introduced in the terms of connections. The covariant form of non-relativistic dynamic equations is written.
math-ph↗
Relativistic and Non-relativistic Equations of Motion
It is shown that any second order dynamic equation on a configuration space $X$ of non-relativistic time-dependent mechanics can be seen as a geodesic equation with respect to some (non-linear) connection on the tangent bundle $TX\to X$ of relativistic velocities. Using this fact, the relationship between relativistic and non-relativistic equations of motion is studied.
gr-qc↗
Covariant Spin Structure
Every Dirac spin structure on a world manifold is associated with a certain gravitational field, and is not preserved under general covariant transformations. We construct a composite spinor bundle such that any Dirac spin structure is its subbundle, and this bundle admits general covariant transformations.
gr-qc↗
Non-Symplectic Geometry of First Order Time-Dependent Mechanics
The usual formulation of time-dependent mechanics implies a given splitting $Y=R\times M$ 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 $Y\to R$ whose fibration $Y\to M$ 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 $Y$ whose fibration $Y\to R$ is not fixed.
dg-ga↗