S. A. Gogilidze

Hamiltonian reduction of SU(2) Dirac-Yang-Mills mechanics

The SU(2) gauge invariant Dirac-Yang-Mills mechanics of spatially homogeneous isospinor and gauge fields is considered in the framework of the generalized Hamiltonian approach. The unconstrained Hamiltonian system equivalent to the model is obtained using the gaugeless method of Hamiltonian reduction. The latter includes the Abelianization of the first class constraints, putting the second class constraints into the canonical form and performing a canonical transformation to a set of adapted coordinates such that a subset of the new canonical pairs coincides with the second class constraints and part of the new momenta is equal to the Abelian constraints. In the adapted basis the pure gauge degrees of freedom automatically drop out from the consideration after projection of the model onto the constraint shell. Apart from the elimination of these ignorable degrees of freedom a further Hamiltonian reduction is achieved due to the three dimensional group of rigid symmetry possessed by the system.

Local symmetries in systems with constraints

For arbitrary systems with first-class constraints the local gauge transformations are constructed in phase and configuration spaces, i.e. a method for obtaining symmetry transformations in the second Noethers theorem is given.

Gauge transformations in constrained theories

A method is proposed for constructing infinitesimal gauge transformations for an arbitrary degenerate Lagrangian without restrictions on the constraint algebra. It is thereby shown that in the general case degeneracy of a theory is due to its gauge invariance.

Higher derivatives in gauge transformations

A mechanism of occurrence of higher derivatives of the coordinates in the symmetry transformation law in Noethers second theorem is elucidated. It is shown that the corresponding transformations in the Hamiltonian formalism are canonical in an extended phase space.

The Ostrogradsky Method for Local Symmetries of Systems with First and Second Class Constraints

In the generalized Hamiltonian formalism by Dirac, the method of constructing the generator of local-symmetry transformations for systems with first- and second-class constraints (without restrictions on the algebra of constraints) is obtained from the requirement for them to map the solutions of the Hamiltonian equations of motion into the solutions of the same equations. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry entirely stipulated by all the first-class constraints (and only by them). A mechanism of occurrence of higher derivatives of coordinates and group parameters in the symmetry transformation law in the Noether second theorem is elucidated. It is shown that the obtained transformations of symmetry are canonical in the extended (by Ostrogradsky) phase space. An application of the method in theories with higher derivatives is demonstrated with an example of the spinor Christ -- Lee model.In the generalized Hamiltonian formalism by Dirac, the method of constructing the generator of local-symmetry transformations for systems with first- and second-class constraints (without restrictions on the algebra of constraints) is obtained from the requirement for them to map the solutions of the Hamiltonian equations of motion into the solutions of the same equations. It is proved that second-class constraints do not contribute to the transformation law of the local symmetry entirely stipulated by all the first-class constraints (and only by them). A mechanism of occurrence of higher derivatives of coordinates and group parameters in the symmetry transformation law in the Noether second theorem is elucidated. It is shown that the obtained transformations of symmetry are canonical in the extended (by Ostrogradsky) phase space. An application of the method in theories with higher derivatives is demonstrated with an example of the spinor Christ -- Lee model.

PHASE SPACE IN SINGULAR THEORIES

The method is formulated for constructing the generators of gauge transformations in the phase space for a given singular Lagrangian. These generators are used to impose the necessary and sufficient conditions on gauge functions. The results are analyzed for particular examples.