Canonical Formulation of A Bosonic Matter Field in 1+1 Dimensional Curved Space

We study a Bosonic scalar in 1+1 dimensional curved space that is coupled to a dynamical metric field. This metric, along with the affine connection, also appears in the Einstein-Hilbert action when written in first order form. After illustrating the Dirac constraint analysis in Yang-Mills theory, we apply this formulation to the Einstein-Hilbert action and the action of the Bosonic scalar field, first separately and then together. Only in the latter case does a dynamical degree of freedom emerge.Comment: 21 page

Two-dimensional metric and tetrad gravities as constrained second order systems

Using the Gitman-Lyakhovich-Tyutin generalization of the Ostrogradsky method for analyzing singular systems, we consider the Hamiltonian formulation of metric and tetrad gravities in two-dimensional Riemannian spacetime treating them as constrained higher-derivative theories. The algebraic structure of the Poisson brackets of the constraints and the corresponding gauge transformations are investigated in both cases.Comment: replaced with revised version published in Mod.Phys.Lett.A22:17-28,200

Comments on ``A note on first-order formalism and odd-derivative actions'' by S. Deser

We argue that the obstacles to having a first-order formalism for odd-derivative actions presented in a pedagogical note by Deser are based on examples which are not first-order forms of the original actions. The general derivation of an equivalent first-order form of the original second-order action is illustrated using the example of topologically massive electrodynamics (TME). The correct first-order formulations of the TME model keep intact the gauge invariance presented in its second-order form demonstrating that the gauge invariance is not lost in the Ostrogradsky process.Comment: 6 pages, references are adde

Analysis of Hamiltonian formulations of linearized General Relativity

The different forms of the Hamiltonian formulations of linearized General Relativity/spin-two theories are discussed in order to show their similarities and differences. It is demonstrated that in the linear model, non-covariant modifications to the initial covariant Lagrangian (similar to those modifications used in full gravity) are in fact unnecessary. The Hamiltonians and the constraints are different in these two formulations but the structure of the constraint algebra and the gauge invariance derived from it are the same. It is shown that these equivalent Hamiltonian formulations are related to each other by a canonical transformation which is explicitly given. The relevance of these results to the full theory of General Relativity is briefly discussed.Comment: Section Discussion is modified and references are added; 19 page

Hamiltonian formulation of tetrad gravity: three dimensional case

The Hamiltonian formulation of the tetrad gravity in any dimension higher than two, using its first order form when tetrads and spin connections are treated as independent variables, is discussed and the complete solution of the three dimensional case is given. For the first time, applying the methods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra of the Poisson brackets among all constraints is calculated. The algebra of the Poisson brackets among first class secondary constraints locally coincides with Lie algebra of the ISO(2,1) Poincare group. All the first class constraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allow us to unambiguously derive the generator of gauge transformations and find the gauge transformations of the tetrads and spin connections which turn out to be the same found by Witten without recourse to the Hamiltonian methods [\textit{Nucl. Phys. B 311 (1988) 46}]. The gauge symmetry of the tetrad gravity generated by Lie algebra of constraints is compared with another invariance, diffeomorphism. Some conclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular, that diffeomorphism invariance is \textit{not derivable} as a \textit{gauge symmetry} from the Hamiltonian formulation of tetrad gravity in any dimension when tetrads and spin connections are used as independent variables.Comment: 31 pages, minor corrections, references are added, to appear in Gravitation & Cosmolog