Quantization of the First-Order Two-Dimensional Einstein-Hilbert Action

A canonical analysis of the first-order two-dimensional Einstein-Hilbert action has shown it to have no physical degrees of freedom and to possess an unusual gauge symmetry with a symmetric field $\xi_{\mu\nu}$ acting as a gauge function. Some consequences of this symmetry are explored. The action is quantized and it is shown that all loop diagrams beyond one-loop order vanish. Furthermore, explicit calculation of the one-loop two-point function shows that it too vanishes, with the contribution of the ghost loop cancelling that of the ``graviton'' loop

Peculiarities of the Canonical Analysis of the First Order Form of the Einstein-Hilbert Action in Two Dimensions in Terms of the Metric Tensor or the Metric Density

The peculiarities of doing a canonical analysis of the first order formulation of the Einstein-Hilbert action in terms of either the metric tensor $g^{\alpha \beta}$ or the metric density $h^{\alpha \beta}= \sqrt{-g}g^{\alpha \beta}$ along with the affine connection are discussed. It is shown that the difference between using $g^{\alpha \beta}$ as opposed to $h^{\alpha \beta}$ appears only in two spacetime dimensions. Despite there being a different number of constraints in these two approaches, both formulations result in there being a local Poisson brackets algebra of constraints with field independent structure constants, closed off shell generators of gauge transformations and off shell invariance of the action. The formulation in terms of the metric tensor is analyzed in detail and compared with earlier results obtained using the metric density. The gauge transformations, obtained from the full set of first class constraints, are different from a diffeomorphism transformation in both cases.Comment: 13 page

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