A Canonical Approach to the Einstein–hilbert Action in Two Spacetime Dimensions

Abstract

The canonical structure of the Einstein-Hilbert Lagrange density L = √ −gR is examined in two spacetime dimensions, using the metric density h µν ≡ √ −gg µν and symmetric affine connection Γ λ σβ as dynamical variables. The Hamiltonian reduces to a linear combination of three first class constraints with a local SO(2, 1) algebra. The first class constraints are used to find a generator of gauge transformations that has a closed off-shell algebra and which leaves the Lagrangian and det(h µν ) invariant. These transformations are distinct from diffeomorphism invariance, and are gauge transformations characterized by a symmetric matrix ζ µν .

Radiative corrections in 2 + 1 dimensional supergravity

“…In this appendix we further illustrate this by deriving the effective action for the first order Einstein-Hilbert (Palatini) action in 1 + 1 dimensions. This theory is manifestly invariant under a diffeormorphism transformation, but it is not this gauge transformation that follows from the first class constraints in the theory [13]. If in the Einstein-Hilbert action…”

Algebraic analysis of a model of two-dimensional gravity

“…This coincidence of the PB between the secondary constraints in the 2DG model and the generators of the SO(2,1)-algebra means that there is a uniform relation between the corresponding representations of these two algebras. The Hamiltonian H c , equation (14), can now be written in the form…”

“…The coefficients in this linear form, (19), are some ξ−numbers which ensure the correct relation with General Relativity (GR) (see below). Note that equations (14) and (19) are the simplest linear forms which are acceptable for the H c Hamiltonian in General Relativity.…”

General relativity in two dimensions: A Hamilton–Jacobi analysis

“…This theory is well explored in the literature within the scope of the Hamiltonian formalism [15][16][17]. Besides, the analysis of the EHA in two dimensions is an interesting subject by itself: several models in lower dimensions are shown to be soluble quantum systems [18][19][20], providing new perspectives on the non-perturbative quantization in higher dimensions.…”