Quadratic curvature theories formulated as Covariant Canonical Gauge theories of Gravity

TL;DR

This paper develops a covariant canonical gauge theory of gravity incorporating all quadratic curvature invariants, extending previous models and enabling potential quantization, especially in conformal gravity contexts.

Contribution

It introduces a Hamiltonian formulation of quadratic curvature gravity theories with independent metric and connection fields, including non-metric and conformal extensions.

Findings

Quadratic tensor invariants are essential for Hamiltonian-Lagrangian transformations.

The theory includes non-metric and conformally invariant extensions.

Framework facilitates potential quantization of quadratic curvature theories.

Abstract

The Covariant Canonical Gauge theory of Gravity is generalized by including at the Lagrangian level all possible quadratic curvature invariants. In this approach, the covariant Hamiltonian principle and the canonical transformation framework are applied to derive a Palatini type gauge theory of gravity. The metric $g_{\mu\nu}$, the affine connection $\gamma\indices{^{\lambda}_{\mu\nu}}$ and their respective conjugate momenta, $k^{\mu\nu\sigma}$ and $q\indices{_{\eta}^{\alpha\xi\beta}}$ tensors, are the independent field components describing the gravity. The metric is the basic dynamical field, and the connection is the gauge field. The torsion-free and metricity-compatible version of the space-time Hamiltonian is built from all possible invariants of the $q\indices{_{\eta}^{\alpha\xi\beta}}$ tensor components up to second order. These correspond in the Lagrangian picture to Riemann tensor invariants of the same order. We show that the quadratic tensor invariant is necessary for constructing the canonical momentum field from the gauge field derivatives, and hence for transforming between Hamiltonian and Lagrangian pictures. Moreover, the theory is extended by dropping metric compatibility and enforcing conformal invariance. This approach could be used for the quantization of the quadratic curvature theories, as for example in the case of conformal gravity.