Treating the Einstein-Hilbert action as a higher derivative Lagrangian: Revealing the missing information about conformal non-invariance

Abstract

The Hamiltonian formulation of conformally invariant Weyl-squared higher derivative theory teaches us that conformal symmetry is expressed through particular first class constraints related to the absence of the three-metric determinant and the trace of the extrinsic curvature from the theory. Any term depending on them which is added to this theory breaks conformal invariance and turns these constraints into second class ones. Such second class constraints are missing in the standard canonical formulation of the conformally non-invariant Einstein-Hilbert theory. It is demonstrated that such constraints do appear if the theory is treated as a higher derivative one: if the extrinsic curvature is promoted to an independent variable, the apparently missing information about conformal behavior is revealed.