Eur. Phys. J. C (2015) 75: 459
https://doi.org/10.1140/epjc/s10052-015-3660-x

Regular Article - Theoretical Physics

Hypersurface foliation approach to renormalization of ADM formulation of gravity

¹ Department of Physics, Hanyang University, Seoul, 133-791, Korea
² Department of Applied Mathematics, Philander Smith College, Little Rock, AR, 72223, USA

^* e-mail: inyongpark05@gmail.com

Received: 4 July 2015
Accepted: 3 September 2015
Published online: 28 September 2015

Abstract

We carry out ADM splitting in the Lagrangian formulation and establish a procedure in which (almost) all of the unphysical components of the metric are removed by using the 4D diffeomorphism and the measure-zero 3D symmetry. The procedure introduces a constraint that corresponds to the Hamiltonian constraint of the Hamiltonian formulation, and its solution implies that the 4D dynamics admits an effective description through 3D hypersurface physics. As far as we can see, our procedure implies potential renormalizability of the ADM formulation of 4D Einstein gravity for which a complete gauge-fixing in the ADM formulation and hypersurface foliation of geometry are the key elements. If true, this implies that the alleged unrenormalizability of 4D Einstein gravity may be due to the presence of the unphysical fields. The procedure can straightforwardly be applied to quantization around a flat background; the Schwarzschild case seems more subtle. We discuss a potential limitation of the procedure when applying it to explicit time-dependent backgrounds.