Foliation, jet bundle and quantization of Einstein gravity

TL;DR

This paper advances the mathematical framework for quantizing Einstein gravity by integrating foliation theory and jet bundle concepts, clarifying the identification of physical degrees of freedom and supporting renormalization.

Contribution

It introduces jet bundle theory into the foliation-based reduction of Einstein gravity, providing a rigorous mathematical foundation for the physical degrees of freedom.

Findings

Clarified the role of foliation in gravity reduction

Integrated jet bundle theory into the quantization framework

Supported the possibility of gravity renormalization through metric redefinition

Abstract

In \cite{Park:2014tia} we proposed a way of quantizing gravity with the Hamiltonian and Lagrangian analyses in the ADM setup. One of the key observations was that the physical configuration space of the 4D Einstein-Hilbert action admits a three-dimensional description, thereby making gravity renormalization possible through a metric field redefinition. Subsequently, a more mathematical and complementary picture of the reduction based on foliation theory was presented in \cite{Park:2014qoa}. With the setup of foliation the physical degrees of freedom have been identified with a certain leaf. Here we expand the work of \cite{Park:2014qoa} by adding another mathematical ingredient - an element of jet bundle theory. With the introduction of the jet bundle, the procedure of identifying the true degrees of freedom outlined therein is made precise and the whole picture of the reduction is put on firm mathematical ground.