Intrinsic time gravity, heat kernel regularization, and emergence of Einstein's theory

TL;DR

This paper develops a Hamiltonian formulation of Intrinsic Time Gravity, demonstrating how Einstein's Ricci scalar potential naturally emerges from a regularized, positive-definite Hamiltonian framework that incorporates spatial diffeomorphism invariance and heat kernel regularization.

Contribution

It introduces a novel Hamiltonian approach to Intrinsic Time Gravity with a natural emergence of Einstein's Ricci scalar potential using heat kernel regularization.

Findings

Einstein's Ricci scalar potential emerges from the Hamiltonian.

The theory maintains spatial diffeomorphism invariance.

Heat kernel regularization isolates divergences effectively.

Abstract

The Hamiltonian of Intrinsic Time Gravity is elucidated. The theory describes Schrodinger evolution of our universe with respect to the fractional change of the total spatial volume. Gravitational interactions are introduced by extending Klauder's momentric variable with similarity transformations, and explicit spatial diffeomorphism invariance is enforced via similarity transformation with exponentials of spatial integrals. In analogy with Yang-Mills theory, a Cotton-York term is obtained from the Chern-Simons functional of the affine connection. The essential difference is the fundamental variable for geometrodynamics is the metric rather than a gauge connection; in the case of Yang-Mills, there is also no analog of the integral of the spatial Ricci scalar curvature. Heat kernel regularization is employed to isolate the divergences of coincidence limits; apart from an additional Cotton-York term, a prescription in which Einstein's Ricci scalar potential emerges naturally from the positive-definite self-adjoint Hamiltonian of the theory is demonstrated.