Gravity, Two Times, Tractors, Weyl Invariance and Six Dimensional Quantum Mechanics

TL;DR

This paper explores a six-dimensional framework for four-dimensional gravity, combining two times physics and tractor calculus to provide new formulations and insights into conformal and gravitational theories.

Contribution

It introduces a novel six-dimensional quantum mechanical approach to gravity by integrating two times physics with tractor calculus, leading to new descriptions of four-dimensional gravity.

Findings

Reformulation of 4D gravity in terms of 6D quantum mechanics.

Derivation of new gravity descriptions from a scalar doublet and tractor vector multiplet.

Establishment of a parent theory unifying various gravity formulations.

Abstract

Fefferman and Graham showed some time ago that four dimensional conformal geometries could be analyzed in terms of six dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently it was shown how conformal geometry provides a description of physics manifestly invariant under local choices of unit systems. Strikingly, Einstein's equations are then equivalent to the existence of a parallel scale tractor (a six component vector subject to a certain first order covariant constancy condition at every point in four dimensional spacetime). These results suggest a six dimensional description of four dimensional physics, a viewpoint promulgated by the two times physics program of Bars. The Fefferman--Graham construction relies on a triplet of operators corresponding, respectively to a curved six dimensional light cone, the dilation generator and the Laplacian. These form an sp(2) algebra which Bars employs as a first class algebra of constraints in a six-dimensional gauge theory. In this article four dimensional gravity is recast in terms of six dimensional quantum mechanics by melding the two times and tractor approaches. This "parent" formulation of gravity is built from an infinite set of six dimensional fields. Successively integrating out these fields yields various novel descriptions of gravity including a new four dimensional one built from a scalar doublet, a tractor vector multiplet and a conformal class of metrics.