2T Physics, Scale Invariance and Topological Vector Fields

TL;DR

This paper develops a classical two-time physics framework incorporating a metric tensor and a vector field, enabling a configuration space formulation of quantum mechanics with gravity in higher dimensions, highlighting scale invariance and duality symmetries.

Contribution

It introduces a novel classical structure with a metric and vector field for quantum mechanics in d+2 dimensions, emphasizing scale invariance and position-momentum duality.

Findings

Constructed a configuration space with metric and vector field

Demonstrated local scale invariance of the Hamiltonian

Connected two-time physics to quantum mechanics with gravity

Abstract

We construct, in classical two-time physics, the necessary structure for the most general configuration space formulation of quantum mechanics containing gravity in d+2 dimensions. This structure is composed of a symmetric Riemannian metric tensor and of a vector field that defines a section of a flat U(1) bundle over space-time. This construction is possible because of the existence of a finite local scale invariance of the Hamiltonian and because two-time physics contains, at the classical level, a local generalization of the discrete duality symmetry between position and momentum that underlies the structure of quantum mechanics.