Gauged Duality, Conformal Symmetry, and Spacetime with Two Times

TL;DR

This paper develops a duality framework linking different physical systems through gauged Sp(2) symmetry, revealing a spacetime with two times that unifies conformal symmetry and various physical models.

Contribution

It introduces a novel duality approach using gauged Sp(2) symmetry to connect diverse systems and interpret spacetime with two times within a unified quantum theory.

Findings

Enlarged spacetime with two times from gauging Sp(2) symmetry.

Duality between free particles and hydrogen atom models.

SO(d,2) conformal symmetry as a gauge symmetry of the theory.

Abstract

We construct a duality between several simple physical systems by showing that they are different aspects of the same quantum theory. Examples include the free relativistic massless particle and the hydrogen atom in any number of dimensions. The key is the gauging of the Sp(2) duality symmetry that treats position and momentum (x,p) as a doublet in phase space. As a consequence of the gauging, the Minkowski space-time vectors (x^\mu, p^\mu) get enlarged by one additional space-like and one additional time-like dimensions to (x^M,p^M). A manifest global symmetry SO(d,2) rotates (x^M,p^M) like d+2 dimensional vectors. The SO(d,2) symmetry of the parent theory may be interpreted as the familiar conformal symmetry of quantum field theory in Minkowski spacetime in one gauge, or as the dynamical symmetry of a totally different physical system in another gauge. Thanks to the gauge symmetry, the theory permits various choices of ``time'' which correspond to different looking Hamiltonians, while avoiding ghosts. Thus we demonstrate that there is a physical role for a spacetime with two times when taken together with a gauged duality symmetry that produces appropriate constraints.