Gauged Duality, Conformal Symmetry, and Spacetime with Two Times

TL;DR

The paper develops a gauge-theoretic framework in which Sp\left(2\right) duality on a phase-space doublet \((X^M,P^M)\) is gauged, enlarging spacetime to \(d+2\) dimensions with two timelike directions and a manifest SO\(d,2\) symmetry. By enforcing Sp\left(2\right) constraints, it demonstrates that multiple familiar systems (e.g., a free massless particle, hydrogen atom, harmonic oscillator) are dual descriptions of a single quantum theory, all corresponding to a unique unitary representation of SO\(d,2\). The quantum analysis establishes that the physical spectrum satisfies \(C_2(\mathrm{SO}(d,2))=1-\frac{d^2}{4}\) with \(C_2(\mathrm{Sp}(2))=0\), and shows how different gauge choices reproduce the standard massless particle in Minkowski space while preserving the duality structure. The work suggests a physical role for two times, provides a path toward interacting theories with background fields, and links the construction to bi-local field formulations and broader duality-inspired frameworks (S-/M-theory, F-theory) with potential implications for higher-dimensional conformal and gauge symmetries.

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^μ, p^μ) 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.