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Register a new userGeometry and Symmetry Structures in 2T Gravity
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2T-gravity in d+2 dimensions predicts 1T General Relativity (GR) in d dimensions, augmented with a local scale symmetry known as the Weyl symmetry in 1T field theory. The emerging GR comes with a number of constraints, particularly on scalar fields and their interactions in 1T field theory. These constraints, detailed in this paper, are footprints of 2T-gravity and could be a basis for testing 2T-physics. Some of the conceptually interesting consequences of the accidental Weyl symmetry include that the gravitational constant emerges from vacuum values of the dilaton and other Higgs-type scalars and that it changes after every cosmic phase transition (inflation, grand unification, electroweak phase transition, etc.). We show that this consequential Weyl symmetry in d dimensions originates from coordinate reparametrization, not from scale transformations, in the d+2 spacetime of 2T-gravity. To recognize this structure we develop in detail the geometrical structures, curvatures, symmetries, etc. of the d+2 spacetime which is restricted by a homothety condition derived from the action of 2T-gravity. Observers that live in d dimensions perceive GR and all degrees of freedom as shadows of their counterparts in d+2 dimensions. Kaluza-Klein (KK) type modes are removed by gauge symmetries and constraints that follow from the 2T-gravity action. However some analogs to KK modes, which we call prolongations of the shadows into the higher dimensions, remain but they are completely determined, up to gauge freedom, by the shadows in d dimensions.
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The relation between motion in $-1/r$ and $r^{2}$ potentials, known since Newton, can be demonstrated by the substitution $rrightarrow r^{2}$ in the classical/quantum radial equations of the Kepler/Hydrogen problems versus the harmonic oscillator. This suggests a duality-type relationship between these systems. However, when both radial and angular components of these systems are included the possibility of a true duality seems to be remote. Indeed, investigations that explored and generalized Newtons radial relation, including algebraic approaches based on noncompact groups such as SO(4,2), have never exhibited a full duality consistent with Newtons. On the other hand, 2T-physics predicts a host of dualities between pairs of a huge set of systems that includes Newtons two systems. These dualities take the form of rather complicated canonical transformations that relate the full phase spaces of these respective systems in all directions. In this paper we focus on Newtons case by imposing his radial relation to find an appropriate basis for 2T-physics dualities, and then construct the full duality. Using the techniques of 2T-physics, we discuss the hidden symmetry of the actions (beyond the symmetry of Hamiltonians) for the Hydrogen atom in $D$-dimensions and the harmonic oscillator in $bar{D}$ dimensions. The symmetries lead us to find the one-to-one relation between the quantum states, including angular degrees of freedom, for specific values of $left( D,bar{D}right) $, and construct the explicit quantum canonical transformation in those special cases. We find that the canonical transformation has itself a hidden gauge symmetry that is crucial for the respective phase spaces to be dual even when $D eqbar{D}$. In this way we display the surprising beautiful symmetry of the full duality that generalizes Newtons radial duality.
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