Scale invariance, and the consequent conformal symmetry of nature, are such natural extensions of the Poincaré symmetry of general relativity that dozens of attempts have been made over the last century to build a coherent conformal theory of gravity. These extensions have held out various hopes ranging from the unification of gravity with electromagnetism (Weyl) to finding deep connections with quantum theory (London), to tools for imposing cosmic censorship (Penrose), to introducing a time-dependent gravitational constant (Dirac). Still later, it was hoped that the extra fields introduced by enlarging the symmetry of general relativity would lead to unification with other fundamental interactions besides electromagnetism.
Unfortunately, these efforts have uniformly been plagued by difficulties. First, the standard gravity theory built on the conformal group does not reduce to general relativity. A theory equivalent to GR is achieved only by introducing extra, compensating fields. Second, it has been shown that the most of the new fields arising from the enlarged symmetry cannot lead to new physics. Third, the one new field that can lead to new physical predictions (the "Weyl vector") predicts expanding or contracting forms of matter, and as a result has defied any interpretation consistent with experience. These failures have even led some to argue that scale invariance cannot be a physical symmetry.
Why does such a clear symmetry of nature resist physical interpretation? We may be starting to find some answers. Our work investigates how a new theory of gravity based on the conformal group solves the three problems listed above, while leading to an interesting class of geometries. These new spaces have a direct interpretation as curved generalizations of phase space. Like quantum mechanics, this model suggests that the true arena for physics is more like phase space than spacetime. Moreover, in the phase space interpretation, the extra dimensions of the theory also require a constant with the units of action. If this constant is Planck's, then these spaces might even shed light on quantum gravity.