Hidden Symmetries, AdS_D x S^n, and the lifting of one-time-physics to   two-time-physics
  The massive non-relativistic free particle in d-1 space dimensions has an action with a surprizing non-linearly realized SO(d,2) symmetry. This is the simplest example of a host of diverse one-time-physics systems with hidden SO(d,2) symmetric actions. By the addition of gauge degrees of freedom, they can all be lifted to the same SO(d,2) covariant unified theory that includes an extra spacelike and an extra timelike dimension. The resulting action in d+2 dimensions has manifest SO(d,2) Lorentz symmetry and a gauge symmetry Sp(2,R) and it defines two-time-physics. Conversely, the two-time action can be gauge fixed to diverse one-time physical systems. In this paper three new gauge fixed forms that correspond to the non-relativistic particle, the massive relativistic particle, and the particle in AdS_(d-n) x S^n spacetime will be discussed. The last case is discussed at the first quantized and field theory levels as well. For the last case the popularly known symmetry is SO(d-n-1,2) x SO(n+1), but yet we show that it is symmetric under the larger SO(d,2). In the field theory version the action is symmetric under the full SO(d,2) provided it is improved with a quantized mass term that arises as an anomaly from operator ordering ambiguities. The anomalous cosmological term vanishes for AdS_2 x S^0 and AdS_n x S^n (i.e. d=2n). The strikingly larger symmetry could be significant in the context of the proposed AdS/CFT duality. 
Itzhak Bars