Hidden symmetries,AdSD×Sn,and the lifting of one-time physics to two-time physics

Abstract

The massive non-relativistic free particle in $d\ensuremath{-}1$ space dimensions, with a Lagrangian $L=(m/2){\mathbf{r\ifmmode \dot{}\else \.{}\fi{}}}^{2},$ has an action with a surprising non-linearly realized $\mathrm{SO}(d,2)$ symmetry. This is the simplest example of a host of diverse one-time-physics systems with hidden $\mathrm{SO}(d,2)$ symmetric actions. By the addition of gauge degrees of freedom, they can all be lifted to the same $\mathrm{SO}(d,2)$ covariant unified theory that includes an extra spacelike and an extra timelike dimension. The resulting action in $d+2$ dimensions has manifest $\mathrm{SO}(d,2)$ Lorentz symmetry and a gauge symmetry $\mathrm{Sp}(2,R).$ The symmetric action 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 ${\mathrm{AdS}}_{d\ensuremath{-}n}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{n}$ curved spacetime will be discussed at the classical level. The last case is discussed at the first quantized and field theory levels as well. For the last case the popularly known symmetry is $\mathrm{SO}(d\ensuremath{-}n\ensuremath{-}1,2)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(n+1),$ but yet we show that the classical or quantum versions are symmetric under the larger $\mathrm{SO}(d,2).$ In the field theory version the action is symmetric under the full $\mathrm{SO}(d,2)$ provided it is improved with a quantized mass term that arises as an anomaly from operator ordering ambiguities. The anomalous mass term vanishes for ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{0}$ and ${\mathrm{AdS}}_{n}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{n}$ (i.e., $d=2n).$ A quantum test for the presence of two-time-physics in a one-time physics system is that the $\mathrm{SO}(d,2)$ Casimir operators have fixed eigenvalues independent of the system. It is shown that this test is successful for the particle in ${\mathrm{AdS}}_{d\ensuremath{-}n}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{n}$ by computing the Casimir operators and showing explicitly that they are independent of n. The strikingly larger symmetry could be significant in the context of the proposed AdS/CFT duality.