Separation of time scales and reparametrization invariance for aging systems

We show that the generating functional describing the slow dynamics of spin-glass systems is invariant under reparametrizations of the time. This result is general and applies for both infinite and short-range models. It follows simply from the assumption that a separation between short time scales and long time scales exists in the system, and from the constraints of causality and unitarity. Global-time reparametrization invariance suggests that the low action excitations in a spin-glass may be smoothly spatially varying time reparametrizations. These Goldstone modes may provide the basis for an analytic dynamical theory of short-range spin glasses.