Transition amplitude for two-time physics

We present the transition amplitude for a particle moving in a space with two times and $D$ space dimensions having an $Sp(2,R)$ local symmetry and an $SO(D,2)$ rigid symmetry. It was obtained from the BRST-BFV quantization with a unique gauge choice. We show that by constraining the initial and final points of this amplitude to lie on some hypersurface of the $D+2$ space the resulting amplitude reproduces well-known systems in lower dimensions. This work provides an alternative way to derive the effects of two-time physics where all the results come from a single transition amplitude.