  We propose an alternative geometric mathematical structure for arbitrary
phase space. The main guide in our approach is the hidden SL(2,R)-symmetry
which acts on the phase space changing coordinates by momenta and vice versa.
We show that the SL(2,R)-symmetry is implicit in any symplectic structure. We
also prove that in any sensible physical theory based on the SL(2,R)-symmetry
the signature of the flat target "spacetime" must be associated with either
one-time and one-space or at least two-time and two-space coordinates. We
discuss the consequences as well as possible applications of our approach on
different physical scenarios.
