  Noninertial transformations on time-position-momentum-energy space {t,q,p,e}
with invariant Born-Green metric ds^2=-dt^2+dq^2/c^2+(1/b^2)(dp^2-de^2/c^2) and
the symplectic metric -de/\dt+dp/\dq are studied. This U(1,3) group of
transformations contains the Lorentz group as the inertial special case. In the
limit of small forces and velocities, it reduces to the expected Hamilton
transformations leaving invariant the symplectic metric and the nonrelativistic
line element ds^2=dt^2. The U(1,3) transformations bound relative velocities by
c and relative forces by b. Spacetime is no longer an invariant subspace but is
relative to noninertial observer frames. Born was lead to the metric by a
concept of reciprocity between position and momentum degrees of freedom and for
this reason we call this reciprocal relativity.
  For large b, such effects will almost certainly only manifest in a quantum
regime. Wigner showed that special relativistic quantum mechanics follows from
the projective representations of the inhomogeneous Lorentz group. Projective
representations of a Lie group are equivalent to the unitary reprentations of
its central extension. The same method of projective representations of the
inhomogeneous U(1,3) group is used to define the quantum theory in the
noninertial case. The central extension of the inhomogeneous U(1,3) group is
the cover of the quaplectic group Q(1,3)=U(1,3)*s H(4). H(4) is the
Weyl-Heisenberg group. A set of second order wave equations results from the
representations of the Casimir operators.
