  The quantum phase leads to projective representations of symmetry groups in
quantum mechanics. The projective representations are equivalent to the unitary
representations of the central extension of the group. A celebrated example is
Wigner's formulation of special relativistic quantum mechanics as the
projective representations of the inhomogeneous Lorentz group. However,
Wigner's formulation makes no mention of the Weyl-Heisenberg group and the
hermitian representation of its algebra that are the Heisenberg commutation
relations fundamental to quantum physics. We put aside the relativistic
symmetry and show that the maximal quantum symmetry that leaves the Heisenberg
commutation relations invariant is the projective representations of the
conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group
and noncommutative structure arises directly because the quantum phase requires
projective representations. We then consider the relativistic implications of
the quantum phase that lead to the Born line element and the projective
representations of an inhomogeneous unitary group that defines a noninertial
quantum theory. (Understanding noninertial quantum mechanics is a prelude to
understanding quantum gravity.) The remarkable properties of this symmetry and
its limits are studied.
