  We present the world-line quantisation of a system invariant under the
symmetries of reciprocal relativity (pseudo-unitary transformations on ``phase
space coordinates" $(x^\mu(\tau),p^\mu(\tau))$ which preserve the Minkowski
metric and the symplectic form, and global shifts in these coordinates,
together with coordinate dependent transformations of an additional compact
phase coordinate, $\theta(\tau)$). The action is that of free motion over the
corresponding Weyl-Heisenberg group. Imposition of the first class constraint,
the generator of local time reparametrisations, on physical states enforces
identification of the world-line cosmological constant with a fixed value of
the quadratic Casimir of the quaplectic symmetry group $Q(D-1,1)\cong
U(D-1,1)\ltimes H(D)$, the semi-direct product of the pseudo-unitary group with
the Weyl-Heisenberg group (the central extension of the global translation
group, with central extension associated to the phase variable $\theta(\tau)$).
The spacetime spectrum of physical states is identified. Even though for an
appropriate range of values the restriction enforced by the cosmological
constant projects out negative norm states from the physical spectrum, leaving
over spin zero states only, the mass-squared spectrum is continuous over the
entire real line and thus includes a tachyonic branch as well.
