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We consider the world-line quantisation of a system invariant under thesymmetries of reciprocal relativity. Imposition of the first class constraint,the generator of local time reparametrisations, on physical states enforcesidentification of the world-line cosmological constant with a fixed value ofthe quadratic Casimir of the quaplectic symmetry group Q(3,1) ~ U(3,1) x H(4),the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberggroup. In our previous paper, J Phys A 40 (2007) 12095--12111, the `spin'degrees of freedom were handled as covariant oscillators, leading to a uniquechoice of cosmological constant, required for projecting out negative-normstates from the physical gauge-invariant states. In the present paper the spindegrees of freedom are treated as standard oscillators with positive normstates (wherein Lorentz boosts are not number-conserving in the auxiliaryspace; reciprocal transformations are of course not spin-conserving ingeneral). As in the covariant approach, the spectrum of the square of theenergy-momentum vector is continuous over the entire real line, and thusincludes tachyonic (spacelike) and null branches. Adopting standard frames, theWigner method on each branch is implemented, to decompose the auxiliary spaceinto unitary irreducible representations of the respective little algebras andadditional degeneracy algebras. The physical state space is vastly enriched ascompared with the covariant approach, and contains towers of integer spinmassive states, as well as unconventional massless representations, withcontinuous Euclidean momentum and arbitrary integer helicity.

Constraint quantisation of a worldline system invariant under reciprocal relativity. II