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Born proposed a unification of special relativity and quantum mechanics thatplaced position, time, energy and momentum on equal footing through areciprocity principle and extended the usual position-time and energy-momentumline elements to this space by combining them through a new fundamentalconstant. Requiring also invariance of the symplectic metric yields U(1,3) asthe invariance group, the inhomogeneous counterpart of which is the canonicallyrelativistic group CR(1,3) = U(1,3) *s H(1,3) where H(1,3) is the HeisenbergGroup in 4 dimensions and "*s" is the semidirect product. This is thecounterpart in this theory of the Poincare group and reduces in the appropriatelimit to the expected special relativity and classical Hamiltonian mechanicstransformation equations. This group has the Poincare group as a subgroup andis intrinsically quantum with the Position, Time, Energy and Momentum operatorssatisfying the Heisenberg algebra. The representations of the algebra arestudied and Casimir invariants are computed. Like the Poincare group, it has alittle group for a ("massive") rest frame and a null frame. The former is U(3)which clearly contains SU(3) and the latter is Os(2) which contains SU(2)*U(1).Comment: 18 pages, PDF, to be published in J. Math. Phys., Mathematica 3.0  computation files available from author at slow@austx.tandem.co

Canonically relativistic quantum mechanics: Representations of the unitary semidirect Heisenberg group, U(1,3)⊗sH(1,3)