 97105 Stephen G. Low
 Canonically Relativistic Quantum Mechanics: Representations of the
Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3)
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Mar 4, 97

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Abstract. Born proposed a unification of special relativity and quantum mechanics
that placed position, time, energy and momentum on equal footing through a
reciprocity principle and extended the usual positiontime and
energymomentum line elements to this space by combining them through a new
fundamental constant. Requiring also invariance of the symplectic metric
yields U(1,3) as the invariance group, the inhomogeneous counterpart of
which is the canonically relativistic group CR(1,3) = U(1,3) *s H(1,3)
where H(1,3) is the Heisenberg Group in 4 dimensions and "*s" is the
semidirect product. This is the counterpart in this theory of the Poincare
group and reduces in the appropriate limit to the expected special
relativity and classical Hamiltonian mechanics transformation equations.
This group has the Poincare group as a subgroup and is intrinsically
quantum with the Position, Time, Energy and Momentum operators satisfying
the Heisenberg algebra. The representations of the algebra are studied and
Casimir invariants are computed. Like the Poincare group, it has a little
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).
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