01-28 Stephen G. Low

Unitary Representations of the Canonical Group, the Semi-Direct Product of the Unitary and Weyl-Heisenberg Group: C(1,3)= U(1,3) x H(1,3). (1022K, Postscript) Jan 17, 01

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Abstract. The unitary representations of the Canonical group that is defined to be the semi-direct product of the unitary group with the Weyl-Heisenberg group, C(1,3)= U(1,3) x H(1,3) are studied. The Canonical group is equivalently written as the semi-direct product of the special unitary group with the Oscillator group C(1,3)= SU(1,3) x Os(1,3). The non-abelian Weyl-Heisenberg group represents the eight physical degrees of freedom, time, position, momentum and energy. The Weyl-Heisenberg group has 4 canonical abelian 4 dimensional subgroups (i.e. translation groups T(4) corresponding to the time-position, momentum-energy, time-momentum and position-energy degrees of freedom respectively. In the embedding into H(1,3) only one of these translation groups is diagonal in a given realization. The canonical group contains four copies of the Poincar group as subgroups that each have invariant translation subgroups corresponding to these four translation groups. The first two are the familiar groups parameterized by rotations and hyperbolic transformations representing velocity addition bounded by c The remaining two are parameterized by the same rotations and hyperbolic transformations representing a new physical concept of force addition bounded by a new constant b . For b large such as if it is the Planck scale b=c^4/G, this approximates the usual Euclidean addition law. These new effects would only be apparent in very strongly interacting systems. The representation theory requires generalized time-like, null, and space-like cases, determined by the value of the Hermitian metric, to be considered. The finite dimensional unitary Little groups are SU(3) for the time-like case and SU(2)xU(1) for the null case. The remaining unitary representations are infinite dimensional.

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