 0128 Stephen G. Low
 Unitary Representations of the Canonical Group, the SemiDirect Product of the Unitary and WeylHeisenberg Group: C(1,3)= U(1,3) x H(1,3).
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Jan 17, 01

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Abstract. The unitary representations of the Canonical group that is defined to be the semidirect product of the unitary group with the WeylHeisenberg group, C(1,3)= U(1,3) x H(1,3) are studied. The Canonical group is equivalently written as the semidirect product of the special unitary group with the Oscillator group C(1,3)= SU(1,3) x Os(1,3). The nonabelian WeylHeisenberg group represents the eight physical degrees of freedom, time, position, momentum and energy. The WeylHeisenberg group has 4 canonical abelian 4 dimensional subgroups (i.e. translation groups T(4) corresponding to the timeposition, momentumenergy, timemomentum and positionenergy 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 timelike, null, and spacelike cases, determined by the value of the Hermitian metric, to be considered. The finite dimensional unitary Little groups are SU(3) for the timelike case and SU(2)xU(1) for the null case. The remaining unitary representations are infinite dimensional.
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