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Deformation of the SO (2, C ) subgroup of the Lorentz group



The two‐dimensional complex sphere S 1 2 + S 2 2 + S 3 2 = S 2 S 1 2 + S 2 2 + S 3 2 = S 2 forms a homogeneous space under the SL (2, C ) group. The little group of a point in this space is the SO (2, C ) group or the horospheric group T (2), according to whether S ≠ 0 or S = 0. Deformation of the SO (2, C ) group into T (2) is investigated and is demonstrated on unitary representations. This deformation is a counterpart to that of the little groups SO (3) or SO (2,1) into E (2). We conclude with a formula relating the matrix elements of unitary representations of the SL (2, C ) group in SO (2) × SO (1,1) = SO (2, C ) basis to those in horospheric basis.



Journal of Mathematical PhysicsAmerican Institute of Physics

Published: May 1, 1974

DOI: 10.1063/1.1666702

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