New Aspects of Angular Momentum Quantization in Curved Geometries

Curvature in velocity space affects angular momentum through the Thomas precession, observable through quantum effects. When position space is curved too, a similar angular momentum effect arises, with an even smaller curvature parameter. In a phase space view of dynamics and group theory the two effects appear through a direct product of two Lorentz groups, one centered on Lorentz boosts and the other on translations in a hyperbolic position space. The usual tensor representation must now be extended to $8\times 8$ matrices arising from position and velocity submatrices. The rotation subgroup becomes a direct product group $R\left( 3 \right)_{\mbox{vel}} \otimes R\left( 3 \right)_{\mbox{pos}} $. Its matrices recouple into a total angular momentum of standard form and a new contra-angular momentum $Q$ represented by $6\times 6$ matrices whose Lie algebra and quantization properties have been derived (Smith, F. T., Ann. Fond. L. de Broglie, \textbf{30}, 179 (2010)). It has quantum numbers $q,m_q $ whose connections with elementary particles are as yet undetermined. Transitions will be highly forbidden except in regions of high gravitational curvature or high relative velocity.