Linearly and nonlinearly transforming fields on homogeneous spaces of the (4,1)-de Sitter group

Abstract

Scalar functions on the homogeneous spaces ℋR of the de Sitter group G=SO(4,1) are studied, where the spaces ℋR are of the form G/K with K being a subgroup of the Lorentz group H=SO(3,1) contained in SO(4,1). The spaces ℋR can be regarded as fiber bundles ER=ER(G/H,H/K), with the base V′4 =G/H being a space of constant negative curvature characterized by a fundamental length parameter R[(4,1)-de Sitter space], and the fiber S=H/K being a homogeneous space of the Lorentz group. The action of G on the spaces ER is a linear action on V4 and a nonlinear action on S, where the latter action is defined by a generalized Wigner rotation. A gauge theory based on the (4,1)-de Sitter group is investigated with matter represented in terms of a generalized wave function Φ(x;ξ,ỹ) [with x∈U4 (Riemann–Cartan space-time), ξ∈V′4, and ỹ∈S] which is defined as a map from a cross section on the bundle E=E(U4, F=ER, G=SO(4,1)) over space-time U4 with fiber F=ER =G/K and structural group G=SO(4,1) into the complex numbers. The introduction of purely nonlinearly transforming fields (N)Φ(x;ỹ) is discussed as well as the nonlinear realization of the SO(4,1) symmetry in terms of transformations of the Lorentz subgroup H (generalized Wigner rotations). The geometric implications of symmetry breaking are pointed out.