POINCARÉ ZERO-MASS REPRESENTATIONS

The zero-mass, discrete-spin, finite-dimensional representations of the proper Poincaré group are discussed, using the nondecomposable — and so nonunitary — representations of the little group, SE(2); matrix elements are thus arbitrary functions. The physical meaning and significance of the results are emphasized. Matrices are not decomposable, so the bases are connections, not tensors, giving gauge invariance — a partial statement of Poincaré invariance for zero mass only. Gravitation, a zero-mass spin-2 field, obeys a nonlinear condition (unlike the zero-mass spin-1 electromagnetic A), the Bianchi identity, which follows from the nature of Γ, and its integrated form, the Einstein equations, resulting in a curvature of space. Gravitation must be and is nonlinear, and electromagnetism linear, because of restrictions on which massless objects can interact with massive ones, these resulting from the differences in their little groups. The nonlinear representation is equivalent to a curvature of space — which thus can be considered a consequence of nonlinearity.