Complete gauge theory for the whole Poincaré group
A gauge theory for the Poincaré group is developed that is based on the point of view that such a theory should possess the essential features of a classical, sourceless Yang-Mills theory. The nature of this requirement is stated precisely, and is essentially the following. The formalism must include (i) a Bianchi identity of the type \(DF=0\), where D is the operator of covariant exterior differentiation, defined in terms of the connection that results from the gauge potential A, while F is the curvature of that connection, and (ii) field equations of the type \(D*F=0\), where * denotes the Hodge duality operator, these equations being derivable from a variational principle. It is emphasized that the absence of a nondegenerate Cartan-Killing metric is not by itself an impediment. A general formalism of this kind for non-semisimple groups is applied to the Poincaré group, the latter being regarded as the Wigner- Inönü contraction of the de Sitter group. In the model thus constructed the field equations are Yang-Mills equations in the Lorentz sector, and the Einstein equations in the translational sector for the case of a torsion-free metric connection.
scientific article; zbMATH DE number 3859944