Nonlinear realizations of the diffeomorphism group in metric-affine gauge theory of gravity

Abstract

The action of diffeomorphisms on coupled metric and spinor fields on a world manifold X is interpreted in terms of nonlinear realizations of the group GL∼+(4,R), the universal twofold covering group of the general linear group GL+(4,R), on the quotient manifold (GL∼+(4,R)×V)/SL(2,C), where SL(2,C) is the spin group and V is the spinor space. By using nonlinear realizations the connection of a metric-affine world manifold couples naturally to standard spinor fields. This enables us not to exceed the scope of usual spinor models as in the case in which infinite-dimensional representations of GL∼+(4,R) are considered. As an application, by starting from the familiar Lagrangian for spin-1/2 models and using the nonlinear realization method, a Lagrangian density for spinor fields which has GL∼+(4,R) as invariance group is constructed. The total Lagrangian density is obtained by adding the Lagrangian of the metric-affine gravity. The energy-momentum current associated with every vector field on the world manifold X is calculated explicitly. It turns out that spinor fields do not contribute to the corresponding superpotential, which takes a form similar to that obtained by Komar.