Gauge formulation of gravitation theories. I. The Poincaré, de Sitter, and conformal cases

The gauge formulations of various gravitation theories are discussed. They are based on the approach in which we have the group DiffR⁴ acting on x^μ and in which we attach to every x^μ a tangent space with the group of action H. Group H does not act on x^μ and plays the role of an internal (global) symmetry group in the standard Yang-Mills theory. The matter fields in the theory transform according to representations of H and are assumed to be scalars of DiffR⁴. The full invariance group of the Lagrangian is then of the form HˡᵒᶜDiffR⁴. Here Hˡᵒᶜ is a local gauge group obtained from H exactly as in the Yang-Mills theory. The approach has two characteristic features: (i) The group Hˡᵒᶜ must be spontaneously broken in order to exclude redundant gauge fields (the Lorentz connections) from the theory in a way covariant with respect to the gauge transformations. (ii) To different H there correspond different gravitational theories, all invariant under DiffR⁴ but differing in backgrounds. Thus if H is isomorphic to the Poincar\'e group the corresponding gauge theory turns out to be equivalent to the usual Einstein or Einstein-Cartan theory of gravity in the Minkowski space as a background. The other choices for H considered in the paper are the de Sitter groups and the conformal group. They yield the Einstein theory with a negative (or positive) cosmological term in the corresponding de Sitter space and the Weyl or Cartan-Weyl theory (depending on realization of the conformal group), respectively.