Broken Weyl Invariance and the Origin of Mass

Abstract

A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, W₄, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields κ_μ, representing the D(1) gauge fields, with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl symmetry is broken explicitly and the corresponding reduction of the Weyl space W₄ to a pseudo-Riemannian space V₄ is investigated assuming the breaking to be determined by an expression involving the curvature scalar R of the W₄ and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus Φ of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with Φ providing a potential for the Weyl vector fields κ_μ. After the Weyl-symmetry breaking, one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying V₄. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on Φ, coupled to the energy momentum tensors of the now massive fields involved together with the (massless) radiation fields.