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,
W
4
, 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 $\ka_\mu$ 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
W
4
to a pseudo-Riemannian space
V
4
is investigated assuming the breaking to be determined by an expression involving the curvature scalar
R
of the
W
4
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 $\ka_\mu$. After the Weyl-symmetry breaking one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying
V
4
. 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.