Restricted Weyl symmetry

We discuss the physics of a restricted Weyl symmetry in a curved space-time where a gauge parameter Ω(x) of Weyl transformation satisfies a constraint Ω=0. First, we present a model of QED where we have a restricted gauge symmetry in the sense that a $U(1)$ gauge parameter θ(x) obeys a similar constraint θ=0 in a flat Minkowski space-time. Next, it is precisely shown that a global scale symmetry must be spontaneously broken at the quantum level. Finally, we discuss the origin of the restricted Weyl symmetry and show that its symmetry can be derived from a full Weyl symmetry by taking a gauge condition $R=0$ in the BRST formalism.