Spontaneously Broken Symmetry and Conformal Invariance

A necessary condition for spontaneously broken dilatational and conformal symmetry is that the length dimension of the field $\varphi \mathrm{}\left(y\right)\mathrm{}$ is different from zero. If we consider the opposite situation in which a chargelike symmetry is broken in the framework of full conformal invariance, we obtain a massive Goldstone mode, which couples conformal-invariantly to $2\gamma$, and which we therefore identify with the $\pi$ or $\eta$. The corresponding local field theory is defined in a five-dimensional space. In (the conformal compactification of) Minkowski space the Goldstone situation cannot arise.