Gauge invariance and mass

The invariance of a theory involving a vector field $A_{\mu }\mathrm{}\left(x\right)\mathrm{}$ under local gauge transformations $A_{\mu }\mathrm{}\left(x\right)\mathrm{}\to A_{\mu }\mathrm{}\left(x\right)+{\partial }_{\mu }\Lambda \mathrm{}\left(x\right)\mathrm{}$, etc., for all $c$--number functions $\Lambda \mathrm{}\left(x\right)\mathrm{}$ in some gauge group $\mathcal{G}$, does not imply that the theory contains a zero-mass gauge particle. It is shown that what is relevant to the existence of zero-mass excitations is not the existence of $\mathcal{G}$ but the presence in $\mathcal{G}$ of the simple gauge functions $\Lambda \mathrm{}\left(x\right)=R\left(x\right)\mathrm{}\equiv r·x$, $r_{\mu }=\mathrm{constants}$, under which $A_{\mu }\mathrm{}\left(x\right)\mathrm{}\to A_{\mu }\mathrm{}\left(x\right)+r_{\mu }$. If $R\left(x\right)\mathrm{}\in \mathcal{G}$, then the transverse gauge particle propagator has a singularity at zero mass. This result and similar results for the other proper vertex functions are deduced by both structural and functional methods. In conventional Lorentz-gauge four-dimensional QED, $R\in \mathcal{G}$ and so the physical photon can be interpreted as a Goldstone boson arising from the spontaneous breakdown of the $R$-transformation invariance. In two-dimensional massless QED (Schwinger model), $R\notin \mathcal{G}$ and so there the photon can be (and is) massive. The point is further illustrated in other two-dimensional soluble models and four-dimensional perturbative models.