Gauge invariance and mass

Abstract

The invariance of a theory involving a vector field ${A}_{\ensuremath{\mu}}(x)$ under local gauge transformations ${A}_{\ensuremath{\mu}}(x)\ensuremath{\rightarrow}{A}_{\ensuremath{\mu}}(x)+{\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{\Lambda}(x)$, etc., for all $c$--number functions $\ensuremath{\Lambda}(x)$ 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 $\ensuremath{\Lambda}(x)=R(x)\ensuremath{\equiv}r\ifmmode\cdot\else\textperiodcentered\fi{}x$, ${r}_{\ensuremath{\mu}}=\mathrm{constants}$, under which ${A}_{\ensuremath{\mu}}(x)\ensuremath{\rightarrow}{A}_{\ensuremath{\mu}}(x)+{r}_{\ensuremath{\mu}}$. If $R(x)\ensuremath{\in}\mathcal{G}$, 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, $R\ensuremath{\in}\mathcal{G}$ 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), $R\ensuremath{\notin}\mathcal{G}$ 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.