Space-time versus world surface conformal invariance for particles, strings and membranes

The action for an extended object (string, membrane, etc) can be written in a form that is worldsheet or worldvolume (super)-Weyl invariant. We show that, as a consequence, such actions are never spacetime conformal invariant. In contrast, we also show that the action for a massless particle of spin 0, 4, or 1, in an arbitrary spacetime background has symmetries corresponding to conformal isometries of the background, precisely because of the absence of worldline (super)-Weyl invariance.