Troubles with Strict Conformal Symmetry

Formal low-energy theorems for matrix elements involving the stress tensor can be used to derive the differential equations that express scale and conformal invariance. We show that these equations, when applied to a single-particle matrix element of two fields, yield an integral representation that violates causality save for the trivial case where it reduces to the tree graphs of a free field. Although it is possible to construct conformally invariant, fully off-mass-shell amplitudes whose restriction to the mass shell yields a nontrivial result, these on-mass-shell amplitudes are not invariant under conformal transformations. This pathological behavior can be traced to an implicit assumption of analyticity when the stress tensor carries off small momenta that is, in general, false.