Conformal anomaly
A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. In quantum field theory when we set to zero we have only Feynman tree diagrams, which is a "classical" theory (equivalent to the Fredholm formulation of a classical field theory). One-loop (N-loop) Feynman diagrams are proportional to (). If a current is conserved classically () but develops a divergence at loop level in quantum field theory (), we say there is an "anomaly." A famous example is the axial current anomaly where massless fermions will have a classically conserved axial current, but which develops a nonzero divergence in the presence of gauge fields. A scale invariant theory, one in which there are no mass scales, will have a conserved Noether current called the "scale current." This is derived by performing scale transformations on the coordinates of space-time. The divergence of the scale current is then the trace of the stress tensor. In the absence of any mass scales the stress tensor trace vanishes (), hence the current is "classically conserved" and the theory is classically scale invariant. However, at loop level the scale current