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 ${\displaystyle \hbar }$ 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 ${\displaystyle \hbar }$ (${\displaystyle \hbar ^{N}}$). If a current is conserved classically (${\displaystyle \hbar =0}$) but develops a divergence at loop level in quantum field theory (${\displaystyle \propto \hbar }$), 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 (${\displaystyle \hbar =0}$), hence the current is "classically conserved" and the theory is classically scale invariant. However, at loop level the scale current can develop a nonzero divergence. This is called the "scale anomaly" or "trace anomaly" and represents the generation of mass by quantum mechanics. It is related to the renormalization group, or the "running of coupling constants," when they are viewed at different mass scales.

While this can be formulated without reference to gravity, it becomes more powerful when general relativity is considered. A classically conformal theory with arbitrary background metric has an action that is invariant under rescalings of the background metric and other matter fields, called Weyl transformations. Note that if we rescale the coordinates this is a general coordinate transformation, and merges with general covariance, the exact symmetry of general relativity, and thus it becomes an unsatisfactory way to formulate scale symmetry (general covariance implies a conserved stress tensor; a "gravitational anomaly" represents a quantum breakdown of general covariance, and should not be confused with Weyl (scale) invariance).

However, under Weyl transformations we do not rescale the coordinates of the theory, but rather the metric and other matter fields. In the sense of Weyl, mass (or length) are defined by the metric, and coordinates are simply scale-less book-keeping devices. Hence Weyl symmetry is the correct statement of scale symmetry when gravitation is incorporated and there will then be a conserved Weyl current. There is an extensive literature involving spontaneous breaking of Weyl symmetry in four dimensions, leading to a dynamically generate Planck mass together with inflation. These theories appear to be in good agreement with observational cosmology.^{[1] [2]}

A conformal quantum theory is therefore one whose path integral, or partition function, is unchanged by rescaling the metric (together with other fields). The variation of the action with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor. As a result, the trace of the stress tensor must vanish for a conformally invariant theory. The trace of the stress tensor appears in the divergence of the Weyl current as an anomaly, thus breaking the Weyl (or Scale) invariance of the theory.