| Weyl anomaly | |
|---|---|
| Name | Weyl anomaly |
| Field | Quantum field theory |
| Discovered | 1970s |
| Associated people | Stephen Hawking, John Preskill, Alexander Polyakov |
Weyl anomaly
The Weyl anomaly, also called the trace anomaly, is the failure of classical local scale (Weyl) invariance to survive quantization in certain quantum field theorys. It manifests as a nonzero expectation value of the trace of the energy–momentum tensor in curved spacetime or in the presence of a regulator, and plays a central role in understanding renormalization, conformal symmetry breaking, and connections between quantum mechanics and general relativity. The anomaly constrains operator product expansions in conformal field theory and enters modern dualities such as the AdS/CFT correspondence.
The Weyl anomaly occurs when a classical symmetry under local rescalings of the metric, g_{μν}(x) → e^{2σ(x)} g_{μν}(x), is not preserved after regularization and renormalization. In flat space many theories are classically scale invariant but develop a trace anomaly due to quantum corrections associated with regularization schemes like dimensional regularization or Pauli–Villars. Its physical significance includes constraints on possible quantum effective actions, contributions to Hawking radiation in black hole physics, and determination of central charges in two-dimensional systems such as those studied by Alexander Polyakov and in applications to condensed matter via critical phenomena.
Mathematically, the Weyl anomaly is expressed by < T^{μ}{}_{μ} > = anomaly functionals[g_{μν}, fields], where < ... > denotes the quantum expectation value. In even spacetime dimensions the anomaly decomposes into combinations of local curvature invariants: in two dimensions the anomaly is proportional to the Ricci scalar R with coefficient the central charge c of the Virasoro algebra; in four dimensions it involves the Weyl tensor squared C^{2}, the Euler density E_4, and scheme-dependent ∇^2 R terms with coefficients often named a and c. These coefficients are constrained by unitarity and monotonicity theorems such as the c-theorem (2D) and the a-theorem (4D) developed by Alexander Zamolodchikov and John Cardy/Zohar Komargodski respectively. The anomaly can be derived from the noninvariance of the regularized path integral measure under Weyl transformations, as demonstrated in calculations by Fujikawa.
Computations of the Weyl anomaly employ techniques from perturbative and nonperturbative quantum field theory: heat kernel methods and Seeley–DeWitt coefficients; zeta function regularization developed in works connected to Stephen Hawking; dimensional regularization; and Fujikawa's path-integral approach. For gauge theories, background field methods and the evaluation of one-loop determinants are standard, while higher-loop contributions require renormalization group analysis. Lattice regularization provides a nonperturbative check, and algebraic renormalization links anomaly coefficients to operator mixing. Important computational references include early papers by Sidney Coleman, Curtis Callan, and Damour on stress tensors and by M. J. Duff on anomalies.
In two-dimensional conformal field theorys the Weyl anomaly is determined by the central charge c appearing in the Virasoro algebra; the Polyakov action gives a nonlocal effective action reproducing the anomaly and is central to string theory quantization as in works by Alexander Polyakov. In four dimensions free fields give characteristic contributions: a real scalar, a Dirac fermion, and a vector field each contribute distinct amounts to the a and c coefficients. In six and higher even dimensions more curvature invariants appear, and classification of independent anomaly terms is a subject of active research involving mathematicians and physicists working on index theorems and gravitational anomalies; connections exist to the Atiyah–Singer index theorem.
The Weyl anomaly is intimately linked to the renormalization group and beta functions: scale anomalies arise when a classically scale-invariant coupling acquires a nonzero beta function, breaking dilatation symmetry. The trace of the energy–momentum tensor relates to the beta functions via operator identities: T^{μ}{}_{μ} = β^i O_i + anomaly curvature terms. This relation underlies derivations of trace identities in perturbation theory and is essential in proving monotonicity theorems such as the c-theorem and a-theorem. Anomalous dimensions, renormalization group flows, and fixed points in theories like Yang–Mills theory and supersymmetric models illustrate the interplay between Weyl anomalies and quantum scaling behavior.
In curved spacetime the Weyl anomaly contributes to the effective action for gravity and affects semiclassical backreaction, including computations of Hawking radiation and black hole entropy corrections. In the context of the AdS/CFT correspondence the holographic Weyl anomaly matches bulk gravity computations: the coefficients a and c are related to parameters of the dual anti-de Sitter space and its gravitational action via holographic renormalization. Studies by Edward Witten, Juan Maldacena, and others established how bulk counterterms reproduce boundary anomalies, providing a bridge between quantum anomalies and classical gravitational dynamics.
While the Weyl anomaly is primarily a theoretical construct, it has indirect experimental relevance: central charges constrained by anomalies control scaling of two-dimensional critical systems observable in condensed matter; anomaly-induced transport coefficients appear in Weyl and Dirac semimetals studied experimentally; and anomaly-related corrections influence cosmological trace contributions in early-universe models. Precision tests of scaling near critical points and measurements of universal transport linked to anomaly coefficients provide arenas where theoretical predictions connect to data from experiments in condensed matter physics and cosmology. Category:Quantum field theory