- Anomaly (physics)
quantum physicsan anomaly or quantum anomaly is the failure of a symmetryof a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physicsan anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking parameter goes to zero. Perhaps, the first known anomaly was the dissipative anomaly in turbulence: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing viscosity.
caling and renormalization
The most prevalent global anomaly in physics is associated with the scaling symmetry, which results in
renormalization. Since regulators generally introduce a distance scale, the classically scale-invariant theories are subject to renormalization group flow, changing behavior with energy scale. For example, the large strength of the strong nuclear forceresults from a theory that is weakly coupled at short distances flowing to a strongly coupled theory at long distances, due to the scaling anomaly.
abelianglobal symmetries pose no problems in a quantum field theory, and are often encountered (see the example of the chiral anomaly). In particular the corresponding symmetries can be fixed by fixing the boundary conditions of the path integral.
Large gauge transformations
Global anomalies in
symmetriesthat approach the identity sufficiently quickly at infinitydo, however, pose problems. In known examples such symmetries correspond to disconnected components of gauge symmetries. Such symmetries and possible anomalies occur, for example, in theories with chiral fermions or self-dual differential forms coupled to gravityin "4k+2" dimensions, and also in the Witten anomaly in an ordinary 4-dimensional SU(2) gauge theory.
As these symmetries vanish at infinity, they cannot be constrained by boundary conditions and so must be summed over in the path integral. The sum of the gauge orbit of a state is a sum of phases which form a subgroup of U(1). As there is an anomaly, not all of these phases are the same, therefore it is not the identity subgroup. The sum of the phases in every other subgroup of U(1) is equal to zero, and so all path integrals are equal to zero when there is such an anomaly and the theory does not exist.
An exception may occur when the space of configurations is itself disconnected, in which case one may have the freedom to choose to integrate over anysubset of the components. If the disconnected gauge symmetries map the system between disconnected configurations, then there is in general a consistent truncation of the theory in which one integrates only over those connected components that are not related by large gauge transformations. In this case the large gauge transformations do not act on the system and do not cause the path integral to vanish.
The Witten Anomaly
gauge theoryin 4 dimensional Minkowski space, a gauge transformation corresponds to a choice of an element of the special unitary groupSU(2) at each point in spacetime. The group of such gauge transformations is connected.
However, if we are only interested in the subgroup of gauge transformations that vanish at infinity, we may consider the 3-sphere at infinity to be a single point, as the gauge transformations vanish there anyway. If the 3-sphere at infinity is identified with a point, our Minkowski space is identified with the 4-sphere. Thus we see that the group of gauge transformations vanishing at infinity in Minkowski 4-space is
isomorphicto the group of all gauge transformations on the 4-sphere.
This is the group which consists of a continuous choice of a gauge transformation in SU(2) for each point on the 4-sphere. In other words, the gauge symmetries are in one to one corresponds with maps from the 4-sphere to the 3-sphere, which is the group manifold of SU(2). The space of such maps is "not" connected, instead the connected components are classified by the fourth
homotopy groupof the 3-sphere which is the cyclic groupof order two. In particular, there are two connected components. One contains the identity and is called the "identity component", the other is called the "disconnected component".
When the theory contains an odd number of flavors of chiral fermions, the actions of gauge symmetries in the identity component and the disconnected component of the gauge group on a physical state differ by a sign. Thus when one sums over all physical configurations in the path integral, one finds that contributions come in pairs with opposite signs. As a result, all path integrals vanish and the theory does not exist.
Anomalies in gauge symmetries lead to an inconsistency, since a gauge symmetry is required in order to cancel unphysical degrees of freedom with a negative norm (such as a
photonpolarized in the time direction). An attempt to cancel them—i.e., to build theories consistentwith the gauge symmetries—often leads to extra constraints on the theories (such is the case of the gauge anomalyin the Standard Modelof particle physics). Anomalies in gauge theories have important connections to the topologyand geometryof the gauge group.
Anomalies in gauge symmetries can be calculated exactly at the one-loop level. At tree level (zero loops), one reproduces the classical theory.
Feynman diagramswith more than one loop always contain internal bosonpropagators. As bosons may always be given a mass without breaking gauge invariance, a Pauli-Villars regularizationof such diagrams is possible while preserving the symmetry. Whenever the regularization of a diagram is consistent with a given symmetry, that diagram does not generate an anomaly with respect to the symmetry.
Vector gauge anomalies are always chiral anomalies. Another type of gauge anomaly is
Anomalies at different energy scales
Quantum anomalies were discovered via the process of
renormalization, when some divergent integrals cannot be regularized in such a way that all the symmetries are preserved simultaneously. This is related to the physics at the UV (i.e. at high energies). However, due to Gerard 't Hooft's anomaly matching condition, any chiral anomalycan be described either by the UV degrees of freedom (those relevant at high energies) or by the IR degrees of freedom (those relevant at low energies). Thus one cannot cancel the anomaly by some UV completionof the theory — an anomalous symmetry is simply not a symmetry of the theory, even though classically it looks as if it is.
conformal anomaly(anomaly of scale invariance)
gravitational anomaly(also known as "diffeomorphism anomaly")
note that all the anomaly cancellation mechanisms result in a
spontaneous symmetry breakingof the symmetry whose anomaly is being cancelled.
[http://ccdb4fs.kek.jp/cgi-bin/img_index?8402145 Gravitational Anomalies] by
Luis Alvarez-Gauméand Edward Witten: This classic paper, which introduces pure gravitational anomalies, contains a good general introduction to anomalies and their relation to regularization and to conserved currents. All occurrences of the number 388 should be read "384".
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