Anomaly (physics)
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In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an 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.
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[edit] Short or long distance effects?
Historically quantum anomalies have often been considered to be short-distance effects (ultraviolet effects, in the language of quantum field theory), as they arise during the process of renormalization, when some divergent integrals cannot be regularized in such a way that all the symmetries are preserved simultaneously.
However the ultraviolet divergences can always be subtracted from an anomaly, or more precisely from the divergence of a classically conserved current. In fact, via such a subtraction, anomalies can always be taken to be finite. In the case of one-loop anomalies, which are the only sources of anomalies of gauge symmetries, one can show that in general anomalies are determined entirely by the quantum numbers of the elementary fields. This implies in particular that anomalies cannot in general be cured by short distance cut offs, and so are in a sense more dangerous than nonrenormalizability. This interpretation of anomalies as infrared (long distance) effects was proposed by Gerard 't Hooft in 1980.
In particular, the effects of anomalies are often felt at long distances, since the lack of this classical symmetry is often manifested in the physics of massless (or nearly massless) particles described by the theory.
[edit] Global anomalies
[edit] Rigid symmetries
Anomalies in abelian global 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.
[edit] Large gauge transformations
Global anomalies in symmetries that approach the identity sufficiently quickly at infinity do, 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 gravity in 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 any subset 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.
[edit] The Witten Anomaly
In SU(2) gauge theory in 4 dimensional Minkowski space, a gauge transformation corresponds to a choice of an element of the special unitary group SU(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 isomorphic to 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 group of the 3-sphere which is the cyclic group of 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.
[edit] Gauge anomalies
Anomalies in gauge symmetries pose problems, since a gauge symmetry is usually required by the phenomenon under study. An attempt to cancel them, ie, to build theories consistent with the gauge symmetries often leads to extra constraints on the theories (such is the case of the gauge anomaly in the Standard Model of particle physics). Anomalies in gauge theories have important connections to the topology and geometry of the gauge group.
Anomalies in gauge symmetries always occur at the one-loop level. At zero loops, one reproduces the classical theory. Feynman diagrams with more than one loop always contain internal boson propagators. As bosons may always be given a mass without breaking gauge invariance, a Pauli-Villars regularization of such diagrams is possible. Whenever the regularization of a diagram is consistent with a given symmetry, that diagram does not generate an anomaly with respect to the symmetry.
[edit] Examples
- consistent anomaly
- covariant anomaly
- conformal anomaly
- chiral anomaly
- gravitational anomaly
- gauge anomaly
- mixed anomaly
- global anomaly
- parity anomaly
- diffeomorphism anomaly
[edit] Anomaly cancellation
- axion
- Chern-Simons
- Green-Schwarz mechanism
- Liouville action
note that all the anomaly cancellation mechanisms result in a spontaneous symmetry breaking of the symmetry whose anomaly is being cancelled.
[edit] References
Gravitational Anomalies by Luis Alvarez-Gaumé and 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".
| Quantum field theory |
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| Field theory • overview of QFT • gauge theory • quantization • renormalization • partition function • vacuum state • anomaly • spontaneous symmetry breaking • condensates Some models: |
