Mixed anomaly
Encyclopedia
In theoretical physics
, a mixed anomaly is an example of an anomaly
: it is an effect of quantum mechanics
— usually a one-loop diagram — that implies that the classically valid general covariance
and gauge symmetry of a theory of general relativity
combined with gauge fields and fermion
ic fields cannot be preserved simultaneously in the quantum theory.
The adjective "mixed" usually refers to a mixture of a gravitational anomaly
and gauge anomaly
.
The anomaly usually appears as a Feynman diagram
with a chiral
fermion
running in the loop (a polygon) with n−k external graviton
s and k external gauge boson
s attached to the loop where
where 
is the spacetime
dimension. Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams.
General covariance and gauge symmetries are very important symmetries for the consistency
of the whole theory, and therefore all gravitational, gauge, and mixed anomalies must cancel out.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a mixed anomaly is an example of an anomaly
Anomaly (physics)
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...
: it is an effect of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
— usually a one-loop diagram — that implies that the classically valid general covariance
General covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
and gauge symmetry of a theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
combined with gauge fields and fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
ic fields cannot be preserved simultaneously in the quantum theory.
The adjective "mixed" usually refers to a mixture of a gravitational anomaly
Gravitational anomaly
In theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics–usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with some other fields. The adjective "gravitational" is derived from...
and gauge anomaly
Gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...
.
The anomaly usually appears as a Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
with a chiral
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
running in the loop (a polygon) with n−k external graviton
Graviton
In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...
s and k external gauge boson
Gauge boson
In particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...
s attached to the loop where
where is the spacetimeSpacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
dimension. Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams.
General covariance and gauge symmetries are very important symmetries for the consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
of the whole theory, and therefore all gravitational, gauge, and mixed anomalies must cancel out.
See also
- gravitational anomalyGravitational anomalyIn theoretical physics, a gravitational anomaly is an example of a gauge anomaly: it is an effect of quantum mechanics–usually a one-loop diagram—that invalidates the general covariance of a theory of general relativity combined with some other fields. The adjective "gravitational" is derived from...
- Green-Schwarz mechanismGreen-Schwarz mechanismThe Green–Schwarz mechanism is the main discovery that started the first superstring revolution in superstring theory.-Discovery:In 1984, Michael Green and John H...