Anomaly (physics)
Encyclopedia
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
.
Technically, an anomalous symmetry in a quantum theory is a symmetry of the action
, but not of the measure
, and so not of the partition function
as a whole.
. 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 force results from a theory that is weakly coupled at short distances flowing to a strongly coupled theory at long distances, due to the scaling anomaly.
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.
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 form
s 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.
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 correspondence 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.
polarized in the time direction). An attempt to cancel them—i.e., 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 can be calculated exactly at the one-loop level. At tree level (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 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 gravitational anomaly
.
, 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 anomaly
can 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 completion
of the theory — an anomalous symmetry is simply not a symmetry of the theory, even though classically it looks as if it is.
Note that all the anomaly cancellation mechanisms result in a spontaneous symmetry breaking
of the symmetry whose anomaly is being cancelled.
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
of a theory's classical action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
to be a symmetry of any regularization
Regularization (physics)
-Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...
of the full quantum theory. In classical physics
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
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
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
.
Technically, an anomalous symmetry in a quantum theory is a symmetry of the action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
, but not of the measure
Measure (physics)
The measure in quantum physics is the integration measure used for performing a path integral.In quantum field theory, one must sum over all possible histories of a system....
, and so not of the partition function
Partition function (quantum field theory)
In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
as a whole.
Scaling and renormalization
The most prevalent global anomaly in physics is associated with the scaling symmetry, which results in renormalizationRenormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
. 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 force results from a theory that is weakly coupled at short distances flowing to a strongly coupled theory at long distances, due to the scaling anomaly.
Rigid symmetries
Anomalies in abelianAbelian
In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:- Group theory :*Abelian group, a group in which the binary operation is commutative...
global symmetries pose no problems in a quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, and are often encountered (see the example of the chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
). In particular the corresponding symmetries can be fixed by fixing the boundary conditions of the path integral.
Large gauge transformations
Global anomalies in symmetries that approach the identity sufficiently quickly at infinityInfinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of 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 form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s 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.
The Witten Anomaly
In SU(2) gauge theoryGauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
in 4 dimensional Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
, a gauge transformation corresponds to a choice of an element of the special unitary group
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
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 correspondence 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
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
of the 3-sphere which is the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
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
Functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions...
, one finds that contributions come in pairs with opposite signs. As a result, all path integrals vanish and the theory does not exist.
Gauge anomalies
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 photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
polarized in the time direction). An attempt to cancel them—i.e., to build theories consistent with the gauge symmetries—often leads to extra constraints on the theories (such is the case of the 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...
in the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physics). Anomalies in gauge theories
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
have important connections to the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
of 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 diagrams with more than one loop always contain internal boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
propagators. As bosons may always be given a mass without breaking gauge invariance, a Pauli–Villars regularization of 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
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
. Another type of gauge anomaly is 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...
.
Anomalies at different energy scales
Quantum anomalies were discovered via the process of renormalizationRenormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
, when some divergent integrals
Ultraviolet divergence
In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy , or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a...
cannot be regularized
Regularization (physics)
-Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...
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
Anomaly matching condition
In quantum field theory, the anomaly matching condition by Gerard 't Hooft states that the calculation of any chiral anomaly by using the degrees of freedom of the theory at some energy scale, must not depend on what scale is chosen for the calculation...
, any chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
can 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 completion
UV Completion
In theoretical physics, ultraviolet completion, or UV completion, of a quantum field theory is the passing from a lower energy quantum field theory to a more general quantum field theory above a threshold value known as the cutoff...
of the theory — an anomalous symmetry is simply not a symmetry of the theory, even though classically it looks as if it is.
Examples
- chiral anomalyChiral anomalyA chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
- conformal anomalyConformal anomalyConformal anomaly is an anomaly i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.A classically conformal theory is a theory which, when placed on a surface with arbitrary background metric, has an action that is invariant under rescalings of the background metric...
(anomaly of scale invarianceScale invarianceIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
) - gauge anomalyGauge anomalyIn 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...
- global anomalyGlobal anomalyIn theoretical physics, a global anomaly is a type of anomaly: in this particular case, it is a quantum effect that invalidates a large gauge transformations that would otherwise be preserved in the classical theory...
- 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...
(also known as diffeomorphism anomaly) - mixed anomalyMixed anomalyIn 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 fermionic fields...
- parity anomalyParity anomalyIn theoretical physics a quantum field theory is said to have a parity anomaly if its classical action is invariant under a change of parity of the universe, but the quantum theory is not invariant....
Anomaly cancellation
- axionAxionThe axion is a hypothetical elementary particle postulated by the Peccei-Quinn theory in 1977 to resolve the strong CP problem in quantum chromodynamics...
- Chern–Simons
- Green–Schwarz mechanism
- Liouville action
Note that all the anomaly cancellation mechanisms result in a spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
of the symmetry whose anomaly is being cancelled.
See also
- AnomalonAnomalonIn physics, an anomalon is a hypothetical type of nuclear matter that shows an anomalously large reactive cross section. They were first noticed in experimental runs in the early 1980s as short tracks in film emulsions or plastic leaf detectors connected to medium-energy particle accelerators...
s, a topic of some debate in the 1980s, anomalons were found in the results of some high-energy physics experiments that seemed to point to the existence of anomalously highly interactive states of matter. The topic was controversial throughout its history.