Conformal anomaly
Encyclopedia
Conformal 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 (Weyl transformations), combined with corresponding transformations of the other fields in the theory.
A conformal quantum theory is one whose partition function is unchanged by rescaling the metric.
The variation of the partition function with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor.
Therefore in the presence of a conformal anomaly the trace of the stress tensor has a non-vanishing expectation.
In string theory
, conformal symmetry on the worldsheet
is a local Weyl symmetry and must therefore cancel if the theory is to be consistent. The required cancellation implies that the spacetime
dimensionality must be equal to the critical dimension
which is either 26 in the case of bosonic string theory
or 10 in the case of superstring theory
. This case is called Critical String Theory. There are alternative approaches known as Subcritical String Theory in which the space-time dimensions can be less than 26 for the bosonic theory or less 10 for the superstring i.e the four dimensional case is plausible within this context.
In quantum chromodynamics
in the chiral limit, the classical theory has no mass
scale so there is a conformal symmetry, but this is broken by a conformal anomaly. This introduces a scale, which is the scale at which colour confinement
occurs. This determines the sizes and masses of hadron
s, including proton
s and neutron
s. Hence this effect is responsible for most of the mass of ordinary matter
. (In fact the quark
s have non-zero masses, so the classical theory does have a mass scale. However, the masses are small so it is still nearly conformal, so there is still a conformal anomaly. The mass due to the conformal anomaly is much greater than the quark masses, so it has a much greater effect on the masses of hadrons.)
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...
i.e. a quantum phenomenon that breaks the conformal symmetry
Conformal symmetry
In theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
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 (Weyl transformations), combined with corresponding transformations of the other fields in the theory.
A conformal quantum theory is one whose partition function is unchanged by rescaling the metric.
The variation of the partition function with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor.
Therefore in the presence of a conformal anomaly the trace of the stress tensor has a non-vanishing expectation.
In string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, conformal symmetry on the worldsheet
Worldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
is a local Weyl symmetry and must therefore cancel if the theory is to be consistent. The required cancellation implies that the spacetime
Spacetime
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...
dimensionality must be equal to the critical dimension
Critical dimension
In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. Above the upper critical dimension the critical...
which is either 26 in the case of bosonic string theory
Bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s.In the early 1970s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory became the real focus...
or 10 in the case of superstring theory
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
. This case is called Critical String Theory. There are alternative approaches known as Subcritical String Theory in which the space-time dimensions can be less than 26 for the bosonic theory or less 10 for the superstring i.e the four dimensional case is plausible within this context.
In quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
in the chiral limit, the classical theory has no mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
scale so there is a conformal symmetry, but this is broken by a conformal anomaly. This introduces a scale, which is the scale at which colour confinement
Colour confinement
Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons. The two types of hadrons are the mesons and the baryons...
occurs. This determines the sizes and masses of hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
s, including proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s and neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s. Hence this effect is responsible for most of the mass of ordinary matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
. (In fact the quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s have non-zero masses, so the classical theory does have a mass scale. However, the masses are small so it is still nearly conformal, so there is still a conformal anomaly. The mass due to the conformal anomaly is much greater than the quark masses, so it has a much greater effect on the masses of hadrons.)