A conformal anomaly, scale anomaly, trace anomaly or Weyl 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 action 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. As a result, the trace of the stress tensor must vanish for a conformally invariant theory. In the presence of a conformal anomaly the trace of the stress tensor can nevertheless acquire a non-vanishing expectation. For this reason, a conformal anomaly is sometimes also called a trace anomaly.

String theory

In string theory, conformal symmetry on the worldsheet is a local

Weyl symmetry :''See also Wigner–Weyl transform, for another definition of the Weyl transform.'' In theoretical physics, the Weyl transformation, named after Hermann Weyl, is a local rescaling of the metric tensor: :g_\rightarrow e^g_ which produces anoth ...

and the anomaly must therefore cancel if the theory is to be consistent. The required cancellation implies that the

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

dimensionality In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...

must be equal to the critical dimension which is either 26 in the case of

bosonic string theory Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the co ...

or 10 in the case of

superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string t ...

. This case is called ''critical string theory''. There are alternative approaches known as ''

non-critical string theory The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invari ...

'' in which the space-time dimensions can be less than 26 for the bosonic theory or less than 10 for the superstring ''i.e.'' the four-dimensional case is plausible within this context. However, some intuitive postulates like flat space being a valid background, need to be given up.

QCD

In quantum chromodynamics in the chiral limit, the classical theory has no

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

scale so there is a conformal symmetry. As a consequence, for example, the proton is massless because the quark's kinetic energy and potential energy cancel by the relativistic virial theorem. However, in the quantum case the symmetry 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 In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...

s, except for

pseudoscalar meson In high-energy physics, a pseudoscalar meson is a meson with total spin 0 and odd parity (usually notated as Pseudoscalar mesons are commonly seen in proton-proton scattering and proton-antiproton annihilation, and include the pion (), ...

s. Hence this effect is responsible for most of the mass of ordinary

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...

. (In fact the quarks have non-zero masses, so the more realistic classical theory does have a mass scale. However, the masses are small so it is still nearly conformal. The mass due to the conformal anomaly is much greater than quark masses, so it has a much greater effect on the masses of hadrons.)

References

* Polchinski, Joseph (1998). ''String Theory'', Cambridge University Press. A modern textbook. ** Vol. 1: An introduction to the bosonic string. . ** Vol. 2: Superstring theory and beyond. . * * * * * * {{String theory topics , state=collapsed Anomalies (physics) Conformal field theory Quantum chromodynamics Renormalization group String theory