Conformal anomaly

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 hadrons, including protons and neutrons. Hence this effect is responsible for most of the mass of ordinary matter. (In fact the quarks 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.)

References

• Polchinski, Joseph (1998). String Theory, Cambridge University Press. A modern textbook.
  • Vol. 1: An introduction to the bosonic string. ISBN 0-521-63303-6.
  • Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4.

• A. M. Polyakov, Phys. Lett. B 103 (1981) 207, Phys. Lett. B 103 (1981) 211.
• T. L. Curtright and C. B. Thorn, Phys. Rev. Lett. 48 (1982) 1309 [Erratum-ibid. 48 (1982) 1768].
• J. L. Gervais and A. Neveu, Nucl. Phys. B 209 (1982) 125.

See also

• Dimensional transmutation.

This string theory-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Anomalies in physics Conformal field theory Quantum chromodynamics Renormalization group String theory String theory stubs Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Yoel Sirkis Independent American Party Look at other dictionaries: Conformal field theory — A conformal field theory (CFT) is a quantum field theory (or statistical mechanics model at the critical point) that is invariant under conformal transformations. Conformal field theory is often studied in two dimensions where there is an… …   Wikipedia Anomaly — An anomaly is any occurrence or object that is strange, unusual, or unique. In the following particular contexts it may refer to:Astronomy* Eccentric anomaly, intermediate value used to compute the position of a celestial object as a function of… …   Wikipedia 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… …   Wikipedia Pioneer anomaly — The Pioneer anomaly or Pioneer effect is the observed deviation from expectations of the trajectories of various unmanned spacecraft visiting the outer solar system, notably Pioneer 10 and Pioneer 11. Both spacecraft are escaping from the solar… …   Wikipedia String theory — This article is about the branch of theoretical physics. For other uses, see String theory (disambiguation). String theory …   Wikipedia Minimal Supersymmetric Standard Model — Beyond the Standard Model Standard Model …   Wikipedia List of string theory topics — See also: List of mathematical topics in quantum theory This is an list of string theory topics. Contents 1 String theory 1.1 String duality 1.2 Particles and fields 1 …   Wikipedia Dimensional transmutation — In particle physics, dimensional transmutation is a physical mechanism that transforms a dimensionless parameter into a dimensionful parameter. In classical field theory such as gauge theory in four dimensional spacetime, the coupling constant is …   Wikipedia Casimir effect — Casimir forces on parallel plates Casimir forces on parallel pl …   Wikipedia Bosonic string theory — is the original version of string theory, developed in the late 1960s. Although it has many attractive features, it has a pair of features that render it unattractive as a physical model. Firstly it predicts only the existence of bosons whereas… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”