Anomaly (quantum field theory)

An anomaly in quantum field theory is the breaking of a classical symmetry of a field theory (on a classical level) through the process of quantization .

From the point of view of quantum field theory, the term anomaly does not actually make sense, because in the classical Limes the symmetries ( belonging to the conservation laws ) do not disappear , but the terms of effect dominate all others. ${\displaystyle \hbar \to 0}$

technical description

From a technical point of view, such anomalous symmetries result from the fact that the action functional of field theory , i.e. H. its Lagrangian , obeys symmetry, but not the measure functions required for quantization and thus also not the generating function of the theory; here are the reduced Planck constant and the exponential function . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {Z}}=\int d\mu \,\exp({\mathcal {A}}/\hbar )}$ ${\displaystyle \,\hbar }$ ${\displaystyle \exp }$

Relevance and Applications

The triangle diagram: the wavy lines stand for photons ; the solid lines directed to the right and left denote electrons and positrons, respectively .

The physical relevance of such anomalies may be expressed. a. in that they make a significant contribution to the decay of the neutral pion ( or ), in connection with the triangle diagram. (The fact that a positron is converted into an electron during complete rotation around the triangle is an explicit signature of the anomalous behavior.) ${\displaystyle \pi ^{0}\rightarrow \gamma \gamma }$ ${\displaystyle \pi ^{0}\rightarrow e^{+}e^{-}\gamma }$

Other examples arise from Ward identities . These are equations for quantum mechanical amplitudes , which in quantized theories take the place of the conservation laws (which have become invalid due to quantization) .

The anomalies also explain the lack of a ninth Goldstone boson , which would otherwise be required by QCD .

Freedom from anomaly of gauge symmetries

While anomalies of global symmetries are harmless and can also be observed in nature, as in the example of decay, anomalies in gauge symmetries that are local would destroy the renormalizability of the theory. Therefore, for reasons of consistency , gauge symmetries must always be free of anomalies. ${\displaystyle \pi ^{0}}$

For the calibration symmetry of the standard model , this is guaranteed by the fact that the anomalous contributions of the various flavors of quarks and leptons cancel each other out as long as the number of generations in the quark and lepton sectors is the same.

literature

• Barry R. Holstein: Anomalies for pedestrians . In: Am. J. Phys. tape 61 , no. 2 , 1993, p. 142-147 , doi : 10.1119 / 1.17328 .
• Reinhold A. Bertlmann : Anomalies in quantum field theory . Oxford University Press, Oxford 2000, ISBN 0-19-850762-3 .
• Roman W. Jackiw: Axial anomaly . In: Scholarpedia . tape 3 , no. 10 , 2008, p. 7302 , doi : 10.4249 / scholarpedia.7302 .

Web links

• hep-ph / 9904347 Anomalous pion decay revisited
• Detailed article on Wikipedia