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13 - Anomalies

Published online by Cambridge University Press:  10 March 2010

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Summary

A characteristic feature of relativistic quantum field theories is that symmetries of the classical theory are not always present after quantization. We do not mean here the spontaneous breaking that is characterized by a non-invariant vacuum and by the presence of the Goldstone bosons. Rather we mean a situation where there is no conserved current for the symmetry despite the absence of any terms in the action that appear to break the symmetry. Such breaking of a symmetry is called anomalous.

If the classical action is invariant, then a naive application of Noether's theorem gives us a conserved current. That is, there is no anomalous symmetry breaking. What prevents the argument from being correct is the presence of UV divergences. The current is a composite operator, i.e., a product of elementary fields at the same point, and to define it, some kind of regularization and renormalization is needed. The renormalization may invalidate the equations used to prove Noether's theorem.

For simplicity, we will consider only global symmetries, as opposed to local, or gauge, symmetries. The simplest cases of global symmetries were considered in Chapter 9. These could be treated by using an ultra-violet regulator that preserved the symmetry. The proof of Noether's theorem can then be made in the cut-off theory.

Type
Chapter
Information
Renormalization
An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion
, pp. 331 - 353
Publisher: Cambridge University Press
Print publication year: 1984

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  • Anomalies
  • John C. Collins
  • Book: Renormalization
  • Online publication: 10 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511622656.013
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  • Anomalies
  • John C. Collins
  • Book: Renormalization
  • Online publication: 10 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511622656.013
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Anomalies
  • John C. Collins
  • Book: Renormalization
  • Online publication: 10 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511622656.013
Available formats
×