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advanced_notions:quantum_field_theory:anomalies [2018/10/11 13:22]
jakobadmin [Intuitive]
advanced_notions:quantum_field_theory:anomalies [2019/07/01 09:37] (current)
jakobadmin [Concrete]
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 <tabbox Concrete> ​ <tabbox Concrete> ​
 +A classical theory possesses a symmetry if the action $S(\phi)$ is unchanged by a transformation $\phi \to \delta \phi$. In a quantum theory, however, we have a symmetry if the path integral $\int D \phi e^{iS(\phi)}$ is invariant under a given transformation $\phi \to \delta \phi$. The key observation is now that invariance of the action $S(\phi)$ does not necessarily imply invariance of the path integral since the measure $D \phi$ can be non-invariant too. In more technical terms, the reason for this is that whenever we change the integration variables, we need to remember that the Jacobian can be non-trivial. ​
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 An anomaly is an obstruction to the construction of a quantum theory that has the same symmetry group as its action. An anomaly is an obstruction to the construction of a quantum theory that has the same symmetry group as its action.
  
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   * see also the discussion in https://​www.math.columbia.edu/​~woit/​QM/​qmbook.pdf and [[https://​physics.stackexchange.com/​questions/​33195/​classical-and-quantum-anomalies|here]]   * see also the discussion in https://​www.math.columbia.edu/​~woit/​QM/​qmbook.pdf and [[https://​physics.stackexchange.com/​questions/​33195/​classical-and-quantum-anomalies|here]]
   * and section 9.2. " Creation of particles by classical fields" ​ [[https://​arxiv.org/​pdf/​hep-th/​0510040.pdf|here]] ​   * and section 9.2. " Creation of particles by classical fields" ​ [[https://​arxiv.org/​pdf/​hep-th/​0510040.pdf|here]] ​
 +  * See also [[https://​philpapers.org/​rec/​FINGTA-2|Gauge theory, anomalies and global geometry: The interplay of physics and mathematics]] by Dana Fine & Arthur Fine
 +  * [[https://​aapt.scitation.org/​doi/​10.1119/​1.17328|Anomalies for Pedestrians]] by Holstein
 +  * See also the section "​Instantons,​ fermions, and physical consequences"​ in the book Classical Solutions in Quantum Field Theory by Erik Weinberg.
  
 <tabbox Abstract>​ <tabbox Abstract>​
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   * see also [[https://​www.tandfonline.com/​doi/​10.1080/​00018739000101531|Geometry and topology of chiral anomalies in gauge theories]] By R. RENNIE   * see also [[https://​www.tandfonline.com/​doi/​10.1080/​00018739000101531|Geometry and topology of chiral anomalies in gauge theories]] By R. RENNIE
   * [[http://​inspirehep.net/​record/​213998/​files/​v15-n3-p99.pdf|Anomalies and Cocycles]] by R. Jackiw   * [[http://​inspirehep.net/​record/​213998/​files/​v15-n3-p99.pdf|Anomalies and Cocycles]] by R. Jackiw
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advanced_notions/quantum_field_theory/anomalies.1539256974.txt.gz · Last modified: 2018/10/11 11:22 (external edit)