advanced_notions:quantum_field_theory:anomalies
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advanced_notions:quantum_field_theory:anomalies [2018/10/11 13:23] jakobadmin [Concrete] |
advanced_notions:quantum_field_theory:anomalies [2019/07/01 09:37] (current) jakobadmin [Concrete] |
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| + | 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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| * 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 | * 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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