advanced_notions:quantum_field_theory:anomalies
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advanced_notions:quantum_field_theory:anomalies [2018/04/15 10:04] ida [FAQ] |
advanced_notions:quantum_field_theory:anomalies [2019/07/01 09:37] (current) jakobadmin [Concrete] |
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| <blockquote>”we must assign physical reality to Dirac’s negative energy sea,because it produces the chiral anomaly, whose effects areexperimentally observed, principally in the decay of the neutral pionto two photons, but there are other physical consequences as well.”<cite>[[https://arxiv.org/pdf/hep-th/9903255.pdf|R. Jackiw]]</cite></blockquote> | <blockquote>”we must assign physical reality to Dirac’s negative energy sea,because it produces the chiral anomaly, whose effects areexperimentally observed, principally in the decay of the neutral pionto two photons, but there are other physical consequences as well.”<cite>[[https://arxiv.org/pdf/hep-th/9903255.pdf|R. Jackiw]]</cite></blockquote> | ||
| + | ----- | ||
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| + | * For more on this intuitive perspective of anomalies, see [[https://arxiv.org/abs/1010.0943|Axial anomaly, Dirac sea, and the chiral magnetic effect]] by Dmitri E. Kharzeev | ||
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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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| Each charge corresponds $Q_a$ to one [[advanced_tools:group_theory|generator]] $G_a$ of the [[basic_tools:symmetry|symmetry]] of the action. These Noether charges represent the generators on our Hilbert space in a quantum theory or on our phase space in a classical theory. | Each charge corresponds $Q_a$ to one [[advanced_tools:group_theory|generator]] $G_a$ of the [[basic_tools:symmetry|symmetry]] of the action. These Noether charges represent the generators on our Hilbert space in a quantum theory or on our phase space in a classical theory. | ||
| - | We can then use the charges and put them into the corresponding Lie bracket. In the classical theory, this is the [[advanced_notions:poisson_bracket|Poisson bracket]], in a [[theories:quantum_mechanics|quantum theory]] the [[equations:canonical_commutation_relations|commutator]]. | + | We can then use the charges and put them into the corresponding Lie bracket. In the classical theory, this is the [[advanced_notions:poisson_bracket|Poisson bracket]], in a [[theories:quantum_mechanics:canonical|quantum theory]] the [[formulas:canonical_commutation_relations|commutator]]. |
| In most cases the Noether charges form a closed algebraic structure which is exactly the same as the algebra of the symmetry of the action. | In most cases the Noether charges form a closed algebraic structure which is exactly the same as the algebra of the symmetry of the action. | ||
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| we deal with local symmetries. A quantum mechanical violation of gauge symmetry | we deal with local symmetries. A quantum mechanical violation of gauge symmetry | ||
| leads to many problems, from lack of renormalizability to nondecoupling of negative norm states. This is because the presence of an anomaly in the theory implies | leads to many problems, from lack of renormalizability to nondecoupling of negative norm states. This is because the presence of an anomaly in the theory implies | ||
| - | that the [[equations:yang_mills_equations:gauss_law|Gauss’ law]] constraint $D · E_A = ρ A$ cannot be consistently implemented | + | that the [[formulas:gauss_law|Gauss’ law]] constraint $D · E_A = ρ A$ cannot be consistently implemented |
| in the quantum theory. As a consequence, states that classically were eliminated by | in the quantum theory. As a consequence, states that classically were eliminated by | ||
| the gauge symmetry become propagating in the quantum theory, thus spoiling the | the gauge symmetry become propagating in the quantum theory, thus spoiling the | ||
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| * For a nice introduction, see http://www.maths.dur.ac.uk/~dma0saa/lecture_notes.pdf | * For a nice introduction, see http://www.maths.dur.ac.uk/~dma0saa/lecture_notes.pdf | ||
| * 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]] | ||
| + | * 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> | ||
| + | |||
| + | **Important Papers:** | ||
| + | |||
| + | * [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.39.693|Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint]] by C. Q. Geng and R. E. Marshak | ||
| + | * [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.41.715|Comment on anomaly cancellation in the standard model]] by J. A. Minahan, P. Ramond, and R. C. Warner | ||
| + | * [[http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/21/068/21068700.pdf|CHARGED NEUTRINOS?]] by R. Foot et. al. | ||
| + | *[[http://xxx.lanl.gov/pdf/hep-th/0006230v1| Particle creation via relaxing hypermagnetic knots]] by C. Adam et. al. | ||
| + | |||
| + | |||
| + | ** Recommended Resources:** | ||
| + | |||
| + | * The best explanation for the idea that anomalies are related to extensions of the corresponding Lie algebra, can be found in www.atlantis-press.com/php/download_paper.php?id=754. In addition, the paper nicely summarizes the various approaches that have been used so far to deal with anomalies. | ||
| + | * An introduction to the geometric picture of anomalies can be found in [[http://inspirehep.net/record/192970|Chiral Anomalies And Differential Geometry]]: Lectures Given At Les Houches, August 1983 by Bruno Zumino | ||
| + | * See also https://www.mathi.uni-heidelberg.de/~walcher/teaching/sose16/geo_phys/Anomalies.pdf | ||
| + | * 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 | ||
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| - | **Important Papers:** | ||
| - | * [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.39.693|Uniqueness of quark and lepton representations in the standard model from the anomalies viewpoint]] by C. Q. Geng and R. E. Marshak | ||
| - | * [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.41.715|Comment on anomaly cancellation in the standard model]] by J. A. Minahan, P. Ramond, and R. C. Warner | ||
| - | * [[http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/21/068/21068700.pdf|CHARGED NEUTRINOS?]] by R. Foot et. al. | ||
| - | *[[http://xxx.lanl.gov/pdf/hep-th/0006230v1| Particle creation via relaxing hypermagnetic knots]] by C. Adam et. al. | ||
| - | |||
| - | |||
| - | ** Recommended Resources:** | ||
| - | |||
| - | * The best explanation for the idea that anomalies are related to extensions of the corresponding Lie algebra, can be found in www.atlantis-press.com/php/download_paper.php?id=754. In addition, the paper nicely summarizes the various approaches that have been used so far to deal with anomalies. | ||
| - | * An introduction to the geometric picture of anomalies can be found in [[http://inspirehep.net/record/192970|Chiral Anomalies And Differential Geometry]]: Lectures Given At Les Houches, August 1983 by Bruno Zumino | ||
| - | * See also https://www.mathi.uni-heidelberg.de/~walcher/teaching/sose16/geo_phys/Anomalies.pdf | ||
| - | * see also [[https://www.tandfonline.com/doi/10.1080/00018739000101531|Geometry and topology of chiral anomalies in gauge theories]] By R. RENNIE | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
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| <tabbox FAQ> | <tabbox FAQ> | ||
| + | ->Are theories with gauge anomalies necessarily inconsistent?# | ||
| + | |||
| + | No! See [[https://www.sciencedirect.com/science/article/pii/000349169190046B?via%3Dihub|Gauge anomalies in an effective field theory]] by JohnPreskill | ||
| + | <-- | ||
| -->Why do we study anomalies with the triangle diagram?# | -->Why do we study anomalies with the triangle diagram?# | ||
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