advanced_notions:quantum_field_theory:anomalies
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advanced_notions:quantum_field_theory:anomalies [2018/05/03 12:55] jakobadmin ↷ Links adapted because of a move operation |
advanced_notions:quantum_field_theory:anomalies [2019/07/01 09:37] (current) jakobadmin [Concrete] |
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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 | * 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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| + | ---- | ||
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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:canonical_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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| * 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> | ||
| + | |||
| + | **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. | ||
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| + | |||
| + | ** 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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