Abstract
So far we did not worry about how the classical symmetries of a theory are carried over to the quantum theory. We have implicitly assumed that classical symmetries are preserved in the process of quantization.
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Notes
- 1.
In any even number of dimensions \(\gamma_{5}\) is defined to satisfy the conditions \((\gamma_{5})^{2}={\bf 1}\) and \(\{\gamma_{5},\gamma^{\mu}\}=0.\)
- 2.
In fact there is a tension between the conservation of the vector an axial vector currents. The calculation of the diagram shown in Eq. (9.31) can be carried out imposing the conservation of the axial vector current , which results in an anomaly for the vector current. Since this would be disastrous for the consistency of the theory, we choose the other alternative.
- 3.
In the real world this makes sense only for the up and down, and perhaps the strange quarks.
- 4.
The normalization of the generators \(T^{I}\) of the global SU(\(N_{f}\)) is given by \({\rm Tr }(T^{I} T^{J})={\frac{1}{2}}\delta^{IJ}.\)
- 5.
An early computation of the triangle diagram for the electromagnetic decay of the pion was made by Steinberger in [11].
References
Álvarez-Gaumé, L.: An introduction to anomalies. In: Velo, G., Wightman, A.S. (eds) Fundamental Problems of Gauge Field Theory, Plenum Press, London (1986)
Jackiw, R.: Topological investigations of quantized gauge theories. In: Treiman, S.B., Jackiw, R., Zumino, B., Witten, E. (eds) Current Algebra and Anomalies, Princeton, NJ (1985)
Adler, S.: Axial-vector vertex in spinor electrodynamics. Phys. Rev. 177, 2426 (1969)
Bell, J.S., Jackiw, R.: A PCAC puzzle: \(\pi^{0}\rightarrow 2\gamma\) in the sigma model. Nuovo Cim. A 60, 47 (1969)
Ynduráin, F.J.: The Theory of Quark and Gluon Interactions. Springer, New York (1999)
Dissertori, G., Knowles, I., Schmelling, M.: Quantum Chromodynamics: High Energy Experiments and Theory. Oxford, New York (2003)
Narison, S.: QCD as a Theory of Hadrons: From Partons to Confinement. Cambridge University Press, Cambridge (2004)
’t Hooft, G.: How the instantons solve the U(1) problem. Phys. Rept. 142, 357 (1986)
Sutherland, D.G.: Current algebra and some nonstrong mesonic decays. Nucl. Phys. B 2, 433 (1967)
Veltman, M.J.G.: Theoretical aspects of high-energy neutrino interactions. Proc. R. Soc. A 301, 107 (1967)
Steinberger, J.: On the use of substraction fields and the lifetimes of some types of meson decay. Phys. Rev. 76, 1180 (1949)
Adler, S.L., Bardeen, W.A.: Absence of higher order corrections in the anomalous axial vector divergence equation. Phys. Rev. 182, 1517 (1969)
Witten, E.: An SU(2) anomaly. Phys. Lett. B 117, 324 (1982)
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Álvarez-Gaumé, L., Vázquez-Mozo, M.Á. (2012). Anomalies. In: An Invitation to Quantum Field Theory. Lecture Notes in Physics, vol 839. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23728-7_9
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DOI: https://doi.org/10.1007/978-3-642-23728-7_9
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