Current algebra and anomalies in gauge field theory

internal algebra of symmetry

• an internal symmetry is defined by the algebra of generators

$$ [I_{\alpha},I_{\beta}]=c_{\alpha \beta \gamma}I_{\gamma} $$

• the generators, in turn, are given by the integral over the time-component of the currents

$$ I_{\alpha}=\int d^3x J_{0,\alpha}(x) $$

• from these equations one obtains the equal-time commutation relation of the currents

$$ [J_{0,\alpha}(\mathbf{x}),J_{0,\beta}(\mathbf{y})]=c_{\alpha \beta \gamma} J_{0,\alpha}(\mathbf{x})\delta(\mathbf{x}-\mathbf{y}) $$

• See [Pietschmann2011] and QCD and quarks for more

related items

• QCD and quarks
• WZW (Wess-Zumino-Witten) model

expositions

• [Pietschmann2011] Pietschmann, Herbert. “On the Early History of Current Algebra.” The European Physical Journal H 36, no. 1 (July 2011): 75–84. doi:10.1140/epjh/e2011-20013-0.
• Weinberg, Steven. “Effective Field Theory, Past and Future.” arXiv:0908.1964 [gr-Qc, Physics:hep-Ph, Physics:hep-Th, Physics:physics], August 13, 2009. http://arxiv.org/abs/0908.1964.
• http://www.worldscientific.com/worldscibooks/10.1142/0131
• http://isites.harvard.edu/fs/docs/icb.topic1146666.files/IV-6-Anomalies.pdf
• Abel, Anomalies

articles

• Alekseev, Anton, and Thomas Strobl. “Current Algebras and Differential Geometry.” Journal of High Energy Physics 2005, no. 03 (March 15, 2005): 035–035. doi:10.1088/1126-6708/2005/03/035.