Current algebra and anomalies in gauge field theory

internal algebra of symmetry

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

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

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

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