current algebra

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Suppose we have some transformation on the fields that we schematically represent as

\phi_i(x) \to U_{ij}(x) \phi_j(x) \approx \phi_i(x) + i \omega^a(x) t^a_{ij} \phi_j(x)\,.

Then we can construct the current

J^{\mu,a}(x) = -\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi(x))} it^a_{ij} \phi_j(x)\,,

which need not be conserved unless \mathcal{L}\, remains unchanged. We may however write

J^{0,a}(x) = \rho^a(x) = -\pi_i(x)i t^a_{ij} \phi_j(x)\,,

which at fixed times obeys

\left[ \rho^a(\mathbf{x}), \rho^b(\mathbf{y}) \right]\, = -\left[ \pi_i(\mathbf{x}) t^a_{ik} \phi_k(\mathbf{x}), \pi_j(\mathbf{y}) t^b_{jm} \phi_m(\mathbf{y}) \right ] \,,
= -t^a_{ik} t^b_{jm} \left[ \pi_i(\mathbf{x}) \phi_k(\mathbf{x}), \pi_j(\mathbf{y}) \phi_m(\mathbf{y}) \right ] \,,
= - t^a_{ik} t^b_{jm}\left( \left[ \pi_i(\mathbf{x}) \phi_k(\mathbf{x}), \pi_j(\mathbf{y}) \right ] \phi_m(\mathbf{y}) + \pi_j(\mathbf{y})\left[ \pi_i(\mathbf{x}) \phi_k(\mathbf{x}), \phi_m(\mathbf{y}) \right ] \right)\,,
=- t^a_{ik} t^b_{jm}\left( \delta_{jk} \pi_i(\mathbf{x}) \phi_m(\mathbf{y}) - \delta_{im} \pi_j(\mathbf{y}) \phi_k(\mathbf{x}) \right) i \delta^{(d)}(\mathbf{x}-\mathbf{y})\,,
= - \left(t^a_{ji} t^b_{ik} - t^b_{ji} t^a_{ik} \right) \pi_j(\mathbf{y}) \phi_k(\mathbf{x}) i \delta^{(d)}(\mathbf{x}-\mathbf{y})\,,
= -[t^a, t^b]_{jk} \pi_j(\mathbf{y}) \phi_k(\mathbf{x}) i \delta^{(d)}(\mathbf{x}-\mathbf{y})\,,
= - i f^{ab}_c t^c_{jk} \pi_j(\mathbf{y}) \phi_k(\mathbf{x}) i \delta^{(d)}(\mathbf{x}-\mathbf{y})\,,
= i f^{ab}_c \rho^c(\mathbf{x}) \delta^{(d)}(\mathbf{x}-\mathbf{y})\,,

The above expression requires some modification, as the currents are composite operators. Therefore we require some normal ordering prescription to get rid of contact terms and render \rho^a(\mathbf{x})\, valid operators. Schematically,

\rho^a(x) = -i : \pi (x) t^a \phi (x) :\,,

so that the expression

\left[ \rho^a(\mathbf{x}), \rho^b(\mathbf{y}) \right] = i f^{ab}_c \rho^c(\mathbf{x}) \delta^{(d)}(\mathbf{x}-\mathbf{y})\,,

is to be interpreted as an operator product expansion.

See also

References

[1]

  1. ↑ Treiman, Sam B. ; Jackiw, Roman ; Gross, David J. (1972). Lectures on current algebra and its applications. Princeton university Press. ISBN 0-691-08118-2. 
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