- #1

AlphaNumeric

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[tex]\delta_{\lambda}\varphi = \frac{1}{2}\lambda^{\mu\nu}M_{\mu\nu}.\varphi[/tex]

for some field [tex]\varphi[/tex], which is one of [tex]S[/tex], [tex]P[/tex] or [tex]\psi[/tex] and [tex]\lambda^{\mu\nu}=-\lambda^{\nu\mu}[/tex]

[tex]M_{\mu\nu}.\psi = -(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})\psi - \Sigma_{\mu\nu}\psi[/tex]

Where [tex]\Sigma_{\mu\nu} = \frac{1}{2}\gamma_{\mu\nu}[/tex], the Dirac matrix thing. I need to show that [tex]\delta_{\lambda}(\bar{\psi}\gamma^{\rho} \partial_{\rho}\psi) = \partial_{\mu}(\lambda^{\mu\nu}x_{\nu}\bar{\psi}\gamma^{\rho} \partial_{\rho}\psi)[/tex]

The notes issue a warning that the algebra of operators, of which [tex]M_{\mu\nu}[/tex] is a part, only asks on fields, so [tex]M_{\mu\nu}.(x^{\rho}\varphi) = x^{\rho}M_{\mu\nu}.\varphi[/tex] and [tex]M_{\mu\nu}.(\partial^{\rho}\varphi) = \partial^{\rho}M_{\mu\nu}.\varphi[/tex]

My problem is that I can't get the [tex]\psi[/tex] field to transform in that nice way. The [tex]\Sigma_{\mu\nu}[/tex] term screws it up and I end up with something I can't write as a total derivative!

If I've made zero sense here, I'm trying to do Exercise I.2 here.

Thanks for any help :)