In trying to derive the Dirac equation in space-time [itex] (1,-a^{2},-a^{2},-a^{2})[/itex], I have read that the Dirac equation is [itex](i\bar{\gamma}^{\mu}(\partial_{\mu}+\Gamma_{\mu})-m)\psi=0 [/itex] where,(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\Gamma_{\mu}=\frac{1}{2}\sum ^{\alpha \beta}e_{\alpha}^{\mbox{ }\nu}(\frac{\partial}{\partial x^{\mu}}e_{\beta \nu}) [/itex]

Is it correct that [itex]e_{\beta\nu}[/itex] is equal to [itex](1,a^{2},a^{2},a^{2}) [/itex], [itex]e_{\alpha}^{\mbox{ }\nu}[/itex] equal to [itex](1,1/a^{2},1/a^{2},1/a^{2})[/itex] and finally [itex]\sum^{\alpha\beta}=\frac{1}{4}(\gamma^{\alpha}\gamma^{\beta} -\gamma^{\beta} \gamma^{\alpha})[/itex]?

With my metric choice [itex]\gamma_{\mu}[/itex] should equal [itex]\frac{3}{2}\frac{\dot{a}}{a}[/itex].

I don't see how with this [itex]\sum[/itex] term that this is possible, have I made a mistake and can anyone help?

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# Spin connection problem

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