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- Thread starter Riotto
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Can you show a few lines of computation because I cannot figure out how are you getting that result. No, I am using $\alpha=1.$

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$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2}(\partial_\mu A^\mu)^2$$

Calculation of canonical momenta is as follows:

$$\pi^\mu \equiv \frac{\partial\mathcal{L}}{\partial \dot{A}_\mu} = -\frac{1}{2}\frac{\partial F_{\nu\rho}}{\partial\dot{A}_{\mu}}F^{\nu\rho} - \partial_\nu A^\nu\eta^{\mu 0}$$

The first term gives:

$$\frac{\partial F_{\nu\rho}}{\partial\dot{A}_{\mu}} = \frac{\partial}{\partial (\partial_0 A_{\mu})}(\partial_\nu A_{\rho} - \partial_\rho A_{\nu}) = \delta^0_\nu \delta^\mu_\rho - \delta^0_\rho \delta^\mu_\nu$$

Finally we obtain:

$$\pi^\mu = F^{\mu 0} - \eta^{\mu 0}\partial_\nu A^\nu$$

This agrees with Tong, so there you go, that's the calculation. It's just basic differentiation though, so I don't know what was the problem there.

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