- #1

- 970

- 3

## Main Question or Discussion Point

For some reason I can't derive the Hamiltonian from the Lagrangian for the E&M field. Here's what I have (using +--- metric):

[tex]

\begin{equation*}

\begin{split}

\mathcal L=\frac{-1}{4}F_{ \mu \nu}F^{ \mu \nu}

\\

\Pi^\mu=\frac{\delta \mathcal L}{\delta \dot{A_\mu}}=-F^{0 \mu}

\\

\mathcal H=\Pi^\mu \dot{A}_\mu -\mathcal L=-F^{0 \mu}\dot{A}_\mu +\frac{1}{4}F_{ \mu \nu}F^{ \mu \nu}

=-F^{0 \mu}\dot{A}_\mu+\frac{1}{4}(2F_{0i}F^{0i}+F_{ij}F^{ij})

\end{split}

\end{equation*}

[/tex]

But F

[tex]

\mathcal H=-F^{0 \mu}\dot{A}_\mu+\frac{1}{2}(-E_{i}^2+B_{i}^2) [/tex]

The Hamiltonian however should be one half the sum of the squares of the electric and magnetic fields. But I can't figure out what I did wrong. I almost have it, as the first term almost adds to the 2nd term to give that, but not quite.

Also, I'm not quite sure when using the (+---) metric whether the canonical momenta is:

[tex]

\Pi^\mu=\frac{\delta \mathcal L}{\partial^0 A_\mu}

[/tex]

or

[tex]

\Pi^\mu=\frac{\delta \mathcal L}{\partial_0 A_\mu}[/tex]

I don't think it matters in the derivation of the Hamiltonian, but which one do you use in the canonical commutation relations for example?

[tex]

\begin{equation*}

\begin{split}

\mathcal L=\frac{-1}{4}F_{ \mu \nu}F^{ \mu \nu}

\\

\Pi^\mu=\frac{\delta \mathcal L}{\delta \dot{A_\mu}}=-F^{0 \mu}

\\

\mathcal H=\Pi^\mu \dot{A}_\mu -\mathcal L=-F^{0 \mu}\dot{A}_\mu +\frac{1}{4}F_{ \mu \nu}F^{ \mu \nu}

=-F^{0 \mu}\dot{A}_\mu+\frac{1}{4}(2F_{0i}F^{0i}+F_{ij}F^{ij})

\end{split}

\end{equation*}

[/tex]

But F

_{0i}=Ei, and F_{ij}=-B_{k}, so this is equal to:[tex]

\mathcal H=-F^{0 \mu}\dot{A}_\mu+\frac{1}{2}(-E_{i}^2+B_{i}^2) [/tex]

The Hamiltonian however should be one half the sum of the squares of the electric and magnetic fields. But I can't figure out what I did wrong. I almost have it, as the first term almost adds to the 2nd term to give that, but not quite.

Also, I'm not quite sure when using the (+---) metric whether the canonical momenta is:

[tex]

\Pi^\mu=\frac{\delta \mathcal L}{\partial^0 A_\mu}

[/tex]

or

[tex]

\Pi^\mu=\frac{\delta \mathcal L}{\partial_0 A_\mu}[/tex]

I don't think it matters in the derivation of the Hamiltonian, but which one do you use in the canonical commutation relations for example?