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## Main Question or Discussion Point

Hello,

I took an Electrodynamics course this semester, where we derived Maxwell's equations from the field's Lagrangian density.

As a motivation, we "looked" for a scalar (in the relativistic sense) having something to do with EM fields - and had we found one we would have declared it a candidate to be our Lagrangian density.

We found two such scalars (in the characteristic polynomial of the EM tensor): [itex]|\mathbf{B}|^2-|\mathbf{E}|^2[/itex] and [itex]\mathbf{E}\cdot\mathbf{B}[/itex]

The professor hand-wavingly claimed that the latter "does not preserve parity" and therefore invalid, so we ignored it for the rest of the course. But now (after the test...) I came back to this dot product, trying to see what "new Maxwell equations" we can theoretically get from it.

I wrote it using the 4-vector-potential and its derivatives, put it in the Euler Lagrange equations - and surprisingly - this element contributes nothing! Putting it in the EL equation just produces zero.

So what is the story of the parity and how does it relate to E dot B?

I took an Electrodynamics course this semester, where we derived Maxwell's equations from the field's Lagrangian density.

As a motivation, we "looked" for a scalar (in the relativistic sense) having something to do with EM fields - and had we found one we would have declared it a candidate to be our Lagrangian density.

We found two such scalars (in the characteristic polynomial of the EM tensor): [itex]|\mathbf{B}|^2-|\mathbf{E}|^2[/itex] and [itex]\mathbf{E}\cdot\mathbf{B}[/itex]

The professor hand-wavingly claimed that the latter "does not preserve parity" and therefore invalid, so we ignored it for the rest of the course. But now (after the test...) I came back to this dot product, trying to see what "new Maxwell equations" we can theoretically get from it.

I wrote it using the 4-vector-potential and its derivatives, put it in the Euler Lagrange equations - and surprisingly - this element contributes nothing! Putting it in the EL equation just produces zero.

So what is the story of the parity and how does it relate to E dot B?

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