# Canonical momentum of electromagnetic field

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## Main Question or Discussion Point

The momentum canonical to the electromagnetic vector field A is straightforward to compute, as is explained in textbooks or webfiles (for example Bjorken-Drell or http://www.physics.buffalo.edu/gonsalves/aqm/lectures/10/lec-10.pdf ). Its time component is zero, while the spatial components are those of the electric field E.

However, this construct is not a 4-vector field, rather it the first row of the electromagnetic field tensor. In particular, irrespective of any Lorentz boost performed, the time component remains zero.

My questions:

- Is there a possibility to get a true 4-vector field?

- In electromagnetism, which 4-vectors fields exist in addition to A in general?

- Is it correct that it makes no invariant sense to speak about „the direction of E“ (in 3-space, at a specified position in 4-space), since this direction changes in general when a Lorentz boost is performed?

- Is it correct that, however, with the direction of time held fixed, E transforms as a 3-vector under spatial rotations? So for a fixed direction of time it makes sense to speak about „the direction of E“?

- Does the 4-divergence of A play any role w.r.t. its canonical momentum?

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