Canonical momentum of electromagnetic field

[gerald V]

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?Many thanks in advance for any answer.

 

[vanhees71]

These difficulties are due to the fact that the Hamiltonian formalism of relativistic field theory is not manifestly Lorentz invariant. So you cannot expect to be able to formulate everything in terms of Lorentz-covariant quantities. That's the reason why the path-integral formalism is so much favorable compared to canonical quantization in relativistic QFT. It very often allows to work in the Lagrangian formalism.

Further, the fact that $\Pi^0=\frac{\partial \mathcal{L}}{\partial \dot{A}^0}=0$ shows that you are working with a model with constraints, and in this case the constraint is due to the gauge invariance of the theory. This has again consequences for how to quantize the electromagnetic field. Either you give up Lorentz invariance and fix the gauge completely or you work in a manifestly covariant gauge and apply the Gupta-Bleuler formalism. The formalism for general systems with constraints has been worked out by Dirac. For a nice introduction to the Gupta-Bleuler formalism see

O. Nachtmann, Elementary Particle Physics, Springer 1990.