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When considering a charged particle in a plane-wave field, it is possible to show that the 2 following quantities are conserved $\boldsymbol{p}_{\perp} - e\boldsymbol{A}_{\perp}$ and $p_z - \gamma$ when $z$ is the direction of propagation of the wave. These conservation laws can be obtained by manipulating the equations of motion but I was wondering if there was a more "beautiful" way to obtain them ? (I have the impression that the conservation of $\boldsymbol{p}_{\perp} - e\boldsymbol{A}_{\perp}$ is linked to the invariance by translation in the perpendicular direction but I don't obtain that by pure application of Noether's theorem. Concerning the other conservation law, I don't see which symmetry it could be...)

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