Abstract
The relativistic two-body problem is considered for spinless particles subject to an external electromagnetic field. When this field is made of the monochromatic superposition of two counter-propagating plane waves (and provided the mutual interaction between particles is known), it is possible to write down explicitly a pair of coupled wave equations (corresponding to a pair of mass-shell constraints) which takes into account also the field contribution. These equations are manifestly covariant; constants of the motion are exhibited, so one ends up with a reduced problem involving five degrees of freedom.
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Notes
But some situations of physical interest are not included in this case: for instance when the external field is a single monochromatic plane wave, it turns out that the wave vector is a direction of strong translation invariance; but it is a null vector, so in this case E L admits no orthonormal frame.
In contrast eq. (8) of that reference holds only if P L is timelike, which is not the case eventually considered in the present paper.
In contrast G fails to be invariant by rotation in \(\mathcal {E}_{12}\), as can be seen by a direct computation.
Space ⊕ three-dimensional hyperbolic, not to be confused with the usual time ⊕ space decomposition.
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Droz-Vincent, P. Relativistic Two-Boson System in Presence of Electromagnetic Plane Wave. Int J Theor Phys 55, 4124–4141 (2016). https://doi.org/10.1007/s10773-016-3040-9
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DOI: https://doi.org/10.1007/s10773-016-3040-9