Lorentz transformation, the solutions in the laboratory frame are found.

The boost required to reach the privileged frame in both scenarios must be taken in the $\mathbf{e}_{2}$ direction. The eigenvalues of $\hat{p}_{3}$ then keep the same constant value in both reference frames. Since we are interested in the solutions to the driven Jaynes–Cummings model on resonance, this operator can be considered zero throughout the following derivation without affecting the final result. Notice, however, that non-zero eigenvalues of $\hat{p}_{3}$ correspond to a non-zero detuning between the two-level system and field mode in the optical analogy, through equation (\[corr\_9\]). We then include this operator for completeness and adopt the notation $k_{\perp} x_{\perp} = k_{2} x_{2} + k_{3} x_{3}$ where required.

Below critical drive: Discrete spectrum
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In the case where $\vert B \vert \geq \vert E \vert$ a boost in the $\mathbf{e}_{2}$ direction with parameter $$\label{eige_4}
\beta_{\mathcal{B}} = \frac{E}{B} \,$$ takes us to a reference frame where the electric field vanishes. Since the Dirac-Pauli equation is second-order in time, both Pauli spinors satisfy $$\label{eige_5}
\left[ \hat{p}_{0}^{\prime 2} -  \hat{p}_{1}^{\prime 2} - \left( \hat{p}^{\prime}_{2}  + \frac{eB'}{c} x^{\prime}_{1} \right)^{2} - \hat{p}^{\prime 2}_{3} - \frac{e\hbar B'}{c} \sigma_{3}  \right] \phi_{P}^{\prime} = 0 \,$$ in this privileged 