in our scenario, there is only one independent quantity to keep track of, which we choose as the surface-subtracted Stokes parameter $\Delta I(\chi_0,x)$.

It is clear from this analysis and especially that an astrophysical understanding of soft and hard X-ray polarized emission from magnetars can constrain or even discover ALPs. Indeed, on the one hand standard astrophysical considerations dictate the expected Stokes parameters $I^\text{exp}$ and $Q^\text{exp}$ at the magnetar’s surface. On the other hand, observations on Earth lead to observed Stokes parameters $I^\text{obs}$ and $Q^\text{obs}$. Thus theoretical and observational astrophysics determine the differences of the Stokes parameters $\Delta I^\text{astro}=I^\text{obs}-I^\text{exp}$ and $\Delta Q^\text{astro}=Q^\text{obs}-Q^\text{exp}$. Non-vanishing values for $\Delta I^\text{astro}$ and $\Delta Q^\text{astro}$ suggest either a misunderstanding of the astrophysical processes at play in magnetars or observational errors. However, from non-vanishing values for $\Delta I^\text{astro}$ and $\Delta Q^\text{astro}$ such that $\Delta Q^\text{astro}=-\Delta I^\text{astro}$ strongly suggest that ALPs exist and ALP-photon oscillations occur in the magnetic field of the magnetar. The relation $\Delta Q^\text{astro}=-\Delta I^\text{astro}$ originating from can thus be seen as a smoking gun signal for ALPs.

Before turning to an analysis of the transformation properties of the surface-subtracted Stokes parameters and their consequences, it is of interest to define a new quantity, 