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The importance of accurately taking into account the energy level shifts due to the blackbody radiation (BBR) in high precision spectroscopy is well known. The work in this area tends to focus on the BBR shift of the ground state and low excited states in view of the needs of current and forthcoming experiments. An early and notable exception is Farley and Wing's calculation of the BBR shift of a number of states and their discussion of the accuracy of the dipole approximation in this context [1]. In particular, they argued that retardation is likely to be significant at temperatures approximately equal or greater than $\alpha m c^2/3 k_{\rm B} n^2$, where $n$ is the principal quantum number of the state, $m$ is the mass of the electron and $k_{\rm B}$ is Boltzmann constant. It has recently been pointed out that a diamagnetic shift proportional to $(k_{\rm B}T)^4/c^5$ may also be significant for sufficiently high Rydberg states, at least at the 1~Hz level at room temperature [2,3]. We re-examine the high-temperature high-$n$ limit of the theory in the present work. We focus on the BBR shift of Rydberg states of hydrogen. We calculate the necessary matrix elements using Sturmian bases [4], estimate the relativistic contributions without retardation using the Breit Hamiltonian, and calculate the electric-, paramagnetic- and diamagnetic- multipole contributions in the Power-Zienau-Wooley gauge. Overall, we confirm that retardation and non-dipole shifts indeed become important at the temperature found by Farley and Wing, and in fact dominate the BBR shift at about 2.5 times that temperatures. We also find that the diamagnetic shift found in Refs. [2] and [3] needs to be corrected by additional contributions of the same order in $(k_{\rm B}T)^4/c^5$.
[1] J. W. Farley and W. H. Wing, Phys. Rev. A {\bf 23}, 2397 (1981)
[2] K. Beloy et al, Phys. Rev. A {\bf 111}, 062819 (2025)
[3] J. J. Lopez-Rodriguez et al, Phys. Rev. A {\bf 112}, 052807 (2025)
[4] M. P. A. Jones, R. M. Potvliege and M. Spannowsky, Phys. Rev. Res. {\bf 2}, 013244 (2020)