On the Hydrogen Atom in the Holographic Universe

We investigate the holographic bound utilizing a homogeneous, isotropic, and non-relativistic neutral hydrogen gas present in the de Sitter space. Concretely, we propose to employ de Sitter holography intertwined with quantum deformation of the hydrogen atom using the framework of quantum groups. Particularly, the $\mathcal U_q(so(4))$ quantum algebra is used to construct a finite-dimensional Hilbert space of the hydrogen atom. As a consequence of the quantum deformation of the hydrogen atom, we demonstrate that the Rydberg constant is dependent on the de Sitter radius, $L_\Lambda$. This feature is then extended to obtain a finite-dimensional Hilbert space for the full set of all hydrogen atoms in the de Sitter universe. We then show that the dimension of the latter Hilbert space satisfies the holographic bound. We further show that the mass of a hydrogen atom $m_\text{atom}$, the total number of hydrogen atoms at the universe, $N$, and the retrieved dimension of the Hilbert space of neutral hydrogen gas, $\text{Dim}{\mathcal H}_\text{bulk}$, are related to the de Sitter entropy, $S_\text{dS}$, the Planck mass, $m_\text{Planck}$, the electron mass, $m_\text{e}$, and the proton mass $m_\text{p}$, by $m_\text{atom}\simeq m_\text{Planck}S_\text{dS}^{-\frac{1}{6}}$, $N\simeq S_\text{dS}^\frac{2}{3}$ and $\text{Dim}{\mathcal H}_\text{bulk}=2^{\frac{m_\text{e}}{m_\text{p}}\alpha^2S_\text{dS}}$, respectively.