Spectroscopy of hydrogen can be used for a search into physics beyond the Standard Model. Differences between the absorption spectra of H$_2$ as observed at high redshift and those measured in the laboratory can be interpreted in terms of possible variations of the proton-electron mass ratio. Investigation of some ten of such absorbers in the redshift range $z= 2.0-4.2$ yields a constraint of $|����/��|< 5 \times 10^{-6}$ at 3$��$. Observation of H$_2$ from the photospheres of white dwarf stars inside our Galaxy delivers a constraint of similar magnitude on a dependence of $��$ on a gravitational potential $10^4$ times as strong as on the Earth's surface. Laser-based precision measurements of dissociation energies, vibrational splittings and rotational level energies in H$_2$ molecules and their deuterated isotopomers HD and D$_2$ produce values for the rovibrational binding energies fully consistent with quantum ab initio calculations including relativistic and quantum electrodynamical (QED) effects. Similarly, precision measurements of high-overtone vibrational transitions of HD$^+$ ions, captured in ion traps and sympathetically cooled to mK temperatures, also result in transition frequencies fully consistent with calculations including QED corrections. Precision measurements of inter-Rydberg transitions in H$_2$ can be extrapolated to yield accurate values for level splittings in the H$_2^+$-ion. These comprehensive results of laboratory precision measurements on neutral and ionic hydrogen molecules can be interpreted to set bounds on the existence of possible fifth forces and of higher dimensions, phenomena describing physics beyond the Standard Model.

Special Review Lecture. Journal of Molecular Spectroscopy