Astronomical Constraints on the Cosmic Evolution of the Fine Structure Constant and Possible Quantum Dimensions

We present measurements of absorption by the 21 cm hyperfine transition of neutral hydrogen toward radio sources at substantial look-back times. These data are used in combination with observations of rotational transitions of common interstellar molecules to set limits on the evolution of the fine structure constant: $\frac{\dot{\alpha }}{\alpha }<3.5\mathrm{}\times {10}^{-15\mathrm{}}{\mathrm{yr}}^{-1\mathrm{}}$, to a look-back time of 4.8 Gyr. In the context of string theory, the limit on the secular evolution of the scale factor of the compact dimensions, $R$, is $\frac{\dot{R}}{R}<{10}^{-15\mathrm{}}{\mathrm{yr}}^{-1\mathrm{}}$. Including terrestrial and other astronomical measurements places $2\sigma$ limits on slow oscillations of $R$ from the present to the epoch of cosmic nucleosynthesis, just seconds after the big bang, of $\frac{\Delta R}{R}<{10}^{-5\mathrm{}}$.