Constraints on Field Theoretical Models for Variation of the Fine Structure Constant

Recent theoretical ideas and observational claims suggest that the fine structure constant alpha may be variable. We examine a spectrum of models in which alpha is a function of a scalar field. Specifically, we consider three scenarios: oscillating alpha, monotonic time variation of alpha, and time-independent alpha that is spatially varying. We examine the constraints imposed upon these theories by cosmological observations, particle detector experiments, and ``fifth force'' experiments. These constraints are very strong on models involving oscillation, but cannot compete with bounds from the Oklo sub-nuclear reactor on models with monotonic time-like variation of alpha. One particular model with spatial variation is consistent with all current experimental and observational measurements, including those from two seemingly conflicting measurements of the fine structure constant using the many multiplet method on absorption lines.