  It is well-known that $f(R)$ theories in Einstein frame is conformally
equivalent to quintessence models in which the scalar field minimally couples
with gravity. If there exists a matter system in Jordan frame, then it
interacts with the scalar field in Einstein frame due to the conformal
transformations. This interaction, in general, may lead to changes of
fundamental constants. Here we will consider possible time variation of fine
structure constant in a general $f(R)$ theory. We will use observational bounds
on these variations and argue that it provides a criterion for constraining
$f(R)$ models.
