  The effect of fractal space time of the quantum particles on the variation of
the fine structure constant $\alpha$ has been studied. The variation of fine
structure constant has been investigated around De Broglie length $\lambda$ and
compton length $\lambda_{c}$ and it has been suggested that the variation may
be attributed to the dimensional transition of the particle trajectories
between these two quantum domains. Considering the Fractal universe with a
small inhomogeneity in the mass distribution in the early universe, the
variation of the fine structure constant have been investigated between matter
and radiation dominated era. The fine structure constant shows a critical
behaviour with critical exponent which is fractional and shows a discontinuity.
It has been suggested that the variation of the fine structure constant may be
attributed to the intrinsic scale dependance of the fundamental constants of
nature.
