  As well known rest masses of elementary particles and physical fields appear
when temperature of Universe become low enough and symmetry broke. Are there
another sources of rest masses that are consequences of other nature of rest
masses or it is the only methods for generating rest masses? What will be with
massless fields when the laws of symmetry are not exact laws but only are very
good approximation and the time is not homogeneous, as it is in the fractal
world? In this paper based on the fractal theory of time and space (developed
by author earlier) a possible source of rest masses caused by the fractional
dimensions (FD) of the time is considered. It gives rest masses (very small)
for all particles and fields including gravitational and electromagnetic
fields. The estimation of the values of the rest masses gives
$m_{g}=\sqrt{\epsilon (tt_{i})^{-1}}$, $m_{f}= \sqrt{(t_{0}t_{i})^{-1}}$ where
$t_{i}$ is the necessary time for photons and gravitons with rest masses
$m_{ph}$ and $m_{g}$ to get the velocity equal the speed of light (in the
fractal Universe the moving with any velocities is possible), $t_{0}$ is the
time of existence of Universe. Under some assumption preliminary values of rest
masses photons and gravitons are obtained: for a rest mass of photons $
m_{f}\sim 10^{-39}g$, for a rest mass of gravitons $m_{g}\sim10^{-43}g$.
