  We analyse the quantization procedure of the spinor field in the Rindler
spacetime, showing the boundary conditions that should be imposed to the field,
in order to have a well posed theory. Because of these boundary conditions we
argue that this construction and the usual one in Minkowski spacetime are
qualitatively different and can not be compared and consequently the
conventional interpretation of the Unruh effect, that is the thermal nature of
the Minkowski vacuum state from the point of view of an accelerated observer,
is questionable. We also analyse in detail the Unruh quantization scheme and we
show that it is not valid in the whole Minkowski space but only in the double
Rindler wedge, and it cannot be used as a basis for a quantum theoretical proof
of the Unruh effect.
