Boundary conditions for the spinor field in Rindler spacetime and the quantum field theoretical basis of the Unruh effect

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.