Majorana fermions in p-wave superconducting wire

The mean-field model of a p-wave superconducting wire has localized zero energy Kitaev Majorana excitations [1]. We show that Majorana fermions appear only in the mean-field model in which the ground state is the superposition of states with different numbers of electrons. In a physical system with fixed number of particles the exact zero energy excitations are not Majorana but two delocalized Dirac fermions which adiabatically tunnel between even and odd condensates. We present a simple model of interacting electrons which in mean-field approximation reproduced Kitaev fermions but when it is solved exactly does not show them.
[1] A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001)