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Ok, i found the (silly) error: $$\bar\Phi\gamma^5\Phi=\Phi^\dagger\gamma^0\gamma^5\Phi$$ so under hermitian conjugation this becomes $$\Phi^\dagger\gamma^5\gamma^0\Phi=-\Phi^\dagger\gamma^0\gamma^5\Phi=-\bar\Phi\gamma^5\Phi$$ that imply $$\bar\Phi\gamma^5\Phi+h.c.=0$$ the same result that we found exploiting the two component structure.

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First, you should sort out if the components of the spinor are c-numbers or Grassmann numbers in your problem. If they are c-numbers, then the pseudoscalar built on the Majorana spinor vanishes, if I am not mistaken.

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For the first part, although it does not quite make sense to ask "the probability to have a Majorana in position x" (there are no "Majorana" in the system; the system only has electrons), the wavefunction $|u|^2$ ($|v|^2$) does have a physical meaning as the weight for a single electron (hole) excitation, see the answers in a closely related question: Can ...

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The Majorana fermions can be written in terms of usual (spinless) fermions by the relation $f_j = c_j^A + i c_j^B$, where $A$ and $B$ refer to sublattice and $j$ to the unit cell. This leads to a spinless superconducting Bogoliubov-deGennes (BdG) Hamiltonian, which means it includes not only fermionic hopping terms, but also terms that create and destroy ...

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