The mapping is defined in terms of four Majoranas per site \(b_i^x,\;b_i^y,\;b_i^z,\;c_i\) such that
\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^z = i b^z c\qquad{(2)}\]
The tildes on the spin operators \(\tilde{\sigma_i^\alpha}\) emphasise that they live in this new extended Hilbert space and are only equivalent to the original spin operators after applying a projector \(\hat{P}\). The form of the projection operator can be understood in a few ways. From a group-theoretic perspective, before projection, the operators \(\{\tilde{\sigma}^x, \tilde{\sigma}^y, \tilde{\sigma}^z\}\) form a representation of the gamma group \(G_{3,0}\). The gamma groups \(G_{p,q}\) have \(p\) generators that square to the identity and \(q\) that square (roughly) to \(-1\). The generators otherwise obey standard anticommutation relations. The well known gamma matrices \(\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}\) represent \(G_{1,3}\) the quaternions \(G_{0,3}\) and the Pauli matrices \(G_{3,0}\).
-The Pauli matrices, however, have the additional property that the chiral element \(\sigma^x \sigma^y \sigma^z = \pmi\) is not fully determined by the group properties of \(G_{3,0}\), but it is equal to \(i\) in the Pauli matrices. Therefore, to fully reproduce the algebra of the Pauli matrices, we must project into the subspace where \(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = +i\). The chiral element of the gamma matrices for instance \(\gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3\) is of central importance in quantum field theory. See [16] for more discussion of this group theoretic view.
+The Pauli matrices, however, have the additional property that the chiral element \(\sigma^x \sigma^y \sigma^z = \pm i\) is not fully determined by the group properties of \(G_{3,0}\), but it is equal to \(i\) in the Pauli matrices. Therefore, to fully reproduce the algebra of the Pauli matrices, we must project into the subspace where \(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = +i\). The chiral element of the gamma matrices for instance \(\gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3\) is of central importance in quantum field theory. See [16] for more discussion of this group theoretic view.
So the projector must project onto the subspace where \(\tilde \sigma^x \tilde \sigma^y \tilde \sigma^z = i\). If we work this through, we find that in general \(\tilde \sigma^x \tilde \sigma^y \tilde\sigma^z = iD\) where \(D = b^x b^y b^z c\) must be the identity for every site. In other words, we can only work with physical states \(|\phi\rangle\) that satisfy \(D_i|\phi\rangle = |\phi\rangle\) for all sites \(i\). From this we construct an on-site projector \(P_i = \frac{1 + D_i}{2}\) and the overall projector is simply \(P = \prod_i P_i\).
Another way to see what this is doing physically is to explicitly construct the two intermediate fermionic operators \(f\) and \(g\) that give rise to these four Majoranas. Denoting a fermion state by \(|n_f, n_g\rangle\) the Hilbert space is the set \(\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}\). We can map these to Majoranas with, for example, this definition
\[\begin{aligned}