# How many dimensions has a Majorana Fermion state space?

For an exercise question, I'm wondering on how I can determine the dimensionality of a state space spanned by Majorana fermions.

A (Dirac) fermionic excitation ($$f , f^\dagger$$) has a state space of dimension 2, spanned by the vacuum state $$|0\rangle$$ and the one-particle-state $$f^\dagger |0\rangle$$.

For Majorana fermions $$\gamma = \gamma^\dagger$$, I can't see how to define a meaningful creation / annihilation operator. The answers on this forum seem to suggest that for a single Majorana operator, the notions of creation/destruction/particle number are not meaningful, however given a set of Majorana operators, the corresponding operators can be constructed by essentially mapping to a (Dirac) fermion. From the answer to this question, I would guess that I get a two-dimensional fermionic state space for every pair of Majorana fermion operators I have. So, three of them ($$\gamma_x, \gamma_y, \gamma_z$$) give me three sets of creation/annihilation operators $$c_{xy}, c_{yz}, c_{zx}$$ and thus an 8-dimensional state space (for context, see the next paragraph).

For concreteness: I am given a fermion representation of a spin-1/2 lattice model and have to answer the questions whether there is a redundancy when describing the model either via a set of two fermionic creation/annihilation operator pairs or via three Majorana fermion operators. My solution would be to determine whether the state spaces of the spin model (two-dimensional at a single lattice site) and the fermionic representation match in order to see whether there can be a one-to-one correspondence of states. For the Schwinger representation (usual fermionic creation/annihilation), I count 4 fermionic base states ( $$|00\rangle , |01\rangle, |10\rangle, |11\rangle$$),but for the Majorana representation I'm not sure how to proceed or whether this line of thought is even valid.