# Defining particles by their commutation/anti-commutation relations

In my studies of many-body physics, I have encountered three types of particles, which can be defined based on their commutation/anti-commutation relations.

1. Fermions, defined by raising/lowering operators $$a_i, a_j^\dagger$$ which anticommute on different sites: $$\{a_i^\dagger, a_j\} = \delta_{ij} \quad \{a_i, a_j\} = \{a_i^\dagger, a_j^\dagger\} = 0,$$

2. Bosons, defined by raising/lowering operators $$b_i, b_j^\dagger$$ which commute on different sites: $$[b_i, b_j^\dagger] = \delta_{ij} \quad [b_i, b_j] = [b_i^\dagger, b_j^\dagger] = 0,$$

3. Spins (particularly, $$1/2$$), defined by raising/lowering operators $$\sigma^i_\pm = \sigma^i_x \pm i\sigma^i_y,$$ which obey fermionic relations on the same site and bosonic relations on different sites: $$\{\sigma_i^+, \sigma_i^-\} = 1 \quad [\sigma_i^+, \sigma_j^-] = 0, i\not=j.$$

Clearly, based on this, one can consider the existence of a fourth type of particle, which is defined by raising/lowering operators $$x_i, x_j^\dagger$$ which obey bosonic relations on the same site, and fermionic relations on different sites: $$[x_i, x_i^\dagger] = 1 \quad \{x_i, x_j^\dagger\} = 0, i\not=j.$$

Does such a particle exist? If not, is there a physical/mathematical reason that forbids such a particle from existing?