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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?

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