Fermionic Algebra and Fock Space

Hilbert Space of a Single Fermionic Mode A single bosonic mode is equivalent to a harmonic oscillator; the relation [â, â†] = 1 gives rise to an infinite-dimensional Hilbert space spanning states |n〉 for n = 0, 1, 2, 3, . . . ,∞. A single fermionic mode is different — its Hilbert space spans just two states, |0〉 and |1〉. In accordance with the Fermi statistics, multiple quanta in the same mode are not allowed. To see how this works, note that the fermionic creation / annihilation operators ↠and â satisfy not just the anti-commutation relation ââ + ââ = 1 (3)