Fermions obey the canonical anti-commutation relations:

• {a†ᵢ, aⱼ} = a†ᵢ aⱼ + aⱼ a†ᵢ = δᵢⱼ (Kronecker delta)
• {aᵢ, aⱼ} = aᵢ aⱼ + aⱼ aᵢ = 0
• {a†ᵢ, a†ⱼ} = a†ᵢ a†ⱼ + a†ⱼ a†ᵢ = 0

When swapping aᵢ past a†ⱼ during normal ordering:

• If i ≠ j: aᵢ a†ⱼ = -a†ⱼ aᵢ (sign flip only)
• If i = j: aᵢ a†ᵢ = 1 - a†ᵢ aᵢ (generates identity term)

This algebra is used by the normal ordering algorithm to properly account for all terms generated during operator rearrangement.