Bosons obey the canonical commutation relations:

• [aᵢ, a†ⱼ] = aᵢ a†ⱼ - a†ⱼ aᵢ = δᵢⱼ
• [aᵢ, aⱼ] = 0
• [a†ᵢ, a†ⱼ] = 0

When swapping aᵢ past a†ⱼ during normal ordering:

• If i ≠ j: aᵢ a†ⱼ = a†ⱼ aᵢ (no sign change)
• If i = j: aᵢ a†ᵢ = 1 + a†ᵢ aᵢ (identity term plus reordered term)