 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2025/3/b/solution

{a_{1}^{†}, a_{2}^{†}} = 0

, where each label includes both momentum and spin. Therefore

∣2, 1 ⟩ = a_{2}^{†} a_{1}^{†} ∣0 ⟩ = - a_{1}^{†} a_{2}^{†} ∣0 ⟩ = - ∣1, 2 ⟩ .

Thus exchanging the complete labels of the two identical fermions changes the sign of their state. If the labels coincide,

(a_{1}^{†})^{2} = 0

; the state vanishes, which is the Pauli exclusion principle. A negative state vector describes the same isolated physical ray, but its relative sign matters when exchange alternatives interfere.

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