Past exam of the quantum field theory course of the University of Oxford / 2025 / 3 / b / Solution

The canonical anticommutation relations give $\{a_1^{†},a_2^{†}\}=0$, where each label includes both momentum and spin. ThereforeThus 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.