Past exam of the quantum field theory course of the University of Oxford / 2022 / 3 / b

Define the constant Dirac gamma matrix combinations ${\alpha }^k={\gamma }^0{\gamma }^k$ and $\beta ={\gamma }^0$. The Hamiltonian operator isEvaluate its commutator with $n(\mathbf{x})$ using the bilocal result in a. The mass term cancels immediately when $\mathbf{z}=\mathbf{y}$, because its two Dirac delta distributions coincide. For the derivative term first keep the two points distinct:The terms in which the derivative acts on ${\psi }_c$ cancel against each other; the derivative of the second Dirac delta distribution remains. ThusThe last equality is integration by parts, or the defining derivative rule for a distribution. Consequently the literal conserved current whose divergence equals $C$ isBecause a commutator of two Hermitian operators is anti-Hermitian, this $\mathbf{J}$ includes $-i$. The physical Hermitian Dirac current is instead $j^k=\bar{\psi }{\gamma }^k\psi$, so $C=-i\nabla \cdot \mathbf{j}$. The Heisenberg equation of motion now givesThis establishes the continuity equation directly from the required commutator, without assuming the Dirac equation or invoking the Noether theorem as a substitute for the computation.