Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 1 / b / Solution

For the local variation $\delta \psi =i\alpha (x)\psi$, $\delta \bar{\psi }=-i\alpha (x)\bar{\psi }$, the terms proportional to $\alpha$ cancel and the differentiated phase givesThus, for a compactly supported $\alpha$, integration by parts givesThis Dirac current is the Noether current in the convention $\delta S=-\int j^{\mu }{\partial }_{\mu }\alpha$. Using the convention with a plus sign reverses the Noether current and the conserved charge together. There is one independent global symmetry in field theory, with four spacetime components of its Noether current. The associated conserved charge is $Q=\int d^{3}x{j}^0$, with vacuum subtraction when required.