Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 301 / 3

Define the spinor generatorsRepeated use of the Clifford algebra givesThe Spinor representation of the Lorentz group isThe adjoint relation for the Dirac matrices impliesExponentiating yields

For $x^{\prime }=\Lambda x$, a Dirac spinor transforms asHenceUsing the result of part i, the Dirac adjoint transforms as

The Dirac action isTo first order,The commutator found in part i givesTogether with $\det \Lambda =1$, the transformations in part ii therefore leave both the kinetic and mass terms invariant through first order. Thus $S_D^{\prime }=S_D+O({\Omega }^2)$.

Under parity, $\overline{\psi }(t,\mathbf{x})\mapsto \overline{\psi }(t,-\mathbf{x}){\gamma }^0$. Since
${\gamma }^0{\gamma }^0{\gamma }^0={\gamma }^0$ and
${\gamma }^0{\gamma }^i{\gamma }^0=-{\gamma }^i$,Moreover ${\gamma }^0{\gamma }^5{\gamma }^0=-{\gamma }^5$, so the axial current is a pseudovector: