Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 1

In four dimensions, $[\psi ]=3/2$, $[A_{\mu }]=1$, and $[F_{\mu \nu }]=2$. The minimal-coupling operator $\bar{\psi }{\gamma }^{\mu }{A}_{\mu }\psi$ has dimension four, so $[e]=0$ and $e$ is a marginal coupling. The Pauli term $\bar{\psi }[{\gamma }^{\mu },{\gamma }^{\nu }]F_{\mu \nu }\psi$ has dimension five, so $[\lambda ]=-1$ and $\lambda$ is an irrelevant coupling by power counting in quantum field theory.

For $x^{\prime }=\Lambda x$, a Dirac spinor and vector field transform aswhere $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. Hence the spinor terms and $F_{\mu \nu }{F}^{\mu \nu }$ are Lorentz scalars. The Maxwell term is real. Integration by parts and $({\gamma }^{\mu }{)}^{†}={\gamma }^0{\gamma }^{\mu }{\gamma }^0$ show that the adjoint of $i\bar{\psi }{\gamma }^{\mu }{D}_{\mu }\psi$ differs from it by $-i{\partial }_{\mu }(\bar{\psi }{\gamma }^{\mu }\psi )$. Finally, $i[{\gamma }^{\mu },{\gamma }^{\nu }]$ is Dirac-Hermitian, so the real coefficient $\lambda$ makes the Pauli bilinear real. Thus the action is real.

Varying $\bar{\psi }$ and $\psi$ independently givesand its adjointwhere $D_{\mu }\bar{\psi }={\partial }_{\mu }\bar{\psi }-ie{A}_{\mu }\bar{\psi }$. Varying $A_{\nu }$ and integrating the Pauli term by parts gives the modified Maxwell equations

The local U(1) gauge symmetry isThe global phase subgroup has Noether current and chargeMultiplying the fermion equation by $\bar{\psi }$, its adjoint by $\psi$, and subtracting shows ${\partial }_{\mu }{j}^{\mu }=0$ on shell. Therefore $\dot{Q}=-\int d^{3}x\nabla \cdot \mathbf{j}=0$ under vanishing boundary flux.

The source in Maxwell's equation isThe second term is an identically conserved magnetization current, because it is the divergence of an antisymmetric tensor. It changes the local source but contributes only a boundary term to the total charge.

For the chiral transformation $\psi \mapsto e^{i\alpha {\gamma }^5}\psi$, one also has $\bar{\psi }\mapsto \bar{\psi }{e}^{i\alpha {\gamma }^5}$. The massless kinetic and vector-current terms are invariant because $\{{\gamma }^5,{\gamma }^{\mu }\}=0$. Since ${\gamma }^5$ commutes with $[{\gamma }^{\mu },{\gamma }^{\nu }]$, the Pauli bilinear transforms with $e^{2i\alpha {\gamma }^5}$ and is not invariant. Thus the continuous classical axial symmetry requires $\lambda =0$.

A fermion mass also breaks the symmetry, so in the massive theory one additionally needs $m=0$, which contradicts a genuinely massive fermion. Making $\alpha$ local produces $({\partial }_{\mu }\alpha )\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. No choice of the vector coupling $e$ or Pauli coupling $\lambda$ cancels this term; gauging it requires an axial gauge field, and at the quantum level one must also address the chiral anomaly.