The free massless <Dirac field> has action
$$
S_{\rm free}=\int d^4x\,i\bar\psi\gamma^\mu\partial_\mu\psi.
$$
Under a finite dilation,
$$
\psi'(x)=\lambda^{-\Delta}\psi(\lambda^{-1}x),
\qquad
\bar\psi'(x)=\lambda^{-\Delta}\bar\psi(\lambda^{-1}x).
$$
The measure, derivative, and fields contribute $\lambda^4$, $\lambda^{-1}$, and $\lambda^{-2\Delta}$, so invariance requires
$$
4-1-2\Delta=0,
\qquad \Delta=\frac32.
$$
For $\lambda=e^\alpha$, the infinitesimal active variations are
$$
\delta\psi=-\alpha\left(x^\nu\partial_\nu+\frac32\right)\psi,
\qquad
\delta\bar\psi=-\alpha\left(x^\nu\partial_\nu+\frac32\right)\bar\psi.
$$
The associated <dilatation current> is
$$
D^\mu=x_\nu T^{\mu\nu},
\qquad
T^{\mu\nu}=i\bar\psi\gamma^\mu\partial^\nu\psi,
$$
up to an improvement term. Its divergence is the on-shell trace $T^\mu{}_{\mu}=0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 301 / 2 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-301/2/i/solution