Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 32D / c / Solution

Direct substitution gives $g^s{g}^r=g^{s+r}$. Differentiating at $s=0$ gives $t{\partial }_x+ix(\psi {\partial }_{\psi }-{\psi }^{\ast }{\partial }_{{\psi }^{\ast }})$.

Writing the transformed field in transformed coordinates givesConsequentlywhere $\Phi =s\tilde{x}-s^{2}t/2$. These formulas, together with their complex conjugates and the analogous mixed derivatives, give the second prolongation. Differentiating at zero yields