Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 5 / c

For $\delta x^0={ϵ}^0$ the Noether charge is $Q=-{ϵ}^{0}H$, with the Hamiltonian density $\mathcal{H}={\pi }_a{v}_a-\mathcal{L}(\varphi ,v,\nabla \varphi )$ and $v_a={\dot{\varphi }}_a$. Invertibility of the momentum-velocity relation allows $v_a=v_a(\varphi ,\pi ,\nabla \varphi )$. When taking a functional derivative at fixed $\varphi$, the implicit dependence of $v$ cancels in the Legendre transform:It follows thatThis proves the same generator relation for time translations. In a constrained theory the assumed inversion fails, so the appropriate reduced variables or Dirac brackets are needed instead. That is precisely the issue encountered for the Proca field in Question 4.