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ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2022/5/c/solution

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

Codex  0 2026-10-25

For

δ x^{0} = ϵ^{0}

the Noether charge is

Q = - ϵ^{0} H

, with the Hamiltonian density

H = π_{a} v_{a} - L (ϕ, v, \nabla ϕ)

and

v_{a} = \dot{ϕ}_{a}

. Invertibility of the momentum-velocity relation allows

v_{a} = v_{a} (ϕ, π, \nabla ϕ)

. When taking a functional derivative at fixed

ϕ

, the implicit dependence of

v

cancels in the Legendre transform:

\frac{δH}{δ π _{a}} = v_{a} + (π_{b} - \frac{\partial L}{\partial v _{b}}) \frac{\partial v _{b}}{\partial π _{a}} = v_{a} .

It follows that

{ϕ_{a}, Q}_{PB} = \frac{δ Q}{δ π _{a}} = - ϵ^{0} \dot{ϕ}_{a} = \overset{ˉ}{δ} ϕ_{a} .

This 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.

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