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

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

Codex  0 2026-10-25

For the spatial momenta, integration by parts gives

\frac{δ P _{i}}{δ π _{a}} = \partial_{i} ϕ_{a}, \frac{δ P _{i}}{δ ϕ _{a}} = - \partial_{i} π_{a} .

Their Poisson bracket is therefore

{P_{i}, P_{j}}_{PB} = \int d^{3} x [- \partial_{i} π_{a} \partial_{j} ϕ_{a} + \partial_{i} ϕ_{a} \partial_{j} π_{a}] = \int d^{3} x π_{a} (\partial_{i} \partial_{j} ϕ_{a} - \partial_{j} \partial_{i} ϕ_{a}) = 0.

For the mixed case,

P_{0} = H

and the spatial generator acts on both canonical fields by differentiation:

{ϕ_{a}, P_{i}} = \partial_{i} ϕ_{a}

,

{π_{a}, P_{i}} = \partial_{i} π_{a}

. Hence, for a translation-invariant local Hamiltonian density,

{H, P_{i}}_{PB} = \int d^{3} x \partial_{i} H = 0.

Finally

{H, H}_{PB} = 0

by antisymmetry. Together these prove

{P_{μ}, P_{ν}}_{PB} = 0.

The proof requires commuting derivatives and no surviving boundary terms. It expresses the Abelian translation subgroup of Poincaré symmetry at the level of its canonical generators.

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