 Solution

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

T^{ρ}_{μ} = \frac{\partial L}{\partial ( \partial _{ρ} ϕ _{a} )} \partial_{μ} ϕ_{a} - δ^{ρ}_{μ} L .

\partial_{ρ} T^{ρ}_{μ} = [\partial_{ρ} \frac{\partial L}{\partial ( \partial _{ρ} ϕ _{a} )} - \frac{\partial L}{\partial ϕ _{a}}] \partial_{μ} ϕ_{a} = 0

on solutions of the Euler-Lagrange equations. The four Noether currents are the four columns of this tensor, whereas their spatial integrals are the conserved four-momentum components:

P_{μ} = \int d^{3} x T^{0}_{μ} = \int d^{3} x [π_{a} \partial_{μ} ϕ_{a} - η_{0 μ} L] .

For

μ = 0

this is the Hamiltonian; for a spatial lower index it is

P_{k} = \int π_{a} \partial_{k} ϕ_{a}

. Under the paper's passive-translation convention, the corresponding generator is

Q = - ϵ^{μ} P_{μ}

. Conservation of the integrated quantities also uses the stated vanishing boundary flux.

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