= Solution
For a first-derivative <Lagrangian density> with no explicit spacetime dependence, introduce the canonical <energy-momentum tensor>
$$
T^\rho{}_{\mu}=\frac{\partial\mathcal L}{\partial(\partial_\rho\phi_a)}\partial_\mu\phi_a-\delta^\rho{}_{\mu}\mathcal L.
$$
Direct differentiation gives
$$
\partial_\rho T^\rho{}_{\mu}=\left[\partial_\rho\frac{\partial\mathcal L}{\partial(\partial_\rho\phi_a)}-\frac{\partial\mathcal L}{\partial\phi_a}\right]\partial_\mu\phi_a=0
$$
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:
$$
\boxed{P_\mu=\int d^3x\,T^0{}_{\mu}=\int d^3x\,[\pi_a\partial_\mu\phi_a-\eta_{0\mu}\mathcal L].}
$$
For $\mu=0$ this is the <Hamiltonian>; for a spatial lower index it is $P_k=\int\pi_a\partial_k\phi_a$. Under the paper's passive-translation convention, the corresponding generator is $Q=-\epsilon^\mu P_\mu$. Conservation of the integrated quantities also uses the stated vanishing boundary flux.
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