{title2=$P^\mu=\int T^{0\mu}\,d^3x$}

For the free <real scalar field> with signature $(+,-,-,-)$,
$$
E=\frac12\int(\dot\phi^2+|\nabla\phi|^2+m^2\phi^2)d^3x,
\qquad \mathbf P=-\int\dot\phi\,\nabla\phi\,d^3x.
$$
The <canonical stress-energy tensor> is $T^{\mu\nu}=\partial^\mu\phi\partial^\nu\phi-\eta^{\mu\nu}\mathcal L$. Its divergence vanishes by the <Klein-Gordon equation>, giving constant integrated charges when integrals and boundary flux are well behaved. The minus sign in contravariant spatial <momentum> is from $\partial^i=-\partial_i$; lower-index spatial charges have the opposite sign. A <plane wave> $\cos(Et-\mathbf k\cdot\mathbf x)$ has <momentum> density pointing along $\mathbf k$.


 Four-momentum of a free real scalar field

ID: four-momentum-of-a-free-real-scalar-field