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## Main Question or Discussion Point

Hi,

I'm trying to compute

[tex]P^{\mu} = \int d^{3}x T^{0\mu}[/tex]

where T is the stress energy tensor given by

[tex]T^{\mu\nu} = \frac{\partial\mathcal{L}}{\partial[\partial_{\mu}\phi]}\partial^{\nu}\phi - g^{\mu\nu}\mathcal{L}[/tex]

for the scalar field [itex]\phi[/itex] with the Lagrangian density given by

[tex]\mathcal{L} = \frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi - m^2\phi^2[/tex]

This is what I get

[tex]T^{\mu 0} = g^{0\mu}\mathcal{H}[/tex]

(using [itex]\mathcal{H} = \Pi\dot{\phi} - \mathcal{L} = \partial^{0}\phi\partial_{0}\phi - \mathcal{L}[/itex])

so

[tex]\int d^{3}x T^{0\mu} = g^{0\mu}H = \frac{1}{2}\int d^{3}p g^{0\mu}E_{p}[a(p)a^{\dagger}(p) + a^{\dagger}(p)a(p)][/tex]

Now, the problem is that if we have

[tex]p^{\mu} = (E_{p}, \vec{p})[/tex]

then [itex]E_{p} = p^{0}[/itex], so

[tex]\int d^{3}x T^{0\mu} = \frac{1}{2}\int d^{3}p g^{0\mu}p^{0}[a(p)a^{\dagger}(p) + a^{\dagger}(p)a(p)][/tex]

Is there some mistake here, because the answer should involve [itex]p^{\mu}[/itex]?

The correct answer is

[tex]\int d^{3}x T^{0\mu} = \frac{1}{2}\int d^{3}p p^{\mu}[a(p)a^{\dagger}(p) + a^{\dagger}(p)a(p)][/tex]

I'm trying to compute

[tex]P^{\mu} = \int d^{3}x T^{0\mu}[/tex]

where T is the stress energy tensor given by

[tex]T^{\mu\nu} = \frac{\partial\mathcal{L}}{\partial[\partial_{\mu}\phi]}\partial^{\nu}\phi - g^{\mu\nu}\mathcal{L}[/tex]

for the scalar field [itex]\phi[/itex] with the Lagrangian density given by

[tex]\mathcal{L} = \frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi - m^2\phi^2[/tex]

This is what I get

[tex]T^{\mu 0} = g^{0\mu}\mathcal{H}[/tex]

(using [itex]\mathcal{H} = \Pi\dot{\phi} - \mathcal{L} = \partial^{0}\phi\partial_{0}\phi - \mathcal{L}[/itex])

so

[tex]\int d^{3}x T^{0\mu} = g^{0\mu}H = \frac{1}{2}\int d^{3}p g^{0\mu}E_{p}[a(p)a^{\dagger}(p) + a^{\dagger}(p)a(p)][/tex]

Now, the problem is that if we have

[tex]p^{\mu} = (E_{p}, \vec{p})[/tex]

then [itex]E_{p} = p^{0}[/itex], so

[tex]\int d^{3}x T^{0\mu} = \frac{1}{2}\int d^{3}p g^{0\mu}p^{0}[a(p)a^{\dagger}(p) + a^{\dagger}(p)a(p)][/tex]

Is there some mistake here, because the answer should involve [itex]p^{\mu}[/itex]?

The correct answer is

[tex]\int d^{3}x T^{0\mu} = \frac{1}{2}\int d^{3}p p^{\mu}[a(p)a^{\dagger}(p) + a^{\dagger}(p)a(p)][/tex]