- #1
maverick280857
- 1,789
- 5
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]