- #1
jfy4
- 649
- 3
I would like to work out the following commutation relations (assuming I have the operators right...)
(1) [tex]\left[\hat{p}^{\alpha},\hat{p}_{\beta}\right][/tex]
(2) [tex]\left[\hat{p}_{\alpha},\hat{L}^{\beta\gamma}\right][/tex]
(3) [tex]\left[\hat{L}^{\alpha\beta},\hat{L}_{\gamma\delta}\right][/tex]
where
[tex]\hat{p}^{\alpha}=i\nabla^{\alpha}[/tex]
[tex]\hat{L}^{\alpha\beta}=i(x^{\alpha}\nabla^{\beta}-x^{\beta}\nabla^{\alpha})[/tex]
where [itex]\nabla[/itex] is the covariant derivative. I have managed to work out (1) I think, I got zero. The others I have started but I can't seem to reduce them down to a pretty form. Do I have the operators right (for general Einstein metric)?
EDIT:
I'll post what I got for (2)
[tex]=\delta_{\alpha}^{\beta}\partial^{\gamma}\psi-\delta_{\alpha}^{\gamma}\partial^{\beta}\psi+\Gamma^{\beta}_{\alpha\delta}x^{\delta}\partial^{\gamma}\psi-\Gamma^{\gamma}_{\alpha\delta}x^{\delta}\partial^{\beta}\psi[/tex]
(1) [tex]\left[\hat{p}^{\alpha},\hat{p}_{\beta}\right][/tex]
(2) [tex]\left[\hat{p}_{\alpha},\hat{L}^{\beta\gamma}\right][/tex]
(3) [tex]\left[\hat{L}^{\alpha\beta},\hat{L}_{\gamma\delta}\right][/tex]
where
[tex]\hat{p}^{\alpha}=i\nabla^{\alpha}[/tex]
[tex]\hat{L}^{\alpha\beta}=i(x^{\alpha}\nabla^{\beta}-x^{\beta}\nabla^{\alpha})[/tex]
where [itex]\nabla[/itex] is the covariant derivative. I have managed to work out (1) I think, I got zero. The others I have started but I can't seem to reduce them down to a pretty form. Do I have the operators right (for general Einstein metric)?
EDIT:
I'll post what I got for (2)
[tex]=\delta_{\alpha}^{\beta}\partial^{\gamma}\psi-\delta_{\alpha}^{\gamma}\partial^{\beta}\psi+\Gamma^{\beta}_{\alpha\delta}x^{\delta}\partial^{\gamma}\psi-\Gamma^{\gamma}_{\alpha\delta}x^{\delta}\partial^{\beta}\psi[/tex]
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