- #1

- 194

- 0

[tex]P^{\mu\nu}=g^{\mu\nu}-\frac{q^\mu q^\nu}{q^2}\hspace{5mm}\text{and}\hspace{5mm}\bar{P}^{\mu\nu}=\frac{q^\mu q^\nu}{q^2}[/tex]

Here, [itex]q^\mu[/itex] is a 4-vector, and [itex]g^{\mu\nu}[/itex] is the metric that goes (1, -1, -1, -1).

I've calculated the following relations:

[tex]q_\mu P^{\mu\nu}=0[/tex]

[tex]P^{\mu\nu} g_{\mu\nu}=-3[/tex]

[tex]P^{\mu\alpha}P_{\alpha}^{\phantom{\alpha}\nu}=P^{\mu\nu}[/tex]

[tex]q_\mu \bar{P}^{\mu\nu}=q^\nu[/tex]

[tex]\bar{P}^{\mu\nu} g_{\mu\nu}=1[/tex]

[tex]\bar{P}^{\mu\alpha}\bar{P}_{\alpha}^{\phantom{\alpha}\nu}=\bar{P}^{\mu\nu}[/tex], and

[tex]P^{\mu\alpha}\bar{P}_{\alpha}^{\phantom{\alpha}\nu}=0\,.[/tex]

I see

*some*symmetry going on here, like -3 corresponding to the three spatial directions, and +1 corresponding to the time direction. And a 'product' of one another yields 0. But, I can't quite pinpoint what [itex]P^{\mu\nu}[/itex] and [itex]\bar{P}^{\mu\nu}[/itex] are in words.