- #1

- 4

- 0

I am trying to show that (for 4x4 matrices) the representation given by equation 3.18 (Peskin and Schroeder, page 39):

[tex]

(J^{\mu\nu})_{\alpha\beta}

=i(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}-\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha})

[/tex]

implies the commutation relations in 3.17:

[tex]

[J^{\mu\nu},J^{\rho\sigma}]

=i(g^{\nu\rho}J^{\mu\sigma}-g^{\mu\rho}J^{\nu\sigma}-g^{\nu\sigma}J^{\mu\rho}+g^{\mu\sigma}J^{\nu\rho})

[/tex]

For some reason I cannot even get this to work for the [tex](\mu,\nu,\rho,\sigma)=(0,1,1,2)[/tex] component:

[tex]

[J^{01},J^{12}]_{\alpha\beta}

=J^{01}_{\alpha\gamma}J^{12}_{\gamma\beta}

-J^{12}_{\alpha\gamma}J^{01}_{\gamma\beta}

=i(\delta^0_{\alpha}\delta^1_{\gamma}-\delta^0_{\gamma}\delta^1_{\alpha})

i(\delta^1_{\gamma}\delta^2_{\beta}-\delta^1_{\beta}\delta^2_{\gamma})

-i(\delta^1_{\alpha}\delta^2_{\gamma}-\delta^1_{\gamma}\delta^2_{\alpha})

i(\delta^0_{\gamma}\delta^1_{\beta}-\delta^0_{\beta}\delta^1_{\gamma})

[/tex]

Now, sums like [tex]\delta^1_{\gamma}\delta^2_{\gamma}[/tex] vanish, whereas [tex]\delta^1_{\gamma}\delta^1_{\gamma}[/tex] is 1, so we get:

[tex]

[J^{01},J^{12}]_{\alpha\beta}=

-\delta^0_{\alpha}\delta^2_{\beta}

+\delta^0_{\beta}\delta^2_{\alpha}

=i(J^{02})_{\alpha\beta}

[/tex]

On the other hand, the right hand side of 3.17 was supposed to give us:

[tex]

i(g^{11}J^{02}-g^{01}J^{12}-g^{12}J^{01}+g^{02}J^{11})_{\alpha\beta}

=i((-1)J^{02}-0-0+0)_{\alpha\beta}=-i(J^{02})_{\alpha\beta}

[/tex]

ugh...

I suspect I messed up with the metric at some point, but I don't see where.

[tex]

(J^{\mu\nu})_{\alpha\beta}

=i(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}-\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha})

[/tex]

implies the commutation relations in 3.17:

[tex]

[J^{\mu\nu},J^{\rho\sigma}]

=i(g^{\nu\rho}J^{\mu\sigma}-g^{\mu\rho}J^{\nu\sigma}-g^{\nu\sigma}J^{\mu\rho}+g^{\mu\sigma}J^{\nu\rho})

[/tex]

For some reason I cannot even get this to work for the [tex](\mu,\nu,\rho,\sigma)=(0,1,1,2)[/tex] component:

[tex]

[J^{01},J^{12}]_{\alpha\beta}

=J^{01}_{\alpha\gamma}J^{12}_{\gamma\beta}

-J^{12}_{\alpha\gamma}J^{01}_{\gamma\beta}

=i(\delta^0_{\alpha}\delta^1_{\gamma}-\delta^0_{\gamma}\delta^1_{\alpha})

i(\delta^1_{\gamma}\delta^2_{\beta}-\delta^1_{\beta}\delta^2_{\gamma})

-i(\delta^1_{\alpha}\delta^2_{\gamma}-\delta^1_{\gamma}\delta^2_{\alpha})

i(\delta^0_{\gamma}\delta^1_{\beta}-\delta^0_{\beta}\delta^1_{\gamma})

[/tex]

Now, sums like [tex]\delta^1_{\gamma}\delta^2_{\gamma}[/tex] vanish, whereas [tex]\delta^1_{\gamma}\delta^1_{\gamma}[/tex] is 1, so we get:

[tex]

[J^{01},J^{12}]_{\alpha\beta}=

-\delta^0_{\alpha}\delta^2_{\beta}

+\delta^0_{\beta}\delta^2_{\alpha}

=i(J^{02})_{\alpha\beta}

[/tex]

On the other hand, the right hand side of 3.17 was supposed to give us:

[tex]

i(g^{11}J^{02}-g^{01}J^{12}-g^{12}J^{01}+g^{02}J^{11})_{\alpha\beta}

=i((-1)J^{02}-0-0+0)_{\alpha\beta}=-i(J^{02})_{\alpha\beta}

[/tex]

ugh...

I suspect I messed up with the metric at some point, but I don't see where.

Last edited: