(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Verify that (2.16) follows from (2.14). Here [tex]\Lambda[/tex] is a Lorentz transformation matrix, [tex]U[/tex] is a unitary operator, [tex]M[/tex] is a generator of the Lorentz group.

2. Relevant equations

2.8: [tex]\delta\omega_{\rho\sigma}=-\delta\omega_{\sigma\rho}[/tex]

[tex]M^{\mu\nu}=-M^{\nu\mu}[/tex]

2.14: [tex]U(\Lambda}^{-1})M^{\mu\nu}U(\Lambda})=\Lambda^\mu_\rho\Lambda^\nu_\sigma M^{\rho\sigma}[/tex]

2.16: [tex][M^{\mu\nu},M^{\rho\sigma}]=i\hbar (g^{\mu\rho}M^{\nu\sigma}-

g^{\nu\rho}M^{\mu\sigma}+g^{\nu\sigma}M^{\mu\rho}-g^{\mu\sigma}M^{\nu\rho})

[/tex]

2.12: [tex]U(1+\delta\omega)=1+{i \over 2\hbar}\delta\omega_{\mu\nu}M^{\mu\nu}[/tex]

3. The attempt at a solution

I assume that [tex]\Lambda[/tex] is a small transformation, as hinted by Srednicki: [tex]\Lambda=1+\delta\omega[/tex] and rewrite the 2.14:

[tex](1-{i \over 2\hbar}\delta\omega_{\rho\sigma}M^{\rho\sigma})M^{\mu\nu}(1+{i \over 2\hbar}\delta\omega_{\rho\sigma}M^{\rho\sigma})=

(\delta^\mu_\rho+\delta\omega^\mu_\rho)(\delta^\nu_\sigma+\delta\omega^\nu_\sigma)M^{\rho\sigma}[/tex].

Then I cross out the [tex]M^{\mu\nu}[/tex] that come on the both sides, throw out the double-omega pieces and rewrite the omegas in the following manner

[tex]\delta\omega^\nu_\sigma=g^{\rho\nu}\delta\omega_{\rho\sigma}[/tex]

I come to

[tex]

[M^{\mu\nu},M^{\rho\sigma}]=2i\hbar (g^{\mu\sigma}M^{\rho\nu}-g^{\rho\nu}M^{\mu\sigma})

[/tex]

which seems to be incorrect. Where do I make the mistakes, if I do? How to derive the (2.16) without involvement of certain expression of the Lorentz generator?

Thanks.

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# [qft] Srednicki 2.3 Lorentz group generator commutator

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