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

Working on an exercise from Srednicki's QFT and something is not clear.

Show that

[tex] [\varphi(x), M^{uv}] = \mathcal{L}^{uv} \varphi(x) [/tex]

where

[tex] \mathcal{L}^{uv} = \frac{\hbar}{i} (x^u \partial^v - x^v \partial^u )[/tex]

2. Relevant equations

(1) [tex] U(\Lambda)^{-1} \varphi(x) U(\Lambda) = \varphi(\Lambda^{-1}x) [/tex]

(2) [tex] \Lambda = 1 + \delta\omega[/tex]

where [tex] \delta\omega [/itex] is an infinitesimal, and

(3) [tex] U(\Lambda) = I + \frac{i}{2\hbar} \delta\omega_{uv} M^{uv} [/tex]

3. The attempt at a solution

Got the left side of (1) equal to

[tex] \varphi(x) + \frac{i}{2\hbar}\delta\omega_{uv}[\varphi(x), M^{uv}] [/tex]

but not sure what to do with the right side and how to get the desired derivatives. I suspect

it has something to do with the transformation (1) of its derivative, but so far no luck.

[tex] U(\Lambda)^{-1} \partial^u \varphi(x) U(\Lambda) = \Lambda^{u}_{ p} \bar{\partial}^p \varphi(\Lambda^{-1}x) [/tex]

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# Homework Help: Lorentz Transformation of Scalar Fields

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