Lorentz Transformation of Scalar Fields

[waht]

Homework Statement

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]

Homework 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]

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]

 

[borgwal]

1) What does Eq. (2) give you for the inverse of \Lambda?

2) use that expression to Taylor expand the right-hand side of your Eq. (1): that should also give you \phi(x) plus something...

 

[waht]

1) What does Eq. (2) give you for the inverse of \Lambda?

Would it be

[tex]\Lambda^{-1} = 1 - \delta\omega [/tex]

and taking the second order [itex] O(\delta\omega^2) [/itex] to zero.

2) use that expression to Taylor expand the right-hand side of your Eq. (1): that should also give you \phi(x) plus something...

If I were to Taylor that then would get something like [itex] \varphi(0) + \varphi^{,}(0) [/itex]

but that doesn't seem right

 

[borgwal]

Would it be

[tex]\Lambda^{-1} = 1 - \delta\omega [/tex]

and taking the second order [itex] O(\delta\omega^2) [/itex] to zero.



If I were to Taylor that then would get something like [itex] \varphi(0) + \varphi^{,}(0) [/itex]

but that doesn't seem right

The first part is correct, the second isn't (but getting close): check your Taylor expansion: what's \phi(x+\epsilon)?