# Lorentz Transformation of Scalar Fields

1. Oct 24, 2008

### waht

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

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

where

$$\mathcal{L}^{uv} = \frac{\hbar}{i} (x^u \partial^v - x^v \partial^u )$$

2. Relevant equations

(1) $$U(\Lambda)^{-1} \varphi(x) U(\Lambda) = \varphi(\Lambda^{-1}x)$$

(2) $$\Lambda = 1 + \delta\omega$$

where $$\delta\omega [/itex] is an infinitesimal, and (3) [tex] U(\Lambda) = I + \frac{i}{2\hbar} \delta\omega_{uv} M^{uv}$$

3. The attempt at a solution

Got the left side of (1) equal to

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

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.

$$U(\Lambda)^{-1} \partial^u \varphi(x) U(\Lambda) = \Lambda^{u}_{ p} \bar{\partial}^p \varphi(\Lambda^{-1}x)$$

Last edited: Oct 24, 2008
2. Oct 24, 2008

### 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....

3. Oct 24, 2008

### waht

Would it be

$$\Lambda^{-1} = 1 - \delta\omega$$

and taking the second order $O(\delta\omega^2)$ to zero.

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

but that doesn't seem right

4. Oct 24, 2008

### borgwal

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