- #1
spaghetti3451
- 1,344
- 33
Hi, the following is taken from Peskin and Schroeder page 36:
##\partial_{\mu}\phi(x) \rightarrow \partial_{\mu}(\phi(\Lambda^{-1}x)) = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x)##
It describes the transformation law for a scalar field ##\phi(x)## for an active transformation.
I would like to work out the intermediate steps by myself as they are missing from the textbook. Can you please correct any mistakes I make?
##\partial_{\mu}\phi(x) \rightarrow \partial_{\mu}(\phi(\Lambda^{-1}x)) = \frac{\partial(\phi(\Lambda^{-1}x))}{\partial x^{\mu}} = \frac{\partial (\Lambda^{-1}x)^{\nu}}{\partial x^{\mu}} \frac{\partial \phi((\Lambda^{-1}x))}{\partial (\Lambda^{-1}x)^{\nu}} = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x)##.
Am I correct?
##\partial_{\mu}\phi(x) \rightarrow \partial_{\mu}(\phi(\Lambda^{-1}x)) = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x)##
It describes the transformation law for a scalar field ##\phi(x)## for an active transformation.
I would like to work out the intermediate steps by myself as they are missing from the textbook. Can you please correct any mistakes I make?
##\partial_{\mu}\phi(x) \rightarrow \partial_{\mu}(\phi(\Lambda^{-1}x)) = \frac{\partial(\phi(\Lambda^{-1}x))}{\partial x^{\mu}} = \frac{\partial (\Lambda^{-1}x)^{\nu}}{\partial x^{\mu}} \frac{\partial \phi((\Lambda^{-1}x))}{\partial (\Lambda^{-1}x)^{\nu}} = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x)##.
Am I correct?