- #1

diegzumillo

- 173

- 18

Maybe you could help me understanding this bit from the beginning of the book (peskin - intro to QFT).

## Homework Statement

In section 2.2, subsection "Noether's theorem" he first wants to show that continuous transformations on the fields that leave the equations of motion invariant (called symmetries) will have a corresponding conserved quantity. We can write a transformation like this:

[tex]\phi (x) \rightarrow \phi (x) + \alpha \Delta \phi (x)[/tex]

More specifically, he arguments that symmetries require the Lagrangian to be invariant under that transformation and writes:

[tex] \mathcal L (x) \rightarrow \mathcal L (x) +\alpha \partial _\mu \mathcal J ^\mu[/tex]

Right, now let's see what this actual variation is using the field variation:

[tex]\alpha \Delta \mathcal L = \frac {\partial \mathcal L}{\partial \phi}(\alpha \Delta \phi) + \left( \frac{\partial \mathcal L}{\partial (\partial _\mu \phi) } \right) \partial _\mu (\alpha \Delta \phi )[/tex]

which becomes (if anyone wants more steps, let me know)

[tex]\alpha \Delta \mathcal L = \alpha \partial _\mu \left( \frac{\partial \mathcal L}{\partial (\partial _\mu \phi) } \Delta \phi \right)[/tex]

So far I follow! but then he writes "we set the remaining term equal to [itex]\alpha \partial _\mu \mathcal J ^\mu[/itex] and find

[tex] \partial _\mu j^\mu (x) = 0[/tex]

where

[tex]j^\mu (x)= \frac{\partial \mathcal L}{\partial (\partial _\mu \phi)} \Delta \phi - \mathcal J^\mu[/tex]

I just don't get his definition of [itex]j^\mu[/itex].

## The Attempt at a Solution

When he says "set the remaining term equal to [itex]\alpha \partial _\mu \mathcal J ^\mu[/itex]" this is what he's saying:

[tex]\alpha \partial _\mu \left( \frac{\partial \mathcal L}{\partial (\partial _\mu \phi)}\Delta \phi \right) = \alpha \partial _\mu \mathcal J^\mu[/tex]

correct? so his definition of [itex]j^\mu[/itex] becomes

[tex]j^\mu = \frac{\partial \mathcal L}{\partial (\partial _\mu \phi)}\Delta \phi - \frac{\partial \mathcal L}{\partial (\partial _\mu \phi)}\Delta \phi = 0[/tex]

wait, what? what am I missing?

I have studied Noether's theorem before and I just don't see why he introduces these two J and j. To me, simply stating that [itex]\Delta \mathcal L = 0[/itex] leads to [itex]\partial _\mu \mathcal J^\mu = 0[/itex] which is the conserved quantity. But I don't get what he is doing.