Calculating Noether Currents in Peskin & Schroeder QFT

[Breo]

Hello folks,

I do not know the calculus behind used to obtain both currents in this page from Peskin and Schroeder QFT book.

$$j^\mu = \partial^\mu \phi $$ and the 2.16 eq.

http://www.zimagez.com/zimage/capturadepantalla-280914-033510.php

How do you calculate them?

 

[Hypersphere]

As Peskin and Schroeder present it, the calculus is essentially that of partial derivatives while treating \phi and \partial_\mu \phi as independent variables. For a given Lagrangian density \mathcal{L}, he defines the current in eq. (2.12). However, the current depends on the symmetry at hand, which enters through the \mathcal{J}^\mu term, defined on the previous page.

If this still looks too opaque, do you remember the treatment of Noether's theorem in classical mechanics of particles?

 

[Breo]

If this still looks too opaque, do you remember the treatment of Noether's theorem in classical mechanics of particles?

No, I didn't learn it in my classical mechanics subject.

My doubt is how it get those results for the currents. I tried to do the calcs but I get different results.

 

[Hypersphere]

Ok... To be honest, that is a bit of a red flag! QFT is a tricky subject in its own right, but it relies heavily on classical mechanics. So you may want to pick up a good book and learn Lagrangians, variational calculus and Noether's theorem properly as soon as possible. Otherwise I think you may be in for a rough ride.

Could you show your work, say for the \mathcal{L}=\left( \partial_\mu \phi \right)^2 Lagrangian under the \phi \rightarrow \phi + \alpha transformation? That way it'll be more clear to us where your problems lie.