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.

 

[sardar]

Hello,

Calculating Noether currents in Peskin & Schroeder QFT involves using the mathematical framework of Noether's theorem, which relates symmetries in a physical system to conserved quantities. In this case, the symmetries are related to the invariance of the action under certain transformations, and the conserved quantities are the Noether currents.

To calculate the Noether currents, you would first need to determine the symmetries of the system and identify the corresponding transformations. Then, using the Lagrangian density of the system, which is given by equation 2.16 in Peskin & Schroeder QFT, you can apply Noether's theorem to find the Noether currents.

The specific example you mentioned involves the symmetry of translations in space and time. The transformation for this symmetry is given by $\delta \phi = a^\mu \partial_\mu \phi$, where $a^\mu$ is a constant vector. By applying Noether's theorem, you can show that the Noether current is given by $j^\mu = \partial^\mu \phi$, as stated in equation 2.16.

I hope this helps clarify the process of calculating Noether currents in Peskin & Schroeder QFT. If you require further assistance, please let me know.