Does the electron field carry information about its charge

[ftr]

Does the electron field has information about its charge(strength) throughout space and if not why not?
I hope this question is not vague.

 

[DaveC426913]

Do you mean: can a proton or other charged particle know the electron's charge? Yes. They interact electromagnetically. I hope I am not interpreting your question too simplistically.

 

[bhobba]

Yes.

But it's tied up with renormalisation.

Heuristically the virtual cloud of electron and positrons around the electron exert a screening effect meaning the closer you get the stronger the charge. Note this is just heuristically - virtual particles actually do not exit - but is a fiction told to beginners so they have a bit of a grasp about what's going on.

In fact its part of the infinities that plague QFT that re-normalization is required to fix (using the usual methods anyway - it can be done without this issue but that as whole new thread).

If we get too close it becomes infinite leading to the so called Landau pole:
https://en.wikipedia.org/wiki/Landau_pole

This is a big problem - or would be except long before the Landau pole is reached QED is replaced by the electro-weak theory.

Interestingly I believe the electro-weak theory has its own Landau pole and how that is resolved someone did explain to me once, but my knowledge wasn't good enough, at least at the time, to understand it. My memory is it had something to do with the Higgs, but don't hold me to it. Also computer calculations show it may not actually exist - but again an expert with greater knowledge than me is needed.

Oh - I forgot to mention if the electro-weak Landau pole does exist and/or is an actual problem it's way below the Plank scale where we know our physics breaks down anyway.

Thanks
Bill

 

[radium]

I think they key idea in this is the Ward-Takahashi identity. For correlation functions, this comes as consequence of the invariance of the functional measure of the path integral under gauge transformations. The Ward-Takahashi identity is basically quantum version of the classical Noether's theorem with a current, but now you also have contact terms.

In the context of renormalization, one direct consequence is the equality Z_{1}=Z_{2}: charge renormalization comes entirely from the photon field renormalization and the charge current is NOT renormalized. Additionally, this implies that the ratio of the charges of a quark and an electron for example are not renormalized.