I Electron repulsion in QFT

How do electrons repel each other in quantum field theory? Virtual photon - is it just a mathematics?
 

Nugatory

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Vanadium 50

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How do electrons repel each other in quantum field theory?
Same way they do in regular QM.
 
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This is not an easy question to answer even at the I level. And I would describe my knowledge of QED as at the intermediate level from books like the following:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

Because of that I usually let others answer this one, but now feel that it gets asked often enough I should really have a take on it. I presume you have read all the usual I level stuff to do with exchange of virtual particles etc. Forget those answers - they are at best a half truth and IMHO actually misleading. Virtual particles are simply not the same as actual particles - technically they are the lines in a pictorial representation of a Dyson series:
https://en.wikipedia.org/wiki/Dyson_series

I came across the the best answer I have seen on another site and will paraphrase it here. Virtual photons are not little balls that two particles shoot at each other. So when two electrons push each other away it’s NOT like two people shooting tennis balls at each other and getting moved by the recoil. Virtual photons are not to be treated like ‘real’ particles - remember what I said they are really just lines that appear in a pictorial representation of a Dyson series. One way of looking at them is they are like “perturbations” in the EM field caused by the presence of a charged particle. The attractive and repulsive force derive from the 'interference' of these perturbations as described in John Baez's excellent article already posted by Nugatory.

That's about the best I can do at the I level. If you want something better then you can have a look at the first few chapters of Zee's textbook on QFT where he gives a fully quantum explanation:
https://www.amazon.com/dp/0691140340/?tag=pfamazon01-20

A word of warning though - I have a copy of that textbook and once thought it very good just as most of the reviews say. Nowadays I am not so sure - IMHO Zee glosses over many important issues and basically oversimplifies a complex subject. Just my view.

My recommendation to learn QFT are (in the following order - although the first two can be read in any order)
Quantum Field Theory for the Gifted Amateur:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20
Student Friendly Quantum Field Theory:
https://www.amazon.com/dp/0984513957/?tag=pfamazon01-20
Mark Srednicki - Quantum Field Theory
https://www.amazon.com/dp/0521864496/?tag=pfamazon01-20

This is just a build up to IMHO the best, but unfortunately more advanced text by Weinberg:
https://www.amazon.com/dp/0521670535/?tag=pfamazon01-20

I personally have read a number of books on QFT and speak from experience in those recommendations. I personally am working on Srednicki right now and hope to get onto Weinberg eventually. But I am 63 and not getting any younger - so we will need to see what eventually happens in my case.

Thanks
Bill
 
Last edited:

king vitamin

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@Philipsmett, are you familiar with path integrals? In my opinion, answering your question is not so hard if one considers electrons to be non-dynamical sources of charge coupled to a free electromagnetic field - then one can show that the potential between two such sources is Coulomb's potential, with the overall sign given by the product of the charges. This is what Zee's book does, if you can get a hold of that.

Of course, this is just an approximation - real electrons are dynamical, and this effect causes QED to be highly nonlinear and nontrivial, leading to corrections to Coulomb's potential. But these corrections turn out to be small enough that they don't change whether the interaction is repulsive/attractive, so this leading-order calculation gives good intuition.

If you're somewhat familiar with how to write down a QFT in a path integral, I can go through the derivation (which I think is very similar to what you'd find in Zee if I remember correctly).
 

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