# Electron and proton charges

tom.stoer
I can recommend some books - but be aware that it's about 20 years ago that I have studied them, so they may be out-dated:
Cheng / Li: Gauge Theory of elementary particle physics
Quigg : Gauge Theories Of Strong, Weak, And Electromagnetic Interactions
Griffiths: Introduction to Elementary Particles
Kane: Modern Elementary Particle Physics

Now that this is answered to everyone but kexue's satisfaction, we are just clearing up what remained of the surrounding questions brought up by the discussion.
What makes you think that the question was answered to everyone but kexue's satisfaction? The OP droped out long ago.

I pointed out to the OP and everybody else in this thread that in the best-selling graduate-level QFT book of Tony Zee the answer to OPs question can be found.

- Why are the bare charges equal?
Because of group theory, check page 394 of Zee (1. ed.).

- How do we know that quantum fluctuations would not make the charges slightly unequal? (After all, the proton participates in the strong interaction and electron does not)
Because charge renormalization depends completely on photon renormalization, check page 189 of Zee(1.ed.).

The last point is also mentioned in Preskill's QFT notes.

Why am I then not contributing to this thread when providing this information?

Does A.Neumaier own PF now?

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There are dummies of different degrees.
What's your background in math and physics?
I was a student back in the 1960s and most of my physics and maths has been forgotten mainly due to lack of use.I am retired now and have the luxury of being able to choose what parts of physics to look at these being the topics that interest me.As far as the S.M. is concerned I just want an overview,ideally with the maths content at a minimum,the sort of non brain straining article that one might read in a magazine such as Focus or Scientific American.
Thanks tom,I have seen Griffiths mentioned so many times in these forums that I am going to have a look at it.

A. Neumaier
2019 Award
As far as the S.M. is concerned I just want an overview,ideally with the maths content at a minimum,the sort of non brain straining article that one might read in a magazine such as Focus or Scientific American.
http://en.wikipedia.org/wiki/Standard_model is a good entry point - the main facts quickly summarized.

Chapters 9–12 of Stenger's book Timeless Reality
(some chapters are free online there) is almost formula-less
(and correspondingly superficial, directed to laymen).

Volume 5.9 of the online book
http://www.motionmountain.net/contents.html
is a bit more demanding but still very casual.

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Thank you so much A.Neumair.

DrDu
Has this argumentation on anomaly freeness also been formulated in a non perturbative setting like algebraic qft or does there exist a soluble toy model?

A. Neumaier
2019 Award
Has this argumentation on anomaly freeness also been formulated in a non perturbative setting like algebraic qft or does there exist a soluble toy model?
There is an algebraic setting, but it is still perturbative.
https://www.amazon.com/dp/0471414808/?tag=pfamazon01-20&tag=pfamazon01-20

Nobody knows how to make sense of the standard model nonperturbatively.

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tom.stoer
I think that in principle it should be possible to formulate the Fujikawa method non-perturbatively.

DarMM
Gold Member
I think that in principle it should be possible to formulate the Fujikawa method non-perturbatively.
Fujikawa's method is nonperturbative since it uses directly the path integral itself, not Feynman diagrams. However it is not rigorous for two reasons:
(a)Nobody knows if the path integral exists in four-dimensions.
(b)He treats the field Lesbesgue measure and the action separately. This is incorrect because the field lesbesgue measure does not exist as a measure on field space and the Action does not exist as a function (it exists technically but is undefined almost everywhere). Only their combination exists as a measure.

(b) isn't too great a problem. Simply prove the anomaly exists on the lattice, where Fujikawa's method is justified since then the Lesbesgue measure and the Action exist in the way he assumes.
However you still run into (a) where we don't know if the continuum path integral exists.

Of course if you are not concerned with issues of rigour, then Fujikawa's method is a satisfying nonperturbative argument.

A. Neumaier
2019 Award
Exactly, in a sense Strocchi's theorem isn't really that surprising. Even Strocchi himself in some of his books makes this point, also see the book by Steinmann "Perturbative QED and Axiomatic Field Theory".

[...] So Strocchi and others such as Nakanishi show that the Gupta-Bluer condition and ghosts arise from trying to work with a field as "Wightman-like" as we can manage.

I don't think Strocchi is really pointing anything out, more just showing where naïve assumptions from formal field theory go wrong and what is really going on behind the scenes.
Thanks for the confirmation.

DrDu
Dear DarMM,

can you give me a picture why and how the gauge symmetry in QED leads to a superselection rule for charge and why this argument breaks down when the symmetry is broken, like in a superconductor?

In "Local quantum field theory" by R. Haag he goes as far as claiming that the usefulness of gauge symmetry results from the fact that we observe charge superselection. I think that is very interesting, especially as I never understood what is the deeper reason behind insisting on gauge theories.

A. Neumaier
2019 Award
The OP droped out long ago.
No, he is still following the thread, as seen from:
There is a long ongoing discussion in this forum on this subject (I started it it).
and doesn't seem to have further questions.

Although I have tried to follow the thread, I got lost in the details of the theoretical physics. I understand that beta decay makes the neutron - proton charge difference equal to the electron charge, but does the neutron neutrality have a fundamental theoretical basis? The question may have been answered, but I have a problem with the details of the physics arguments.

tom.stoer
Short summary:
There are processes in the SM violating certain symmetries that are valid classically (Noether theorem - current conservation) via so-called anomalies (essentially triangle diagrams in Feynman diagrams). There are anomalies which are welcome b/c they explain certain physical effects (pion decay, eta' mass); these anomalies are usually due to global symmetries. Then there are anomalies which must not exist as they would spoil the consistency of the SM; these anomalies are due to local gauge symmetries. In QED there is no gauge anomaly as the left- and right-handed fermions contribute with opposite sign and therefor the anomalies cancel exactly. But in the el-weak interactions the left- and right-handed fermions couple differently to the gauge bosons which means that the anomalies do no longer cancel trivially but that there are non-trivial consistency conditions, a set of algebraic relations between particle-type specific parameters which are essentially the charges of these particles. Solving this consistency conditions results (besides other physical predictions) in a relation saying
q(u) = 2/3 q(e)
q(d) = -1/3 q(e)
where the 1/3 is due to the fact that each quark is counted three times b/c it exists in three different colors. That means that due to these algebraic relation q(proton) = -q(electron). Then there was a last statement that the algebraic relations itself are valid at higher looporder, i.e. that the electric charges of the individual particles scale identically under the renormalization group. That means that once the ratio between two charges is fixed, it remains fixed at all orders in perturnation theory.

A. Neumaier
2019 Award
Solving this consistency conditions results (besides other physical predictions) in a relation saying
q(u) = 2/3 q(e)
q(d) = -1/3 q(e)
where the 1/3 is due to the fact that each quark is counted three times b/c it exists in three different colors. That means that due to these algebraic relation q(proton) = -q(electron).
And this also implies that the neutron is exactly neutral, answering mathman's question.

DrDu
As yet I got no reply to my posting #111, I went on reading and think I found some explanations which are nicely in line with the current discussion.
@article{wightman1995superselection,
title={{Superselection rules; old and new}},
author={Wightman, AS},
journal={Il Nuovo Cimento B (1971-1996)},
volume={110},
number={5},
pages={751--769},
issn={0369-3554},
year={1995},
publisher={Springer}
}

and

@article{strocchi1974proof,
title={{Proof of the charge superselection rule in local relativistic quantum field theory}},
author={Strocchi, F. and Wightman, A.S.},
journal={Journal of Mathematical Physics},
volume={15},
pages={2198},
year={1974}
}

The first article by Wightman is an easy read also for the non-specialist in field theory (like me) while the second one is highly technical.
The basic argument ( as far as I understood it) is that a global gauge symmetry leads to the existence of a conserved charge. If the gauge symmetry is furthermore local, this does not lead to any new conserved quantity but the charge current vector can be written as $$j^\mu=\partial_\nu F^{\mu \nu}$$ (forgive me potential sign errors) which encompasses Gauss law for the charge density.
Now as we already discussed Gauss law allows to express the charge inside a volume to be expressed in terms of the electric field on the boundary. But the electric field on the boundary will commute with all operators localized inside the region. Hence the charge commutes with all local operators which and is thus a classical quantity. That is precisely the statement of supersymmetry. In formulas:

$$\int dV [\rho(x), A]=\int dV [\text{div} E(x), A]=\int dS\cdot [E, A]=0$$
where A is any (quasi-) local operator.

Now this argument is not precise as Gauss law does not hold as an operator equation. Hence in the second article Strocchi and Wightman use the Gupta Bleuler formalism.
The argument still assumes that the total charge can be represented as a unitary operator. This statement breaks down if the symmetry is broken.

After Goldstones theorem there was a lot of discussion how it can be avoided leading eventually to the Higgs mechanism. As far as I understand, the condition for Higgs mechanism to apply coincide with the presence of a superselection rule in the unbroken case.

DarMM