# Electron and proton charges

tom.stoer
When I first read the above opening question in this thread my original thought was(and tentatively still is) that within experimental error the charges on the electron and proton are numerically equal and that charge is conserved and that total charge is zero,these things being based on observations and measurements.In other words the answer to the question is that it is a "fact of nature" based on experiments.
Now this thread has been going on for a long time with people trying to answer the question by referring to various theories ,models and equations.Most of what has been written goes enormously above my present knowledge but at least I have been able to scan and get a rough feeling of what the discussions have been about.
Throughout the discussion I have had the uncomfortable feeling,and at present it is only a feeling,that some or all of the theories referred to are based on the relevant experimental observations such as those listed above.If that's the case is it legitimate to use a theory based on experimental observations in order to explain those observations?Surely it is the observations that should inform the theories and not the other way round.
Anyway,when I get time,I will try to research the origins of these theories and the experimental evidence on which they are based.In the meantime if anyone can enlighten me I will be grateful.
Of course I am not able to comment on ALL models and explanations (mentioned in this thread) you are referring to, but of course a few general comments are in order.

There is a rather satisfactory theory called "the standard model of elementary particle physics" which subsumes our present knowledge regarding electro-weak and strong interactions as relativistic quantum field theories. Currently there is no evidence (at least not for experimentally accessible energies at colliders including the first LHC results) that this SM is at odds with known experimental facts.
[There are hints from cosmology and astrophysics that dark matter may exist which would be certainly a strong hint towards physics beyond the SM; there are numerous questions that cannot be answered withing the SM, e.g. the values of the masses and coupling constants of SM particles; but even if the SM is expected to be replaced some day by a deeper theory explaining some of the SM ad hoc inputs, nevertheless the SM passed all direct experimental tests as of today]

Afaik all above mentioned models and explanations are nothing else but certain applications, interpretations or formulations of the SM. That means we tried to show how the SM explains (or at least motivates) the experimental fact that the electric charge of electron and proton are numerically equal. There was and still is a strong and fruitful interplay between theory an experiment that led over decades to the formulation of the SM. Therefore you should not worry about the observations and the theoretical explanations we discussed.
[coming back to DM and a possible explanation via SUSY: I don't think that the arguments presented here would change so much when taking SUSY into account]

A. Neumaier
2019 Award
there are not so many physicists able to understand Strocchi ...
Well, I don't understand everything but at least some things in his papers.

Strocchi echoes (in many of his papers) more or less your arguments about why under certain assumptions the Gauss law implies the absence of charged states, but puts them into the framework of axiomatic field theory (where the completely different notation and terminology makes things look very different). This results in theorem that precisely specify the assumptions that go into the conclusions.

Prompted by a remark in a different thread,
it is well known that Wightman axioms are very difficult to satisfy, and actually impossible in gauge field theories (http://arxiv.org/abs/hep-th/0401143).
(for my first reactions to this remark see the discussion there), I started to look into the evidence Strocchi referenced there. I am still reading, and hope to present my findings later in this thread.

His formal exposition is generally rigorous (if one ignores somewhat looser talk in the introductions), but I find fault with his informal conclusions, since they are based on interpreting assumptions (stated in his theorems) that are far from trivial and by no means only formal translations of properties necessary for the real thing.

In particular, at present I don't think his no-go theorems are relevant for theories (like QCD) expected to have a mass gap. The situation may be different (i.e., not of Wightman form) for theories like QED that have massless asymptotic fields, because then the asymptotic states carrying the scattering physics cannot be described in a Fock representation but need more general coherent representations of the CCR (and different such representations for asymptotic states of different velocity).

The article mentioned by
I found an interesting article on that subject (although I basically don't understand a word of it), by the specialists on that field:
http://arxiv.org/abs/hep-th/9705089

A superselection sector is essentially an orbit of the algebra of local observables on a representing Hilbert space. (This is usually expressed by saying that superpositions between different superselection sectors are forbidden. See, e.g., http://en.wikipedia.org/wiki/Superselection) Thus it characterizes the asymptotic structure of a theory, capturing in particular the boundary conditions at infinity that we had been discussing.

The OP asked a very interesting and obvious question that QFT definetely should be able to answer. And of course it does.

But the same people that yelled at me in some other thread that 'virtual' particles explain nothing, can not answered it, but digress in some obsure discussions, whereas exactly these very virtual particles explain very, very well what goes on here.

Why is the charge of a proton and electron equal? It is because the charge renormalization of the proton and electron is completely determined by the photon field renormlization!

That's all. That is the explanation, the only explanation.

Check the John Preskill notes, or even better one of the leading textbook of QFT, Zee "QFT in a Nuttshell", the chapter called "Polarizing the vacuum and renormalizing the charge".

A. Neumaier
2019 Award
Why is the charge of a proton and electron equal? It is because the charge renormalization of the proton and electron is completely determined by the photon field renormalization!

That's all. That is the explanation, the only explanation.
That's no explanation at all. Your argument would also ''explain'' that the charge of an alpha particle is equal to that of an electron, since it doesn't make any use of any special property of the proton.

Stop polluting serious discussions with your superficial views!

To the OP, the answer to your question can be found in "QFT in a nutshell" of Anthony Zee, chapter III.7. or in John Preskill QFT lecture notes, second part of chapter five.

(And could someone, maybe a mentor explain to me if it is ok on PF to be accused of polluting the discussion when providing the standard textbook answer to an asked question?)

DrDu
Kexue, from my limited understanding I would say that your explanation only explains why the ratio of the bare charges of the electron and the proton equals the ratio of the physical charges. However, the discussion of A Neumaier and tom stoer is about why this ratio is 1.

Of course I am not able to comment on ALL models and explanations (mentioned in this thread) you are referring to, but of course a few general comments are in order.

There is a rather satisfactory theory called "the standard model of elementary particle physics" which subsumes our present knowledge regarding electro-weak and strong interactions as relativistic quantum field theories. Currently there is no evidence (at least not for experimentally accessible energies at colliders including the first LHC results) that this SM is at odds with known experimental facts.
[There are hints from cosmology and astrophysics that dark matter may exist which would be certainly a strong hint towards physics beyond the SM; there are numerous questions that cannot be answered withing the SM, e.g. the values of the masses and coupling constants of SM particles; but even if the SM is expected to be replaced some day by a deeper theory explaining some of the SM ad hoc inputs, nevertheless the SM passed all direct experimental tests as of today]

Afaik all above mentioned models and explanations are nothing else but certain applications, interpretations or formulations of the SM. That means we tried to show how the SM explains (or at least motivates) the experimental fact that the electric charge of electron and proton are numerically equal. There was and still is a strong and fruitful interplay between theory an experiment that led over decades to the formulation of the SM. Therefore you should not worry about the observations and the theoretical explanations we discussed.
[coming back to DM and a possible explanation via SUSY: I don't think that the arguments presented here would change so much when taking SUSY into account]
Thanks tom,I know very little about the SM but your post has triggered my interest and I intend to have a closer look at the subject.Let me try to clarify the point I tried to make in my previous post:
As I understand it is experiments such as deep inelastic scattering that provide evidence for SM and the existence of,for example,quarks and that these quarks have charges such as plus or minus 1/3e or plus or minus 2/3e.Now the scattering experiments do not measure these charges(as far as I am aware) but it is experiments carried out separately and which started before the advent of SM which measured(measure) these charges.When it was concluded that quarks have the fractional charges that they are assumed to have was this not based on the previously gained evidence that the electron charge and proton charge are numerically equal?In other words can the SM or any other theory explain the equality of these charges when the theory itself is partly based on the experimental evidence that the charges are equal?

A. Neumaier
2019 Award
When it was concluded that quarks have the fractional charges that they are assumed to have was this not based on the previously gained evidence that the electron charge and proton charge are numerically equal?In other words can the SM or any other theory explain the equality of these charges when the theory itself is partly based on the experimental evidence that the charges are equal?
The values of the fractional charges for the quarks u and d come from the fact that p=uud and n=udd must have charge e and 0, respectively, where e is the proton charge. This is independent of the question whether the electron charge is -e. The latter follows in the standard model from the fact (discussed earlier in this thread) that the triangle anomaly cancels.

There is no consistent variant of the standard model in which this could be relaxed. Thus the anomaly cancellation provides a theoretical explanation for the initially only empirically observed fact that the hydrogen atom seems to be exactly neutral, equivalently that electron and proton charges are equal and opposite.

A. Neumaier
2019 Award
To the OP, the answer to your question can be found in "QFT in a nutshell" of Anthony Zee, chapter III.7. or in John Preskill QFT lecture notes, second part of chapter five.

(And could someone, maybe a mentor explain to me if it is ok on PF to be accused of polluting the discussion when providing the standard textbook answer to an asked question?)
Pollution referred to bringing virtual particles into a completely unrelated discussion.

That I answered at all was to clarify that your alleged standard textbook answer was a spurious argument that didn't contribute anything to answering the question of the OP.

Those participating in discussions should contribute answers only if they really understood the problem discussed (which is surely not the case if a serious thread seems ''almost comical''), while ignorance (or hearsay knowledge such as yours) should be contributed in the form of questions (or requests for correction).

Pollution referred to bringing virtual particles into a completely unrelated discussion.

That I answered at all was to clarify that your alleged standard textbook answer was a spurious argument that didn't contribute anything to answering the question of the OP.
..
The question of the OP was

The charge of an electron is exactly equal in magnitude to that of a proton (2 up quarks plus down quark). What is the theoretical basis for this, or is essentially a fact of nature that is accepted?
I gave him the answer and the reference where he can check it.

Still in post 56 the OP (mathman by the way) asked

As I interpret this statement, the proton - electron charge magnitude agreement is basically observational. Are there any fundamental theoretical bases for this?
So he was still in the dark after 55 posts why protons and electrons have equal charges. Again, if someone polluted this thread with disgressions into mathematical arguments and contributed very little in answering the OP answer, it must be clearly you.

Comical is that Preskill and Zee who provide the answer in their texts both use virtual particles to explain this curious fact why protons and electrons have equal charges.

Comical because you and other posters in that thread who so fiercly reject any usefulness of the concept of virtual particles could not answer the question.

DarMM
Gold Member
In case anybody is wondering Strocchi's theorem proves a very restricted statement. In QED we have the Electron fields and the Photon Fields. Strocchi shows that if you assume:
(a) $$\Psi$$ and $$A_{\mu}$$ are Wightman Fields.
(b) They obey Maxwell's equations
(c) They are covariant
(d) Gauge invariance holds.

Then physical charge vanishes.

However this isn't really a problem. What the theorem is "really" proving is that there are no gauge invariant, local, covariant Wightman fields. QED can be described in terms of Wightman fields which are not covariant or in terms of Wightman fields which describe the dynamics directly, the "physical fields", instead of the Lagrangian fields. The problem with these is that non-covariant fields lose manifest covariance. The physical fields would make the Hamiltonian look hideous and you can't see gauge-invariance (similar to describing QCD with proton fields, the Hamiltonian would be infinitely long and gauge invariance of quarks and gluons would be invisible). You could also work with non-local objects like "Wilson loops" whose algebra would satisfy the Haag-Kastler axioms, but this would be even more difficult.

This is a problem for perturbation theory where you would like a covariant, local field for doing calculations. So if you want to do that you need to drop some assumption. Commonly we drop (b) and obtain an enlarged Hilbert space of states on which Maxwell's equations do not hold. In some subspace of this space they hold, the physical Hilbert space. This subspace is then specified by the Gupta-Bluer condition. (In Yang-Mills theories the interaction makes the condition more subtle and you need to enlarge the Hilbert space even further to obtain a simple linear condition. The correct enlargement is to include fermions with incorrect statistics, which you will know as ghosts.)

So we perform calculations in this enlarged Hilbert space, where we are allowed use a local, covariant field and compute physical state -> physical state processes.

Of course one could just calculate in the physical Hilbert space, but standard perturbation theory would be impossible, but the Wightman axioms do hold for the physical fields.

If anybody is afraid of the lack of rigour here, since I assume QED exists*, just pretend I am talking about QED in 2,3 dimensions where it does exist. Or consider the electron field to be classical in 4D. Or perhaps take my remarks in the Yang-Mills case.

*Which some doubt due to triviality.

DarMM
Gold Member
The charge of an electron is exactly equal in magnitude to that of a proton (2 up quarks plus down quark). What is the theoretical basis for this, or is essentially a fact of nature that is accepted?
A silly answer maybe, but a quantum field theory containing SU(3) quarks and U(1) leptons will only be sensible (that is have no infinities) if the quarks in each generation have +2/3 and -1/3 charge. The quarks will always be able to produce a particle with spin-1/2 and charge +1 by group theory. So your question can be reduced to why does the world involve SU(3) and U(1).

tom.stoer
A silly answer maybe, but a quantum field theory containing SU(3) quarks and U(1) leptons will only be sensible (that is have no infinities) if the quarks in each generation have +2/3 and -1/3 charge. The quarks will always be able to produce a particle with spin-1/2 and charge +1 by group theory. So your question can be reduced to why does the world involve SU(3) and U(1).
Isn't this a simplification of the "anomaly cancellation argument"?

A. Neumaier
2019 Award
In case anybody is wondering Strocchi's theorem proves a very restricted statement. In QED we have the Electron fields and the Photon Fields. Strocchi shows that if you assume:
(a) $$\Psi$$ and $$A_{\mu}$$ are Wightman Fields.
(b) They obey Maxwell's equations
(c) They are covariant
(d) Gauge invariance holds.

Then physical charge vanishes. [...]

This is a problem for perturbation theory where you would like a covariant, local field for doing calculations. So if you want to do that you need to drop some assumption. Commonly we drop (b)
I wonder why the assumption (a) is reasonable: Since A(x) is an unobservable, gauge-dependent field, I don't see any reason to suppose that it must be a Wightman field.
Both the requirement of causal commutation rules for A(x) and the requirement of Lorentz invariance for A(x) seem to be not gauge covariant, hence can hold, if at all, only in special gauges. But both are part of the assumption that A(x) is a Wightman field.

Is there anything left from Strocchi's assertions in his many papers on the subject if one drops these two assumptions?

A. Neumaier
2019 Award
I gave him the answer and the reference where he can check it.

Still in post 56 the OP (mathman by the way) asked

So he was still in the dark after 55 posts why protons and electrons have equal charges.
The appropriate answer was clear after post number #65-#67.

You entered the discussion in post #73 after the question had been fully settled (and the thread had continued with discussing some other issues raised on the way), and provided an ''answer'' that doesn't hold water since it applies equally to arbitrary charged particles in place of the proton, no matter what their charge. But you are so sure of yourself that you don't even admit it after it was pointed out by Dr.Du that you made an obvious mistake.

I knew already that it is futile to argue with you, and this will be my last comment on this.

DarMM
Gold Member
I wonder why the assumption (a) is reasonable: Since A(x) is an unobservable, gauge-dependent field, I don't see any reason to suppose that it must be a Wightman field.
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".

Is there anything left from Strocchi's assertions in his many papers on the subject if one drops these two assumptions?
No not really. Strocchi is mainly concerned with issues that arise in a rigorous study of gauge theories that don't occur in other field theories. For example the theorem above simply shows that $$A_{\mu}$$ isn't a Wightman field so a rigorous treatment will not be as straight forward. Theorems like the above are also used to show where certain objects from formal field theory orginate from in a rigorous approach. 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.

DarMM
Gold Member
Isn't this a simplification of the "anomaly cancellation argument"?
Yes, indeed it is. I should have read the first page, I see you said pretty much the same thing.

DrDu
I would like to learn more about the connection between gauge fields and charge superselection; a problem which has turned up in this thread, too. However I don't want to hijack it. Therefore I started a new one in the quantum theory forum:
How does a gauge field lead to charge superselection?

Maybe someone can give me an idea of this anomaly cancellation argument.

The appropriate answer was clear after post number #65-#67.

You entered the discussion in post #73 after the question had been fully settled (and the thread had continued with discussing some other issues raised on the way), and provided an ''answer'' that doesn't hold water since it applies equally to arbitrary .....
Well, why the bare charges equal exactly, Zee says in grand unified theories it can be shown that this follows from group theory.

But I was refering to that given the bare charges are completetly equal, and given that a proton is a composite of quarks and participates in strong interaction whereas an electron does not, how come that both their charges are still exact equal.

The answer to that was not given in this thread before my post.

And where and when was the discussion fully settled regarding OP's question? What now is the fully settled answer to OP's question according to you?

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tom.stoer
Maybe someone can give me an idea of this anomaly cancellation argument.
The anomalies in 4-dim. quantum field theory are usually contained in so-called triangle graphs with three external gauge boson lines and three inner fermion lines forming a triangle. This graph is divergent and has to be renormalized. It contributes to two different continuity equations (in quantum field theory replaced by so-called Ward identities). Now one finds that in order to keep one current conservation law one must violate the other one; the renormalization is not able to protect both conservations laws which means that one current becomes anomalous.

Usually one choses the renormalization such that the gauge current (e.g. the electromagnetic current) derived from a local gauge symmetry remains conserved whereas the other current (axial current) derived from a global symmetry becomes anomalous (the two currents are due to the fact that one can project to left- or right-handed fermions; therefore instead of calling in axial anomaly sometimes one refers to it as chiral anomaly). The reason for gauge current conservation is renormalizibility, i.e. consistency of the theory. The anomaly itself has physical effects which can be seene.g. in pion decay and the mass of the eta-prime meson.

Now in electroweak interactions the left and the right handed currents become gauge currents which are conserved seperately in classical field theory. But due to the above arguments that means that one can no longer protect both gauge symmetries in the current conservation b/c one must necessarily break gauge invariance either in the left or in the right handed sector.

That would mean that the theory becomes inconsistent, but there is one way to protect both gauge symmetries in the left- and in the right-handed sector. Roughly speaking each fermion species comes with its own triangle anomaly. But the external gauge bosons do not carry any fermion information which means that in order to calculate the total contribution of the triangle graphs to the current conservation one has to sum over all triangle graphs. Each triangle comes with a pre-factor that is related to the (electroweak) charges of the inner fermion in that graph. So the sum over all graphs vanishes iff the sum over these pre-factors vanishes which results in a constraint for the electroweak charges of the fermions.

In the SM the anomaly has to cancel in each generation, which essentially means that given the electric charge of the fermions (up, down, e, e-neutrino) and the multiplicity of the fermions in the graph (e.g. counting different colors) the electric charges must fulfill certain consistency conditions. In addition it means that one generation has to be complete. That was one reason for the existence of the top quark: an incomplete 3rd generation (., bottom, tau,tau-neutrino) would cause the gauge current to become anomalous whereas a complete 3rd generation (top, bottom, tau,tau-neutrino) saves the consistency.

So once one knows the electric charge of the electron the charges of up and down are essentially fixed: afaik one can derive both Q(d) = -Q(u)/2 and Q(d) = Q(e)/3; the last equation is related to the fact that there are three colors - and the different colors are counted individually in the triangle diagrams.

But using these equations one automatically finds that Q(proton) = 1 and Q(neutron) = 0. In addition one finds Q(proton) = - Q(electron).

I'lltry to find some references where all this is derived rigorously.

DrDu
Thank you Tom, that's quite interesting.

A. Neumaier
2019 Award
Well, why the bare charges equal exactly, Zee says in grand unified theories it can be shown that this follows from group theory.

But I was refering to that given the bare charges are completetly equal, and given that a proton is a composite of quarks and participates in strong interaction whereas an electron does not, how come that both their charges are still exact equal.

The answer to that was not given in this thread before my post.
Because nobody had asked about that. Reducing the OPs question about real particles to that for bare (nonexistent) particles is not a useful contribution to the discussion.

And where and when was the discussion fully settled regarding OP's question? What now is the fully settled answer to OP's question according to you?
Charges are equal up to sign because the triangle anomaly of the standard model must cancel, a consistency condition without which the SM would not be well-defined.

Yes, it follows from group theory, and in a way that makes all your virtual particle talk look silly, because the cancellation says that effects involving Feynman diagrams in which (according to you) virtual particles are created and destroyed are in fact completely absent (and must be so for consistency reasons).

Thanks A.Neumaier(post 83) and thanks tom.I'm struggling with this but as I understand it cancellation of the anomaly results in the charges on uud minus the charge on e being zero.That's impressive stuff.
I want to know more so can anyone recommend a book or article on the "S.M. for dummies"

A. Neumaier
2019 Award
Thanks A.Neumaier(post 83) and thanks tom.I'm struggling with this but as I understand it cancellation of the anomaly results in the charges on uud minus the charge on e being zero.That's impressive stuff.
I want to know more so can anyone recommend a book or article on the "S.M. for dummies"
There are dummies of different degrees.
What's your background in math and physics?

A. Neumaier