Electron and Proton Charges: A Fundamental Mystery or a Natural Phenomenon?

[DarMM]

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]

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.

 

[kexue]

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 referring 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?

 

[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 separately 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]

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 referring 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).

 

[Dadface]

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]

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]

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?

It is not really hijacked since the questions are related - charges are particular superselection rules, and there would have been little point in moving a discussion in progress to a new place.

Indeed, the discussion was part of answering the original post starting this thread. 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.

The superselection stuff is part of Strocchi's work, and DarMM is much more competent here than I or Tom Stoer. I hope he'll be able to explain it to our all satisfaction.

 

[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

 

[kexue]

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?

 

[Dadface]

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]

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
http://www.colorado.edu/philosophy/vstenger/void.html
(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.

 

[Dadface]

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]

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

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

 

[tom.stoer]

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

 

[DarMM]

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]

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]

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.

 

[mathman]

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]

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.
Specifically I read
@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]

As yet I got no reply to my posting #111, ...

I'll put together a reply. It is taking some time because your question has a very deep answer, linking into the nonperturbative definition of the Higgs phenomena.

 

[DrDu]

I'll put together a reply. It is taking some time because your question has a very deep answer, linking into the nonperturbative definition of the Higgs phenomena.

I am already biting my nails ...