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

[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 ...