# Electron and proton charges

tom.stoer
I agree with the lattice-idea, but I woudn't integrate over all copies = ober R³ but over R³/Z³ which means. That's the idea of the 3-torus: elimination of the IR divergence.

Regarding mass or energy: no, this is different as there is no Gauss law coming from gauge invariance which requires E=0. In GR the total energy cannot be defined via a volume integral in case of arbitrary spacetimes. This is one big issue in GR - unique defintion of energy!

In GR reformulated as a gauge theory (see Ashtekar's variable in loop quantum gravity) something like that indeed happens.

DrDu
I agree with the lattice-idea, but I woudn't integrate over all copies = ober R³ but over R³/Z³ which means. That's the idea of the 3-torus: elimination of the IR divergence.
Could you elaborate on this? To avoid misunderstanding: I would also integrate only over one cell, e.g. to calculate total charge, but in this cell, there are field contributions from charges in other cells (or due to paths of non-zero winding number). To make things clearer lets consider a single point charge Q at $$R$$:
The potential at r is then $$\phi(r)\propto Q \sum_{lmn} |r-R-(al,bm ,cn)^T|^{-1}$$
where a,b, c are the dimensions of the torus and i,j,k are in Z.

tom.stoer
I understand your idea of the copies.

But as we saw this configuration (non-vanishing total charge) is rules out.

DrDu
I would be really interested about the opinion of some mathematician like A.Neumaier, who recently joined this forum, on that discussion so I thought I bring it up on the agenda once more.

A. Neumaier
2019 Award
I would be really interested about the opinion of some mathematician like A.Neumaier, who recently joined this forum, on that discussion so I thought I bring it up on the agenda once more.
(i) on some of the posts of Tom Stoer, where I disagree or have questions,
(ii) on mathman's original posting.
Note that the fact I don't comment the others does not mean that I agree with what they wrote.

As we discussed in the other thread, Tom Stoer's derivation of the neutrality of the universe implicitly assumes boundary conditions at infinity that smuggle in the desired conclusion as an assumption.

This whole operator G becomes ill-defined on a torus for Q ne 0.
...
so $$\mathbf{E}=-i \mathbf{K} \rho(\mathbf{K})/K^2+\mathbf{E}_\perp$$
I think this is not true. E lives the bosonic sector of the Hilbert space whereas the charge density lives in the fermionic sector. That means you can't solve the equation as an operator equation.
This would be the case in a free theory. But in the interacting theory, all fields act (densely, after smearing) on the whole Hilbert space of the interacting representation. Thus solving equations makes at least formally sense, as long as noncommutativity is respected. Thus your criticism does not hold water.

The [temporal] gauge isn't unsuitable. It's pÃ¼erfectlywell defined and in the context of canonical quantization it's the gauge that makes most sense! The problem is that most people are not familiar with it as standard QFT text books do only talk about Lorentz gauge.
One can find it in the QFT book by Bjorken and Drell (Vol. 2).

Now the requirement is to have gauge-invariant physical states, i.e. every physical state is a color-singulet! But this automatically means that all Q's have zero eigenvalue on physical states! Please note that this shortcut is not possible in QED as the gauge group has an abelian structure which does not immediately single out color singulets.
The quantum numbers (masses, charges, and flavors) of a particle are a property of its single particle representation, not of a particular state. Charge-neutrality means transforming to the trivial representation, while nontrivial charge says transforming according to a nontrivial representation. Thus it is not a question about eigenvectors. Physical states need not transform trivially; gauge invariance only requires that the states psi and U psi describe the same physical situation when U is a local gauge transformation. (Otherwise we wouldn't even have photons....)

The standard model requires that certain charges of different types of leptons and quarks add up to zero b/c of anomaly cancellation. w/o this perfect match the theory would have triangle anomalies in the chiral el.-weak sector which would spoil its mathematical consistency.
For the sake of definiteness, could you please write down this constraint explicitly?

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?
Phenomenologically, electrons and many ions are stable in isolation (i.e., with nothing else in the universe), so these are observable charged states of QED (when enhanced with nuclei in case of ions).

Also, we observe on a daily basis that the bigger a metal object the more charge it can hold without being unstable and discharge. Thus a universe with small but nonzero net charge is consistent with experiment (and presumably therefore also with the standard model). If the charge imbalance is too large, particles of the excess charge would move away from each other until their fields will hardly be noticable to each other. This is also consistent with QFT, due to the cluster decomposition property of particles and bound states (discussed, e.g., in Weinberg's Vol. I). Thus locally, one expects to see a rough but not exact charge balance. Which is what we actually observe.

Phenomenologically, electrons and many ions are stable in isolation (i.e., with nothing else in the universe), so these are observable charged states of QED (when enhanced with nuclei in case of ions).

Also, we observe on a daily basis that the bigger a metal object the more charge it can hold without being unstable and discharge. Thus a universe with small but nonzero net charge is consistent with experiment (and presumably therefore also with the standard model). If the charge imbalance is too large, particles of the excess charge would move away from each other until their fields will hardly be noticable to each other. This is also consistent with QFT, due to the cluster decomposition property of particles and bound states (discussed, e.g., in Weinberg's Vol. I). Thus locally, one expects to see a rough but not exact charge balance. Which is what we actually observe.
As I interpret this statement, the proton - electron charge magnitude agreement is basically observational. Are there any fundamental theoretical bases for this?

tom.stoer
The quantum numbers (masses, charges, and flavors) of a particle are a property of its single particle representation, not of a particular state. Charge-neutrality means transforming to the trivial representation, while nontrivial charge says transforming according to a nontrivial representation. Thus it is not a question about eigenvectors. Physical states need not transform trivially; gauge invariance only requires that the states psi and U psi describe the same physical situation when U is a local gauge transformation.
Before gauge fixing yes. But after full gauge fixing there is no (continuous) gauge symmetry left (it has been reduced to the identity in the physical sector of the Hilbert space - except for discrete topological gauge transformations / Gribov copies); the physical states are identical with the kernel of the associated generator of gauge transformations (here: generalized Gauss law).
My argument is of course used only in the physical sector. One could "rotate back" introducing unphysical states again, but that is not the intention.
Conclusion: after complete gauge fixing + implementation of the Gauss law constraint the kernel of the Gauss law operator is identical with the physical subspace and is identical with the singulet of the gauge symmetry.

Regarding my statement "the standard model requires that certain charges of different types of leptons and quarks add up to zero b/c of anomaly cancellation. w/o this perfect match the theory would have triangle anomalies in the chiral el.-weak sector which would spoil its mathematical consistency." you wrote
For the sake of definiteness, could you please write down this constraint explicitly?
I have no comprehensive list; I would have to compile it. One has to count all triangle anomalies generated by the chiral fermions in the standard model. One has to distinguish between different axial currents (only axial currents associacted to local gauge symmetries are relevant; the anomaly in the flavor sector is uncritical). The sum of all these anomalies have to cancel, therefore the coupling constants involved (incl. symmetry factors etc.) have to sum to zero.

A. Neumaier
2019 Award
Now the requirement is to have gauge-invariant physical states, i.e. every physical state is a color-singulet! But this automatically means that all Q's have zero eigenvalue on physical states!
Charge-neutrality means transforming to the trivial representation, while nontrivial charge says transforming according to a nontrivial representation. Thus it is not a question about eigenvectors. Physical states need not transform trivially; gauge invariance only requires that the states psi and U psi describe the same physical situation when U is a local gauge transformation.
Before gauge fixing yes. But after full gauge fixing there is no (continuous) gauge symmetry left (it has been reduced to the identity in the physical sector of the Hilbert space - except for discrete topological gauge transformations / Gribov copies)
But if there is no gauge symmetry left, your original argument breaks down since the resulting physical states (representatives of the gauge orbits) can no longer be required to have gauge-invariant physical states!

Regarding my statement "the standard model requires that certain charges of different types of leptons and quarks add up to zero b/c of anomaly cancellation. w/o this perfect match the theory would have triangle anomalies in the chiral el.-weak sector which would spoil its mathematical consistency." you wrote
For the sake of definiteness, could you please write down this constraint explicitly?
I have no comprehensive list; I would have to compile it. One has to count all triangle anomalies generated by the chiral fermions in the standard model. One has to distinguish between different axial currents (only axial currents associated to local gauge symmetries are relevant; the anomaly in the flavor sector is uncritical). The sum of all these anomalies have to cancel, therefore the coupling constants involved (incl. symmetry factors etc.) have to sum to zero.
A reference to the details would be enough. I'd simply like to check whether this implies that electron and proton charge must have equal magnitude.

tom.stoer
I'll send you some references to read the details.

DrDu
I personally was more interested in Tom's argument that on a torus the boundary conditions would exclude the existence of total charge. I had the feeling that this may be one of the arguments that zero times infinity is zero.

A. Neumaier
2019 Award
I personally was more interested in Tom's argument that on a torus the boundary conditions would exclude the existence of total charge. I had the feeling that this may be one of the arguments that zero times infinity is zero.
It seems to me that for a torus, his argument is sensible though perhaps not rigorous,
whereas in R^4 it should be wrong.

tom.stoer
On T³ there is no boundary at all; therefore integrating the Gauss law constraint equation G(x)|phys> = 0 (which is identical with gauge invariance of physical states)
is exactly Q|phys> = 0. That means that the requirement of vanishing total charge is a special case of gauge invariance.

The argument is rigorous for T³.

tom.stoer
plus references therein, esepecially
[1] F. Lenz, H.W.L. Naus, K. Ohta, and M. Thies, (1994a). Ann. Phys., 233, 17.
[2] F. Lenz, H.W.L. Naus, and M. Thies, (1994b). Ann. Phys., 233, 317.

In [1] an simple qm toy model is dicussed and an application to QED on T³ is presented. In [2] the approach is applied to QCD on T³ in axial gauge.

Then I found the following diss. http://tobias-lib.uni-tuebingen.de/volltexte/2006/2358/pdf/diss.pdf five minutes ago (in German, but I guess its OK for Arnold Neumaier :-) Looking at the table of contents I guess it provides a good introduction to the methods applied to QCD in Coulomb gauge.

A. Neumaier
2019 Award
plus references therein, esepecially
[1] F. Lenz, H.W.L. Naus, K. Ohta, and M. Thies, (1994a). Ann. Phys., 233, 17.
[2] F. Lenz, H.W.L. Naus, and M. Thies, (1994b). Ann. Phys., 233, 317.

In [1] an simple qm toy model is dicussed and an application to QED on T³ is presented. In [2] the approach is applied to QCD on T³ in axial gauge.

Then I found the following diss. http://tobias-lib.uni-tuebingen.de/volltexte/2006/2358/pdf/diss.pdf five minutes ago (in German, but I guess its OK for Arnold Neumaier :-) Looking at the table of contents I guess it provides a good introduction to the methods applied to QCD in Coulomb gauge.
Thanks, but this was not quite what I asked for. I wanted to see details for your statement that non-equal magnitude of proton and electron charge would cause an anomaly that cancels in the case of equality.

tom.stoer
Thanks, but this was not quite what I asked for. I wanted to see details for your statement that non-equal magnitude of proton and electron charge would cause an anomaly that cancels in the case of equality.
I thought you need both. Anyway - you should read the Lenz et al. papers if you are interested in canonical quantization of QCD. In addtrion I guess the Jackiw papares are very interesting. I met him a couple of times and was always very impressed.

Let's se if I can find something regarding anomaly cancelation (realted to the ABJ anomaly :-). It was always used as a reason why the top-quark MUST exist.The argument works on the level of fundamental fermions (quarks and leptons), not on the level of protons of course.

tom.stoer
@A.Neumaier: I found this paper which describes the mechanism of anomaly cancellation in section 2

http://arxiv.org/abs/hep-ph/0303191
The Top Quark, QCD, and New Physics
Authors: S. Dawson (BNL)
(Submitted on 21 Mar 2003 (v1), last revised 21 Mar 2003 (this version, v2))
Abstract: The role of the top quark in completing the Standard Model quark sector is reviewed, along with a discussion of production, decay, and theoretical restrictions on the top quark properties. Particular attention is paid to the top quark as a laboratory for perturbative QCD. As examples of the relevance of QCD corrections in the top quark sector, the calculation of $e^+e^-\to t {\bar t}$ at next-to-leading-order QCD using the phase space slicing algorithm and the implications of a precision measurement of the top quark mass are discussed in detail. The associated production of a $t {\bar t}$ pair and a Higgs boson in either $e^+e^-$ or hadronic collisions is presented at next-to-leading-order QCD and its importance for a measurement of the top quark Yukawa coupling emphasized. Implications of the heavy top quark mass for model builders are briefly examined, with the minimal supersymmetric Standard Model and topcolor discussed as specific examples.

A. Neumaier
2019 Award
@A.Neumaier: I found this paper which describes the mechanism of anomaly cancellation in section 2
http://arxiv.org/abs/hep-ph/0303191
Yes. This explains why the fractions between quark charges and the electron charge has the standard values. In particular, it explains why hydrogen is exactly neutral,
and answers the original poster''s question.

It leaves open, however, the question whether the total charge of the universe is zero.

Thanks for the discussion and the references.

tom.stoer
It leaves open, however, the question whether the total charge of the universe is zero.
Do you agree in the meantime that I have answered this question for a closed universe with compact topology?

A. Neumaier
2019 Award
Do you agree in the meantime that I have answered this question for a closed universe with compact topology?
Yes, on the level of rigor customary in theoretical physics.

tom.stoer
pooh

DrDu
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

tom.stoer
... although I basically don't understand a word of it ...
don't care; there are not so many physicists able to understand Strocchi ...

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?
It is because the charge renormalization of the proton and electron is completely determined by the photon field renormlization. Look for example Preskill QFT notes, chapter 5, pages 52-121 starting at page 12 especially.

But beware, he says that the charge renormalization is completly determined by photon field renormalization may be interpreted to mean that charge renormalization is entirely an effect of the "dielectric properties of the vaccum." Virtual pairs of charged particles tend to screen the bare charge...

Vanadium, Tom, A.Neumaier would like to have banned such loose and unrigorous talk, some would say physical reasoning, from this site. This also leads to such great answers to your questions, I guess.

tom.stoer