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

[tom.stoer]

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.

 

[tom.stoer]

So, when I find discussion of Lorentz invariance and Gauss' Law, I am not surprised to find conflict.

There is no conflict.

The problem seems tobe that chosing e.g. the temporal gauge (or the Coulomb gauge) the equations are no longer Lorentz invariant explicitly; one has to check Lorentz invariance explicitly. This has to be done on the level of the operator algebra for the Poincare generators H, Pⁱ, Lⁱ and K_i. After very many pages of boring calculations one finds that the algebra still closes w/o anomaly.

So Lorentz invariance still holds.

btw.: I don't know whether there's confision between Lorentz invariance and Lorentz gauge.

 

[DrDu]

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.

I was arguing classically.
You may use \mathbf{E}_{||}+i \mathbf{K} \rho(\mathbf{K})/K^2=0 as the constraint.
It can be seen that the longitudinal part of the electric field has to be divergent.
My question is whether this really indicates that solutions with Q ne 0 aren't admissible or whether this is an artifact of splitting a non-divergent field E into divergent longitudinal and transverse parts as a consequence of imposing an unsuitable gauge?

 

[FizzyWizzy]

I apologize. I didn't mean to suggest there was conflict between Gauss' law and Lorentz invariance, but that conflict in discussions concerning these topics tends to exist. (Conflict having a more social than mathematical meaning.) In fact, I often refer to the "Gauss model" (terminology adopted by my advisor) in my dissertation work, and my model being in conflict with it. At the time, I could not explain all the differences, but in the past few months I have converged on a substantial understanding and have pinpointed it to a single single term that approaches a negligible fraction of the total (potential) energy of the system as N grows very large -- hence, in agreement with the textbook.

I, therefore, wanted to draw out the notion that Gauss' law and Lorentz invariance may include implicit assumptions about the nature of a given system. Quantum theory, for instance, appears almost exclusively concerned with dynamic (implicitly read as "statistical") properties while electrostatics is purely static (and perhaps somewhat non-physical in its assumption as well -- though my position is that the electrostatic configuration is that toward which a given N-charge system is driven).

Further still, I see clearly that a distinction must be made between large N and few N systems -- perhaps insofar as all these models/theories are concerned. DFT works well in some cases, while QMech. works better in others. My model (which could potentially develop into a nice, new theory) appears to be a bit more like DFT, but instead of merely knowing (trial) density or wave functions, we might look for a more fundamental spatial symmetry function (of point charges) from which both a wavefunction and density functional may be obtained with greater precision -- if not, dare one say, exactness.

So, when we discuss electron and proton charge -- as is the intent of this thread, do we wish to expound on mathematics concerning continuous charge distributions? or should we constrain ourselves to the discrete nature of charge? If so, to which mathematics and theories are we to resort? I am not convinced that Gauss' Law is explicitly meant to be concerned with discrete, few N systems. I can see how it may apply to N=1, but even then, I personally see a factor of 2 that must be involved. -- though I haven't worked through Gauss' law carefully enough to see that it applies explicitly to N=1. At a glance, I think we must make some further assumptions about the single charge itself in order to fully justify Gauss' law in the discrete regime. One such assumption is that an electron or proton's charge is continuously distributed -- or perhaps not. Does it matter if it is a twisted "string" of charge? or a shell/sheet? or a broken sheet that spins incessantly about an axis and generates a magnetic moment? ...whereupon in free space, if an electron spins in this classical manner, does it really have a magnetic field (if we do not have a reference frame)? Key to this line of thought is interaction. Now one may be considered in reference to the other, and the magnetic moment certainly plays a role.

I just think there is so much more to be learned and understood. Perhaps existing models and ideas need to be carefully revisited, reworked, -- or something new from scratch.

 

[tom.stoer]

You may use \mathbf{E}_{||}+i \mathbf{K} \rho(\mathbf{K})/K^2=0 as the constraint.
It can be seen that the longitudinal part of the electric field has to be divergent.

I still can't see why there should be something divergent.

OK, let's do it that what. First we restrict to compact one-dim. space = to a circle which is T¹ = S¹ instead of T³. You can introduce creation and annihilation operators for the E and the fermion fields. Then can transform to momentum space. The constraint

G(x) = \partial_x E(x) + \rho(x)

is translated as follows

G(x) = \sum_n G_n e^{ip_nx} = \sum_n \left[(ip_n) E_n + \rho_n\right] e^{ip_nx}

with p_n \sim n/L

\rho_n is bilinear in the b_{k+n} and b^\dagger_k

Of course - as you already said -

\rho_0 \sim Q

The constraint is translated as follows

G_n|\text{phys}\rangle = 0 \quad \forall n

But for n=0 the E-field drops out due to the combination (nE_n)_{n=0} and one finds

Q|\text{phys}\rangle = 0

My question is whether this really indicates that solutions with Q ne 0 aren't admissible or whether this is an artifact of splitting a non-divergent field E into divergent longitudinal and transverse parts as a consequence of imposing an unsuitable gauge?

The gauge isn't unsuitable. It's püerfectlywell 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 textbooks do only talk about Lorentz gauge.

The aim is to eliminate the longitudinal part of the gauge field (which means to move it to the unphysical sector of the Hilbert space). The presence of the constraint G(x) ensures that these unphysical degrees of freedom of the A- and E-field stay within this sector under time evolution (w.r.t. to the physical Hamiltonian); so an unphysical state stays unphysical and doesn't mix with the physical sector.

 

[tom.stoer]

@FizzyWizzy: I hope it becomes clear what the context of my arguments is: it's QED with both electrons and photons being quantized [it my no longer be if one studies quantized electric particles in a classical el.-mag background field; and it my break down if one couples quantized photons to static - infinitly heavy - electric charges]. But QED is the most general context I can think about.

I think one can show that my argument is valid in all cases (1-dim., 3-dim., different topolgies etc.) All what happens is that in non-compact cases one may get surface charges and somekind of background fields. I think this is the only way to escape from the Q=0 conclusion.

 

[tom.stoer]

One can use even simpler (!) arguments in the case of non-abelian gauge theories.

Usually the calculations in QCD become awfully complicated. But I think I can provide a short cut. Again one finds a Gauss law constraint which now lives in color space. It reads

G^a(x) = \partial_x E^a(x) + \rho^a(x)

where a=1..8 is the SU(3) color index and the charge density has a quark and a gluon contribution (the latter one being the special ingredient of the non-abelian gauge group)

The Gauss law operators satisfy a local SU(3) algebra, i.e.

[G^a(x), G^b(y)] = if^{abc}G^c(x) \delta(x-y)

Again one can integrate the Gauss law constraint and derive the global SU(3) algebra

[Q^a, Q^b] = if^{abc}Q^c

Now comes the funny thing: As G(x) generates "topologically small" local gauge transformations, Q simply generates "global" gauge transformations, i.e. gauge transformations where the gauge parameter is space-time independent.

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.

So in QCD the color-neutrality is an almost algebraic property following directly from the local algebra of the "color-electric" Gauss law.

------

Of course this reasoning remains valid in 3+1 dim. spacetime

 

[DrDu]

Sigh. You don't even try to understand my argument. Did you ever do classical electrodynamics?

Ok, let's try me another way of argumentation: You say that this argumentation applies to all kind of massless gauge bosons. However, if a symmetry gets broken, the symmetry broken state is one of unsharp charge. If only the state with Q=0 is available, how can I end up by symmetry breaking in a state in whicha measurement of Q may yield something different from 0?

 

[tom.stoer]

You don't even try to understand my argument. Did you ever do classical electrodynamics?

I tried to, but I think it's not relevant as soon as you quantize the theory. The crucial point is that you are no longer allowed to "solve" the equation KE = ... as E = (...)/K. Yes, I studied classical electrodynamics, but I don't know whether it says soemthing different. If you run into a contradiction with QED it's the classical reasoning that must be wrong.

Ok, let's try me another way of argumentation: You say that this argumentation applies to all kind of massless gauge bosons. However, if a symmetry gets broken, the symmetry broken state is one of unsharp charge. If only the state with Q=0 is available, how can I end up by symmetry breaking in a state in whicha measurement of Q may yield something different from 0?

I always knew that you wouldcome up with this question :-) I have to admit that I haven't studied this case in detail, so I can't say what happens to the physical states.

Can you explain where you think my argument fails? Is it because the vacuum may be no longer a singulet state?

 

[DrDu]

Actually, the proof of the Goldstone theorem which is intimately related to broken symmetry is very similar to your argumentation why total charge has to vanish, i.e. it also relates the total charge operator (or better to say the limit of a local operator approximating the latter) to an integral over the boundary. The limit is somewhat intricate, that's why I am questioning so hard your argumentation

On the other hand, in a finite system the ground state is unique, hence in a closed topology symmetry can never be broken in the strict sense. So following your argumentation, symmetry breaking is also unnatural in open topologies?

 

[tom.stoer]

I think it depends if you talk about global or local symmetries. Global symmetries can be broken via the Goldstone mechanism. There is no Gauss law associated with global symmetries (the Gauss law is the relict of the local gauge symmetry and reflects the fact that A° is not a dynamical degree of freedom but a Lagrange multiplier).

b/c there is no equation like the Gauss law for global symmetries my argument isn't valid.

For local symmetries it's different as I do not see that the gauge symmetry is really broken. I think this is - strictly speaking - not true. You can derive U(1) and SU(2) Gauss law constraints from the variation with respect to the A° and B° gauge fields. The SU(2) Gauss law constraint has again an non-abelian gauge field current term plus a Higgs term. But nevertheless it must violate the physical states in the same way as the abelian Gauss law.

I didn't check all the details but I am pretty sure that the action of the Gauss law isn't that much different from the SU(3) or QCD case.

 

[enotstrebor]

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 note in many of the answers to this question, there is one significant problem.

There seems to be a consistent errant view, that within the present theory, fundamentals like charge equivalence have a theoretical basis, as if theory defines nature rather than the theory and mathematical models are the result of nature (based on measured/experimental evidence).

If nature did not first present an equivalence of charge experience, then the theory would not either, or the theory would fail to match experience/experiment.

As it turns out, the continued extensions of mathematical model(s) of nature consistently evolved yielding the resulting base CPT symmetry of the present theory (Not to getting into violations of this symmetry in nature) and THUS the manipulation of these equations end in charge equivalence, but they are not the source of nature's behavior.

This experimental equivalence of charge like the experimental equivalence of a particles mass to energy does not have a theoretical base within the present theory.

In order to have a "theoretical basis" requires that, for the present theory's "point particle" there exists an underlying (theoretical) source model where the underlying source produces an equivalence of positive and negative charge (and the wave behavior and the point behavior) and answers the question what is the underlying reason a massed particle resists a change to velocity and why the energy content (as seen in particle anti-particle annihilation), and the particle's mass (as measured by resistance to a change in velocity) is directly proportional to the resultant photon energy.

But as the present theory denies that an underlying source can exist, no theoretical basis can exist within the present theory.

 

[tom.stoer]

Is I said in one post: there is indeed no direct reason why the charge of electron and proton match, but afair there is some support.

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.

 

[DrDu]

I pondered about our interesting discussion and wanted to propose the following alternative solution to vanishing total charge:
In my view, the problem is due to the long range instantaneous and hence artificial nature of the Coulomb interaction in the constraint. Hence, before disccussing this constraint, it should be regularized.
A simple way would be to start from the Proca equation and then consider the limit m->0.
The constraint becomes:
\nabla \cdot \mathbf{E}-\rho+m^2 \phi =0
For a toroidal topology, a potential problem arises especially for the k=0 Fourier component of the constraint. Reproducing your argument, the k=0 component of \rho -m^2 \phi has to vanish. In the limit m->0, this can either be achieved by \rho(k=0)=0 and finite phi (your proposal) or by an arbitrary total charge and \phi(k=0)->\infty.
I don't see that an infinite constant value of the potential \phi makes any problem as it does not influence the fields which are the only observables in the m=0 limit.

This resembles the resolution of the paradox with the negative energy states in the Dirac equation. If these states are filled, the electrons would provide a (negative) infinite mass. However, leaving gravity aside, this would be unobservable and corresponds only to a shift of the zero point of energy.

 

[tom.stoer]

I still think that the long range instantaneous Coulomb interaction is not artificial but physical. This gauge is widely used in some QED and even in QCD calculations, the latter one making good progress towards the explanation of confinement. I agree that there are scenrios where different gauges are easier to handle, but the gauge itself is not a problem in principle.

Regarding the Proca equation: the potential is still instantaneous and seems to violate causality Lorentz invariance in the same way (of course one can show that it doesn't). So you do not get rid of an instantaneous interaction. All what you get is an exponential decay instead of an 1/r decay, but I do not see the benefit. If you try to regularize the infinite IR contributions due to 1/r for non-compact space, I think compactification (e.g. the 3-torus) is easier to handle.

There is one severe issue with the Proca equation, namely that it explicitly breaks the gauge invariance due to the mass term A²(x). This introduces a third physical polarization and alters the theory completely. Especially the form and the meaning of the of the Gauss law is completely different. I don't think that you can recover QED with massless photons from the Proca theory.

 

[DrDu]

I don't think that you can recover QED with massless photons from the Proca theory.

But that's the way used e.g. by Zee in "QFT in a Nutshell".

 

[tom.stoer]

Interesting; how does he get rid of the longitudinal photon?

anyway - I think Proca theory is irrelevant here; I still do not understand your problem with the 1/r potential and/or the A°=0 & div A = 0 gauge.

 

[DrDu]

Apparently, the longitudinal photon decouples from matter in the limit m->0. See e.g.
http://archive.numdam.org/article/AIHPA_1972__16_1_79_0.pdf
where also different forms of the constraint are discussed.

 

[tom.stoer]

Thanks, I have to check the details. I guess it will not work in non-abelian gauge theories; I haven't seen massive gluons and Pauli-Willars for QCD.

Nevertheless: I do not understand the problem with the 1/r potential and/or the A°=0 & div A = 0 gauge in massless QED.

 

[DrDu]

In contrast to what I have been writing on the beginning of this thread I don't think anymore that my problem has to do with a specific choice of gauge but only with the long range nature of the Coulomb potential. In massaging the constraint you implicitly assume the electric field to be well defined (which indeed it is in the case Q=0). For finite Q it isn't well defined so you also cannot argue that it's divergence will make no contribution using some differential geometric identities. The 3 torus can also be viewed at as an infinite periodic array (a "crystal"). In the case of a Yukawa interaction (or more generally a short range interaction), the electric field at some point in the crystal can be approximated by summing over the fields generated by the charges which are subsequently at further and further distance. For a Coulomb potential, this sum won't converge.

Btw, shouldn't your argument also show that in a closed topology total mass (or better the energy momentum tensor) has to be 0?

 

[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 let's 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]

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.

OK, I read the whole thread, and comment below
(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 püerfectlywell 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 textbooks 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.

 

[mathman]

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]

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.