You write: "a mass density isn't useful: either because it seems to be infinite (for the electron),"

This is your opinion, but it makes no physical sense that its density could be infinite.

The electron cannot be a mathematical point. Mathematical points do not have charge, mass and spin.

We know for certain that its invariant rest mass is 9.10938188E-31 kg. Since it physically exists, the energy making up its invariant rest mass also has physical existence as a substance. If it has physical existence as a substance, then even if it is in constant oscillation locally, its "substance" has to occupy some physical volume.

This is my opinion.

Best Regards

André

When we treat a particle as a "point", we are saying that its dimension is irrelevant and / or indeterminate, and information such as densities need to be treated differently (not just mass density but charge density, etc.). A convenient way to define mass density is as an amount that, when integrated throughout the space, provides the total mass of the system. To describe the mass density of a point particle, we generally multiply the mass of the particle by a Dirac delta function, whose pole is the position vector of the particle. Thus, by integrating density throughout the space, we have the total mass of the particle. If we only define density as infinite, integration through all space will lead to an infinite mass, which makes no sense. So we use the delta function. The same treatment can be used for other types of densities.

When describing the movement of the Earth around the Sun, it is also convenient to consider the Earth as a point-like object. But, the Earth does not have infinite density (Thank God, otherwise we could not live here).

There are different experiments which assess the electron radius. For instance, it has to be less than the Compton wavelength 10-16cm. But, the best thing is to browse in Internet - there is a rich bibliography.

The density of the electron simply doesn't make sense, that's all.

Everything that carries a rest-mass has a density, a nucleus has density, a barion has density. The problem, with these tiny objects is that their radiuses are uncertain. Different types of measurement lead to different values. Thus, the density is not a precise concept for them.

The stimation of the electron radius is to be less than the Compton length 10-16cm. Thus, we have only an upper bound.

Dividing the electron mass, 10-27g, by the (very loosely estimated) volume ~ 10-48cm, one gets the density 1021g/cm3. For comparison, the proton has a density of the order of magnitude of only 1014g/cm3.

The extremely high electron density (uncertain as it may be) points to the extraordinarily strong forces that confine its charge to such a small volume. it may be that this is why until now, no experiment succeeded to break the electron into pieces.

Dear Jayaram as.

Actually, the electron is observed during scattering experiment as "behaving" point-like, not that it is a point particle in the mathematical sense. Just like the Earth and Moon are treated as if their masses were concentrated at their point-centers to calculate their orbits about the Sun.

This is the explanation.

Assuming that the electromagnetically oscillating energy making up the rest mass of the electron has physical existence, then if it is metaphorically immobilized in the smallest isotropic sphere, then we end up with a finite volume that allows calculating its inner electric and magnetic fields.

If interested, if you plug the electron Compton wavelength into equation (40ii) of the following paper, you will obtain this smallest isotropic volume for the electron rest mass energy.

http://www.gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/2257

Then establishing the ratio of the known electron mass energy over this volume, you will obtain the maximum isotropic density of the electron energy.

Best Regards

André

That the electron is a "point particle" is a mathematical convention. However, the elctron is a localizable smeered out concemtration of energy that, travelling less than the velocity of light, has mass as well as electric charge. The electron, having mass, participates in interactions with other particles of the Standard Model. Though the electon has mass it cannot be assigned a specific volume in space so that the ratio of mass to volume is undeterminable.

That the electron is a point particle is the mathematical description of a physical property-that it isn't the bound state of other point-like objects (at least up to the energies probed).

Before going on and on, it might be useful to reflect on just what the notion of the mass density of an electron could ever be used for, in practice, to immediately understand that it can't. There's no way to consistently define it-any more than it's possible to define the density of the proton.

(There's no problem in describing electrically charged spheres in classical electrodynamics, of course; just that this isn't the way to describe the properties, already, of the proton, that is known not to be a point particle.)

There's a perfectly consistent mathematical description of the electron as a point particle, so it's not the case that there's any reason to talk about its density. Similarly there's a perfectly consistent mathematical description of the proton as a bound state of quarks.

That the mass density of these objects turns out not to be a useful concept for describing their properties doesn't mean anything more profound than realizing that this, classical notion, doesn't have a useful counterpart for such objects.

If the electron were a point particle, i.e. occupying a geometrical, dimensionless point, its density would be infinite. Nothing is infinite in the nature, the concept of infinite is an idealization.

The electron has a charge and a magnetical momentum, which shows that there is an internal dynamics in the electron. The electrostatic repulsion in the electron is tremendous - normally the electron should explode. If it doesn't do so, means that the forces that keeps it entire are terribly high.

There is even a theory - but I am not competent in it - which says that the electron is a composite object, consisting in a bare particle lacking properties, and surrounded by virtual photons that give the electron the properties known to us from expeiments.

All this does not impede treating the electron as a point particle in models of phenomena at scales far greater than the assumed radius of the electron.

Dear Stam,

You wrote: "There's a perfectly consistent mathematical description of the electron as a point particle, so it's not the case that there's any reason to talk about its density."

Why not?

We do treat the Earth as if it was a mathematical dimensionless point to calculate its trajectory.

This does not make anybody confuse the Earth with a mathematical point with an infinite mass.

Why this insistence in confusing the real electron and its mathematical representation?

That we mathematically represent it as a point to calculate its trajectory and its confirmed and recorded scattering encounters with other submicroscopic particles does not prevent us from also calculating its density and talk about it just as we do for the Earth mass.

If the electron physically exists, we have every right to investigate all of its properties.

Best Regards

André

That the electron has charge doesn't mean it's made up of other particles (that's what internal structure means); neither the fact that it has mass and spin. And, for the moment, everything that's been measured about it indicates that it's not a bound state of other particles.

There isn't any situation-experimental or theoretical-where the density of the electron (or the proton, that does have a finite size, and so on) comes up.

Indeed the density of the Earth, that's been mentioned, is, also, not a very useful notion, in practice; that's why it doesn't make sense bringing it up.

What this whole discussion shows, in fact, is all sorts of situations, where the notion of a mass density isn't useful: either because it seems to be infinite (for the electron), or because its value, while finite, doesn't mean anything either (for the Earth).

The notion of a mass density, mass per unit volume, makes sense, in certain contexts, for materials that are ``homogeneous enough'' that the ratio m(V)/V, as V is taken smaller and smaller, first of all, is finite and, second, doesn't fluctuate too much. For the electron the first statement doesn't hold, for the Earth it's the second.

There's no reason for the limit, as V->V_cutoff (above the molecular scale, in practice, for bulk matter), of the ratio m(V)/V, to be uniquely defined. And even less, if one allows that V_cutoff=0.

You write: "a mass density isn't useful: either because it seems to be infinite (for the electron),"

This is your opinion, but it makes no physical sense that its density could be infinite.

The electron cannot be a mathematical point. Mathematical points do not have charge, mass and spin.

We know for certain that its invariant rest mass is 9.10938188E-31 kg. Since it physically exists, the energy making up its invariant rest mass also has physical existence as a substance. If it has physical existence as a substance, then even if it is in constant oscillation locally, its "substance" has to occupy some physical volume.

This is my opinion.

Best Regards

André

Indeed André Michaud - as we surely can make a conservative estimate of its upper and lower size. As it's neither a Black Hole nor larger than around 10-18 m, my conservative estimate of its radial size is re = α3+x · ħ / me / c ; where α is the Fine Structure Constant and x is a small number between zero and say 5.

Equipped with this radius re = 0.150 [am] for x = 0 and the well known mass me, we calculate:

ρe = 3 · me / 4π / re3 = 6.436 1025 [kg/m3 ] .

Taking x = 1 yields re = 1.095 [zm] giving its density as:

ρe = 1.656 1032 [kg/m3].

To refresh readers on SI-prefixes, [a] is atto = 10-18 , while [z] is zepto = 10-21.

All the best, GKO.

Dear Gudlaugur Kristinn Ottarsson

I completely agree with you that an estimate of the isotropic density of the energy "substance" of which the electron rest mass is made can be made from its known parameters.

From the electromagnetic perspective and known parameters from the method I mentioned previously, if the inner electromagnetic oscillation of its "energy substance" was stopped so that it regroups into the smallest isotropic sphere, the figure for its density falls into the same range as your own 1.656E32 figure.

In electromagnetism, the small number x that you mention turns out to be 2:

Volume = λc3 α5/ 2π2 and Energy=moc2, then E/V=5.457571092E33 j/m3.

Very far from infinite density.

Best Regards

André

Hi all,

If you really want to know the electron, then read the first part of the Daon theory; electron.pdf

Thank you Stellan for your interesting paper.

Your mathematical development is quite interesting.

Best Regards

André

We consider that the electron is just a point, because that is mathematically convenient. But the electron has an internal dynamics - it has a magnetic momentum.

I tend to agree with Sophia, it has what is known as a finite Classical radius.

In the quantum it also has a radius, but which seems indefinite, depending on the experiment. I really do not know why some particle physicists continue to say its point with no internal structure. Maybe more advanced research is needed.

Juan, my friend,

I'll tell you a couple of things with the hope that you won't get a headache. In QFT the electron is not considered exactly a single particle. It is a complex consisting in some miserable particle with no properties, "dressed" with a coat of virtual photons that award to the electron its properties.

I am no specialist in the issue, I only told you what's written in different articles that I saw. Now, an electron can be at rest, but photons cannot. So, I can't imagine how those photons, virtual as they may be, can dress a particle at rest. Again, I am no specialist in QFT. Anyway, this complex does not seem to inspire the idea that the electron is a point.

With kind regards

UGHG, this is the second time today I heard about virtual photons, presumably the phantom particles allowed by the uncertainty principle, stabalized by something.

Im not up to date in this territory.

Thanks Sophia

classical electron radius is still amongst the physical constant and interrelated to other constants. I think that we should be less proud of our limited knowledge and speak about models of things rather than what things really are. For certain things some models are better than others and can be used interchangeably depending on purpose. So if one decides the electron is a point particle as if in point mass model of classical physics one should take density parameter appropriate to the model, in this case NOT APPLICABLE. No one asked in classical physics about material point mass density for a reason. Infinity is not a number to be compared with real numbers but a symbol of a convergence process where a number grows indefinitely big.

@ Juan,

You see, the electron as described by the non-relativistic QM, and the one described by QFT is the same object in the nature. So, if QFT, which is the wiser brother of QM, tells us that the electron is such a complication, then it's to QFT that we have to believe. So I think. Bottom line, QFT says that the electron has a structure, it can't be a dimensionless point.

With kind regards!

A useful or perhaps intuitive estimate of the density of an electron is the space density of its probablilty function, muiltiplied by its mass.

This might, for example, be close to 1 electron mass per cubic Angstrom, if it is is an atomic orbital.

I think of an electron as spread out in space, usually without sharp edges, as described by its probablility function. As a consequence it can be very large, for example in interference experiments, and so have very low density. So the density of an electron is not a constant.

Do anyone know what is the higher experimental density measured nowadays?