@federico

The cases we discuss in QFT deal with electron as object and not the source. In any such calculations in order to ignore effects due to electon itself we assume it to be point like.

If in any phenomenon electron itself is a source then its dimensions have to be included.

Indeed the electron is ALWAYS a source of electromagnetic field, so its self-field cannot be excluded. Actually it is also always a source of other fields, as well as of gravity and torsion.

Quantum fluctuations of some property of a particle don't affect the property that it's a particle in spacetime; they affect the properties of the states of the particle in phase space.

The Compton wavelength doesn't imply any lower limit on the size of an object, that is subject to quantum fluctuations. It implies a lower limit on the uncertainty of the position of a particle.

The one-particle excitations of quantum fields have all the properties of point objects.

Uncertainty in the electron position does not say about its size but just about the probability of it being there and this, of course, affects some measurements, like g for instance. An extended electron is not compatible with relativity. An elementary particle must be a point singularity. The old vision associating infinity and other inconsistencies to a point charge, although very common, is not correct. There is published literatura about.

Manoelito Souza to say that the size of a particle is less than d, you have to confine it in a trap with a size smaller than d, but this is not possible because there are phenomena such as tunneling and pair production. So to say that particles are point-like is a metaphysical statement.

An extended electron is of course compatible with general relativity. On the contrary point-like particles are inconsistent because they have infinite self-energy.

Confining is not size measuring. Position uncertainties are related to probabilty, anything else is opinion.

Reducing position uncertainty has a price in energy, quantum effects put a limit on this reduction, say an upper bound, which is not the particle size.

An extended electron is not compatible with RESTRICT relativity.

Classical models for the electron was a very hot research subject, early in the XX century. A well known Rohrlich's book is a classical reference. Just for mentioning an example: What holds together the parts of an electron against electrostatic repulsion?

About infinite self-energy and others CED inconsistences I reafirm the last paragraph of my original answer. They come from misinterpretations of solutions to Maxwell's equations.

Only a point electron produces an entirely consistente CED. This is not a metaphysical argument. I can provide the due references.

There are several senses in which the electron is not a pointlike object, but there are also several senses in which it is. To answer the initial question, I will concentrate on the latter. First, the FIELD operator of the electron is a local field defined at a point. Second, the width of the WAVE FUNCTION of the electron can be smaller than the Compton wavelength.

Indeed, that the Compton wavelength should not be interpreted as the size of the object is particularly clear for the photon. The photon's Compton wavelength is infinite (because its mass is zero), but it certainly doesn't make much sense to think of photon as an object with infinite size.

Manoelito Souza and Hrvoje Nikolić I didn't say that the Compton length is the size of the particle, I said that it is a lower bound on the measurable size of the particle. There are no other *direct* ways to measure the size of a particle, then to confine it in a trap.

P.s. Hrvoje Nikolić for massless particles the measure of localizability is just the wavelength since the Compton length is not defined. The field operator is not a physical stuff, only its mean value on a state has physical meaning.

Manoelito Souza classical electrodynamics of point particles is not consistent at all, since the energy of the field goes to infinity near the charge, then you would have a singularity in the filed and an infinite mass for the particle. This problem has never been solved, but neither has it ever been proven that it cannot be solved. Indeed there are localized stable solutions of the classical Maxwell-Dirac-Einstein system (it is very well possible that Poincaré stresses could have a general relativistic origin).

Federico Comparsi What if a particle had such a small mass that its Compton wavelength was one million kilometers. Would you say then that it cannot be trapped in a trap smaller than that?

Hrvoje Nikolić yes

Dear Federico Comparsi ,

It is out of any reasonable doubt that a point like particle, endowed with a finite mass, has "infinite" energy density so it has to be considered as a sort of a singularity. Then you should assume that diverging energy density and the vanishing size of the particle have to converge to finite values of charge and mass. I guess this was one of Einstein's attempt to consider electrons as "singularities" in gravitation...

If the size of the particle is identically 0 then it is a bit hard to imagine any limit.

The only way out is to provide one degree of freedom, as you said. It is meaningless to talk about "a particle" , the uncertainty principle forbids this simplistic and paradoxical concept. It solves this "conundrum" by smearing the particle same as what occurs for any electrons combined to form an atom with its wave-function, which does not live only in Phase-space useful for somebody to make "calculations" but it is its ontology.

this is a quite interesting point, but you should modify GR not a little bit. I would interpret it as a suggestion that something like a heavily locally distorted "metric" around the charge could manage to exert a tension to hold it together..

but does it make sense to talk about parts of an electron held together.?..then it is not anymore an elementary particle and we go back to the same issue for each of its sub-parts then...

Stefano Quattrini there is a problem with infinite energy (or charge) density, since the field too goes to infinity. Indeed Einstein's attempt was to search for REGULAR (not singular) solutions of the non-linear system.

As regards the last point, there have been attempts in this direction in the torsional modifications of GR.

Dear Federico Comparsi

>

If you refer to Cartan's attempts, they were mostly devote to provide a place for "spin" in Gravitation, as far as I know, or you refer to others?

Dear Stefano Quattrini , Federico Comparsi ,

To answer such questions one has to consider a theory of quantum gravity. No particle can be smaller than the Planck length, because even space-time is not well defined on smaller scales.

Dear Jose Gaite ,

I totally agree, it is infact meaningless to even imagine point like particles if it is accepted that the smallest meaningful length is Planck length.

The model of point like particles is a classical "first approximation" and it works very well in most problems which do not need particular levels of accuracy.

Federico Comparsi And what if we lived in a Universe smaller than the Compton length of some very light particle? Wouldn't then the particle be confined in a trap smaller than the Compton length?

Hrvoje Nikolić I think that this is not possible.

Federico Comparsi Why do you think that it is not possible?

Hrvoje Nikolić because I think that there could be a deep relationship between the Hubble radius and the mass of the lightest particle.

The “beginning“ is unphysical. If you have an electron without charges, then it’s not an electron.

Something like this has been said, but it is good to insist: the electron can be described by a local wave-function. This is not possible for a composite and hence not point-like particle. On the other hand, at high energy, a secon-quantized theory is mandatory, because there are quantum fluctuations of the electron field that manifest themselves as particles, e.g., virtual electron-positron pairs. This phenomenon must be understood as non-localization rather than compositeness, and it is actually a consequence of Heisenberg's uncertainty principle.

Dear Christian Baumgarten,

If I understand your objection, it is about the wave function of a free electron being non normalizable and hence being indistinguishable from a similar many-electron wave function. I suppose this is so, provided that the electrons do not interact, as in a free electron beam. The many-electron wave function should be a function of many variables but this would be irrelevant, as long as all the electrons were in the same state of motion. However, the Pauli exclusion principle forbids it. It could happen if the electron were a boson. At any rate, I do not see a definite objection to localizability or the lack thereof.

Dear Christian Baumgarten,

Sorry, I do not understand. If the electron has a center of mass, it is because it is a composite particle. If it is not composite, we can say that it is point like, assuming the pertinent restrictions on localizability of point-like particles. Right?

Dear all,

It is evident and indisputable that there cannot be any point-like electron, and that is a comfortable approximation that works to a certain extent.

On the other hand, the electron is an elementary entity and it is meaningless to think about other material "parts".

Since it is mathematically the resonance of two "spinors" emerging out of nothing coupling with the Higgs, it comes naturally that it is confined in a certain volume of space which is small but in average huge in comparison with the Planck length.

Dear Christian Baumgarten ,

I stand by my statement: "If the electron has a center of mass, it is because it is a composite particle." This has nothing to do with phase-space considerations.

Of course, any point-like particle has a sort of minimal phase-space extension in accord with the uncertainty principle.

Dear Christian Baumgarten,

I do not want to enter into a one-on-one argument, because other people may not be interested.

Of course, the exact meaning of point-like particle has changed with time, especially in the 20th century. Only in the last century has been understood the need of a high-energy experiment to locate a particle. The uncertainty principle and the creation of virtual particles are physical realities, not mathematical constructions.

Why some physicists persist in saying that in quantum field theory electrons are point-like?

Just because any physical field theory (aiming to describes matter's behavior) ends up being based on Maxwellian general relations. However, Maxwell's equations, besides treating electrons as point particles, account for its local fluctuations. In effect, the Heaviside ellipsoid (equipotentials of a free electron) only can be explained in terms of randomly oriented orbits (at the Compton’s domain.) of a massless elementary electrical charge, which, therefore, are capable of being contracted.

especially, because the mass of an electron is unequal zero and hence its density would be infinity, which is nonsense.

Dear Baumgarten

Dear Albert

Very well! Then explain to me how the scalar potential of a free electron is obtained.

Any such argument based on the uncertainty principle is just a manifested opinion, an extrapolation of its physical content. Not measuring is not equivalent to no-existing.