# Theoretical Models for Sizes of Fundamental Particles

I am aware that according to the Standard Model of Particle Physics, fundamental particles such as electrons and quarks are treated as point-like particles. However, if fundamental particles are indeed 0-dimentional points with no spatial extent, it creates problems (i.e. fundamental particles would be black holes with infinite mass-density with radii far smaller than the Planck length that would evaporate almost instantaneously due to Hawking radiation).

From what I understand, due to the limitations of modern technology, we are currently unable to probe distances small enough to directly measure the sizes of fundamental particles. But surely there are theoretical models that have predicted the sizes of these particles. To be certain, these predictions can't be tested due to the aforementioned limitations of current technology. But I'm curious as to what numerical values these theoretical models predict.

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mfb
Mentor
i.e. fundamental particles would be black holes with infinite mass-density with radii far smaller than the Planck length that would evaporate almost instantaneously due to Hawking radiation
Please give a reference for that claim.

The Standard Model has some theoretical problems, but that is not among them.
From what I understand, due to the limitations of modern technology, we are currently unable to probe distances small enough to directly measure the sizes of fundamental particles.
We can set upper limits, and the upper limits are constantly improving. No sign of a finite size yet.

String theory has something like a finite size for particles - of the order of the Planck length. Way too small to test it experimentally.

dextercioby
Please give a reference for that claim.
Let's take the electron for example, which has a mass of 9.109×10-31 kilograms.

Density is defined as mass/volume. If the electron were a point particle, then it's volume would be 0. Any object with a non-zero mass compressed into a volume of 0 would have infinite density.

The Schwarzchild radius of a black hole is calculated to be 2GM/c2. Plugging in the mass of the electron to that equation, it results in a Schwarzchild radius of 1.353×10-57 meters. It's a small number, but it's still greater than 0, meaning that a particle with the mass of an electron with no spatial extent would indeed be a black hole. The Planck length is 1.616×10-35 meters, which is more than 22 orders of magnitude larger than the Schwarzchild radius of a black hole with the mass of an electron.

According to Stephen Hawking, black holes emit radiation and will evaporate over time. The time it will take for a black hole to evaporate is calculated to be 5120πM3G2/ħc4. Again, plugging in the mass for the electron, we get an evaporation time of 6.36×10-107 seconds. The Planck time is 5.391×10-44 seconds, a whopping 63 orders of magnitude larger than the evaporation time of a black hole with the mass of an electron.

I did all of my calculations using Wolfram Alpha.
Way too small to test it experimentally.
That's exactly why I'm curious about theoretical (i.e. non-experimental) models.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
What makes you think that general relativity is valid on the quantum scale? (Hint: It most likely isn't)

mfb
Mentor
By reference, I meant a peer-reviewed publication.
It is easy to plug in values into WolframAlpha, that doesn't mean that would lead to a meaningful result.

By the way: A charged black hole would start emitting electrons (or positrons) in a late stage of the evaporation process. Do you see a problem with your approach of an electron as black hole that would evaporate?

By reference, I meant a peer-reviewed publication.
That's the thing. I've been searching for weeks and I haven't found any. That's why I've resorted to creating a profile on this site and posing the question to actual people who could potentially read the question. I would have preferred to find the answers myself, but even after weeks, I haven't found any results that I'm comfortable with.

I did find a theoretical model for the size of an electron in the form of an article on ResearchGate from 2009 that derived the radius of an electron to be 0.011802 femtometers.

The reason I'm not comfortable with that derivation is because that radius of 1.18×10-17 meters is ten times larger than the smallest distance that we can probe with our current technology, 10-18 meters.
http://www.physicsoftheuniverse.com/numbers.html

I know that theoretical (non-experimental) models exist because I was able to find one. I was just hoping that there are other models that predict sizes smaller than what we're currently able to measure, because anything that predicts a larger size is just plain wrong. I'm by no means a physics major, I'm just doing this for fun.

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mfb
Mentor
That's the thing. I've been searching for weeks and I haven't found any.
If there is no publication discussing an obvious idea, then it is not a good model.
Physics Essays is a very questionable journal, and that publication is even more questionable. Just ignore it.

vanhees71
Gold Member
The entire thread starts with a wrong assumption, namely that relativistic QFT would treat particles as pointlike. To the contrary, it describes particles in terms of quantized fields, that's why it's called quantum field theory in the first place. The provided "picture" of "particles" is even farther away from the notion of classical particles as small (but never really pointlike!) bodies than the picture provided by non-relativistic quantum theory.

The extreme case are photons, which do not even admit the definition of a position observable!

Despite some work of theoretical heavy-ion physicists on mini blackholes in heavy-ion collisions and the look of the experiments at LHC for them, there's no hint whatsoever for their existence. So the speculation about mini black holes seem not to be favored at the moment, let alone the speculation that some known elementary particles might be mini black holes. This is already ruled out by the fact that there is at least one stable elementary particle, the electron. If it was a mini black hole it should decay quite quickly by Hawking radiation, which obviously it not the case.

There are a handful of papers studying electrons and other charged particles as Kerr-Newman black holes (Kerr-Newman, rather than Schwarzschild, is the metric for a charged spinning object). But the important thing is that the repulsive force due to the charge overwhelms the attractive force due to the mass, so there would be no event horizon and thus no Hawking radiation. Instead, the electron would be a naked singularity.

In string theory none of this happens because (like Orodruin suggests) you don't just have gravitons, there are other string modes that dominate interactions over very short distances. I don't know how it looks in other approaches to quantum gravity.

Let's take the electron for example, which has a mass of 9.109×10-31 kilograms.

Density is defined as mass/volume. If the electron were a point particle, then it's volume would be 0. Any object with a non-zero mass compressed into a volume of 0 would have infinite density.
This mental picture assumes that elementary particles are "objects", like tiny balls. The point is, this macro-world intuition fools us. Particles are not tiny balls. They are excitations of fields. "Size of an elementary particle" (thus, size of an excitation in a field) is not the same concept as "size of a tiny ball".

According to Stephen Hawking, black holes emit radiation and will evaporate over time. The time it will take for a black hole to evaporate is calculated to be 5120πM3G2/ħc4. Again, plugging in the mass for the electron, we get an evaporation time of 6.36×10-107 seconds. The Planck time is 5.391×10-44 seconds, a whopping 63 orders of magnitude larger than the evaporation time of a black hole with the mass of an electron.
Let's assume the above is a realistic picture. _What_ can be emitted from an electron-mass black hole of charge -1 e and spin 1/2? One electron? Hmm. Thus, this "electron black hole" will decay into one electron? Are these two things observationally different?

The entire thread starts with a wrong assumption, namely that relativistic QFT would treat particles as pointlike.
This is a step in the right direction. I get the feeling that part of the confusion comes from the fact that QFT and SM are used for different purposes. QFT is used to explain what gives rise to particles, SM is used to explain how particles interact. That's the sense that I get, at least. Correct me if I'm wrong.

I'm beginning to wonder if my question has more to do with QFT than with high-energy particle physics.
Let's assume the above is a realistic picture. _What_ can be emitted from an electron-mass black hole of charge -1 e and spin 1/2? One electron? Hmm. Thus, this "electron black hole" will decay into one electron? Are these two things observationally different?
The reason that I brought it up was to point out the absurdity of the results attained from the 0-dimentional point-particle model.

This is a step in the right direction. I get the feeling that part of the confusion comes from the fact that QFT and SM are used for different purposes. QFT is used to explain what gives rise to particles, SM is used to explain how particles interact. That's the sense that I get, at least. Correct me if I'm wrong.
Not quite. QFT is the general mathematical framework, in which we formulate physical models/theories. It is highly flexible and finds application not only in high energy physics, but also in condensed matter theory and statistical mechanics.
The standard model is a particular(!) quantum field theory, which describes the fundamental interactions (electromagnetic, weak and strong interactions) and the fundamental particle-fields that couple to them. The best way to think about particles is indeed in terms of excitations of the basic fields.

Maybe as a really simple example from condensed matter physics:
Think about a solid crystal, ie. a huge number of atoms arranged on a lattice, held together via springs. Now laying a coordinate system above this crystal, we might assign some number φ(x, y, z) to each spring, describing their compression/extension from the equilibrium position. Until now, the possible coordinates (x,y,z) are discrete (we had a lattice), but, if we zoom out far enough, we can choose to forget about this discreteness and just think about φ as being a continuous function (a "field"). Now, in an equilibrium situation, the field value should just vanish everywhere, φ = 0. But if i act on my crystal with a force on a few atoms, eg. i displace a single atom from its equilibrium position, pressure waves will travel through it. In the φ-language, it means that φ(x,y,z,t) is now some wave-like function of time. These would be the excitations of this "spring field". It is most useful to write φ(x,y,z,t) now as a sum of planar waves, ie. fourier decomposition. These planar waves are in some sense the most elemenatry excitations of φ, as they have a definite momentum (or wave-vector). In the particle language, we call these excitations phonons, or more generally quasi-particles (as they are not excitations of some fundamental field).
In this example, it is obvious, that these phonons are not pointlike at all. But the creation of them might look point-like (the displacement of a single atom was point-like in the continuum limit, but if we zoom back in, we see that it's not really point-like at all). So the interaction of the phonon-field with "sources" can be point-like, but for the phonons themself, the term doesn't really make sense.

Going back to particle physics: The fundamental fields are, as far as we know, not just an emergend property of an underlying crystal of some sort, but the concepts stay the same. The basic excitations of the electron field, ie. the electrons, are no more point-like than the phonons were. But the interaction with another field (eg. the electromagnetic field/photon field) can be point-like (replace the displacement of a single atom in the above example with the interaction between the fields). So that's what people mostly mean when they speak about point-like particles: Interactions between fields are point-like, and there is no "zooming in" like in the crystal/example, that would resolve this point-likeness (At least to the experimental precision that we have achieved so far).

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krater and vanhees71
haushofer
The entire thread starts with a wrong assumption, namely that relativistic QFT would treat particles as pointlike. To the contrary, it describes particles in terms of quantized fields, that's why it's called quantum field theory in the first place. The provided "picture" of "particles" is even farther away from the notion of classical particles as small (but never really pointlike!) bodies than the picture provided by non-relativistic quantum theory.

The extreme case are photons, which do not even admit the definition of a position observable!
I've never really understood this "are there pointlike paricles in qft"-business. Are you refering to the distinction between "bare particles" vs "dressed particles" and renormalization? Could you elaborate on your post? :)

vanhees71
Gold Member
Well, I never understood what it means to have a "point particle" in relativistic QFT either. Relativistic QFT (theory!) has as primary objects local field operators, used to define various unitary representations of the Poincare group, and what has a clear particle definition are the one-particle Fock states of free particles.

Free particles in the strict sense are not observable, because they are free, i.e., they don't interact with anything, including detectors. So what's described by QFT in terms of observables are scattering-matrix elements to determine the interaction rates in the sense leading from a given initial asymptotic free states (usually of two particles) to some given final state of asymptotic free ##N##-body states.

That's what can be unambiguously related to real-world experiments, and that's the way you have to define observable quantities to make sense within relativistic QFT. Now the question is, what's the "size" of a particle, and that's not so easy to answer, given the picture about particles provided by QFT. You have to define it in tems of some scattering experiments. One way for charged particles, e.g., are the rms-charge radii which can be determined, e.g., by scattering an electron on the particle in question. Already for a proton, this is difficult, and there's some debate about its actual value. Here not only scattering states in the simple sense above are used but also bound-state properties like the Lamb shift of usual and muonic hydrogen atoms, and the result is amazingly uncertain.

https://arxiv.org/abs/1301.0905

If you think that classical point particles are simple, you'll be surprised either. There's no consistent model for them at all, given the still unsolved problem to find a self-consistent description of a charged point particle moving in electromagnetic fields including radiation reaction. Wrt. this problem you are better off with the QFT picture, where you can at least define a systematic perturbative solution of this problem.

protonsarecool and haushofer
haushofer
Thanks. So I guess confusion arises because people mistakenly take local field operators for point particles.

If you happen to know some nice reference about the classical point particle problem you mention, I'd be happy to see it. That does surprise me, to be honest. It reminds me of regularisation issues in GR for 'point particles following geodesics', which is an issue I thought to be resolved.

vanhees71
Gold Member
For the problem with radiation reaction you can start with Jackson. What comes quite close to a solution is to use the Landau-Lifshitz approximation of the Abraham-Lorentz equation (see the vol. 2 of their textbook). A very nice paper, using a Born rigid body as a model of a particle with finite extension is

R. Medina, Radiation reaction of a classical quasi-rigid extended particle, J. Phys. A 39, 3801 (2006)
https://arxiv.org/abs/physics/0508031

Concerning the motion of particles with an electric and/or magnetic moment, as an example see

P. Grassberger, Classical charged particles with spin, J. Phys. A 11, 1221 (1978)

haushofer

vanhees71
Going back to particle physics: The fundamental fields are, as far as we know, not just an emergend property of an underlying crystal of some sort, but the concepts stay the same. The basic excitations of the electron field, ie. the electrons, are no more point-like than the phonons were. But the interaction with another field (eg. the electromagnetic field/photon field) can be point-like (replace the displacement of a single atom in the above example with the interaction between the fields). So that's what people mostly mean when they speak about point-like particles: Interactions between fields are point-like, and there is no "zooming in" like in the crystal/example, that would resolve this point-likeness (At least to the experimental precision that we have achieved so far).
This is a helpful piece of information. Do the excitations of quantum fields have some spatial extent? If so, can that spatial extent be used as a definition of the size of a particle?

dextercioby
Homework Helper
No, the spatial extent is assigned only to macroscopic quantities, such as traces in a Wilson chamber or spots on a dark screen. But this involves macroscopic measurement. You cannot infer from these (visible to the human eye) traces the size/extent of their cause. The most successful theory in physics in terms of observables' prediction with respect to their experimental measurements is quantum electrodynamics, albeit a mathematically ill-defined theory, but one whose success depends heavily on the assumption of no size either to the quantum fields or their excitation (which as said, from a mathematical pov don't really exist).

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vanhees71
It's clear that I know very little about quantum physics. But in my searches, I came across an article on QFT that made a statement about the point-like nature of particles:

"the Compton wavelength is the distance at which the concept of a single pointlike particle breaks down completely."

What does it mean for the concept of a point-like particle to break down?

dextercioby
Homework Helper
What you need to understand is that "a pointlike particle" is just a theoretical/mathematical model/assumption. It's not really the accurate description of (human perceived) reality. This model is known to hold for all "elementary" (i.e. with no theorized/experimentally detected internal structure) blocks of matter (i.e. to avoid using the word particle again). Another useful model is the one of "quantum/quantized field". It is postulated that the elementary excitations of these fields are pointlike particles. This is how the connection between two descriptions of matter (matter as a quantized field, matter as a collection of particles) is built.

The concept of "pointlike particle" won't break down, unless we find experimental evidence that the underlying quantum fields (which, as I told you are intimately linked with the model of a pointlike particle) won't offer a better description/explanation of the (human perceived/mathematically theorized) reality. That said, I believe David Tong made that assertion without too much thought.

What you need to understand is that "a pointlike particle" is just a theoretical/mathematical model/assumption. It's not really the accurate description of (human perceived) reality. This model is known to hold for all "elementary" (i.e. with no theorized/experimentally detected internal structure) blocks of matter (i.e. to avoid using the word particle again). Another useful model is the one of "quantum/quantized field". It is postulated that the elementary excitations of these fields are pointlike particles.
I'm not sure there is anything resembling "pointlike" in, for example, an electron wavefunction with definite momentum.

My understanding is that "electron is pointlike" can be interpreted that "it is physically possible to localize electron wavefunction so that it has high probability (say, 99.999%) of being in any beforehand specified arbitrarily small volume". Extended objects such as atomic nucleus don't have this property.

vanhees71