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We are told that elementary particles has dimension zero and take up no space. For example, the electron is a point particle that have a negative unit charge, also has mass and spin, but no size.

My question is what bad consequence does it lead if we assume that electron has size? Is there any direct or indirect experiments that verify this assumption (or facts?), i.e. elementary particles have no size. Or what theoretical contradiction will it have if we do not assume point particle?

Qmechanic
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an offer can't refuse
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    Size relative to what? The electron does not have an internal structure on the scale accessible with current accelerators. That's a fundamentally different (but correct) statement compared to the oversimplifying statement that the electron has no size. So how would we find such a structure, if it existed? The same way we found it for protons: by using a very powerful microscope called an accelerator. Eventually we will very likely have an accelerator that will find structure inside electrons... we just don't have one, yet. – CuriousOne May 08 '15 at 15:10
  • @CuriousOne Why are you so sure we will find structure inside electrons? On what are you basing this claim? There is really nothing that requires electrons to have a physical size. They are quantum fields, not hard balls. I'd say that as we get accelerators with higher energies, we will simply have ways of measuring the position of an electron better, but there's nothing that says we should find structure of any kind – Jim May 08 '15 at 15:11
  • @ACuriousJim: If you like, we have already found structure inside electrons: keep shooting them together and photons and electrons and muons and all kinds of other stuff comes "out". We know that the charge of an electron is not the elementary charge as we thought in the past. The current number for that is one third of an electron charge. Now, it may be that the symmetries of the standard model hold for any energy scale, but how likely is that for what is, in effect, simply one of the mean field theories of the real thing? – CuriousOne May 08 '15 at 15:16
  • @CuriousOne Thanks for your explanation. I take the word "point particle" too literally. – an offer can't refuse May 08 '15 at 15:17
  • The idea of point particles, in general, stems from a misinterpretation of the concept of center of mass in Newtonian physics. A center of mass does not imply that the body it belongs to is infinitely small, it merely means that for all practical purposes one does not have to care about its internal structure, including its size. For electrons that is, so far, true: we do not have to care about their internal structure because they seem to be "small enough" to be described by this old center of mass concept. As you can see from the discussion with ACuriousJim that's not quite as simple... – CuriousOne May 08 '15 at 15:21
  • @CuriousOne Colliding something at high energies and seeing other things made does not mean those other things are part of the structure of the original particles. Nor does having a non-elementary charge mean that there must be something deeper. – Jim May 08 '15 at 15:21
  • As a quantum field, talking about the electron's size does not even make sense. We can talk about the probable location of an electron, but at that location it has no size, it's still just a quantum field. That's where the notion of point-particles stems from – Jim May 08 '15 at 15:23
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    @CuriousJim: What makes you think that it's not turtles all the way down? Certainly not an honest extrapolation of the history of science. The only constant there is that every time we look with sharper sights, we find keep finding new stuff. The standard model is a purely phenomenological model. It explains exactly nothing, even though it describes the data well. – CuriousOne May 08 '15 at 15:24
  • @ACuriousJim: A hydrogen atom is also just "a quantum field", or better, it is a different solution of the same quantum field, and yet, it defines a physical size scale just fine. You and I are stuff made from the very same quantum field and we have a rather large size (I certainly do). What makes you think that the electron is special and that it does not have a, so far, undiscovered scale? – CuriousOne May 08 '15 at 15:26
  • Related: http://physics.stackexchange.com/q/7322/2451 , http://physics.stackexchange.com/q/39590/2451 and links therein. – Qmechanic May 08 '15 at 15:31
  • @CuriousOne Nothing makes me think one way or the other about the structure of electrons. But there is no evidence for structure, which does tend to support theories that there is not internal structure. Statements that it has a structure should be founded on more than speculation. At least statements that there is no structure are founded on the lack of evidence for any. Just like statements that there is no sky are valid within a community of people that have never had contact with anything from the outside. – Jim May 08 '15 at 15:48
  • @ACuriousJim: I didn't say that the electron has observed structure, I said that we can't stop looking for it, if we want to be scientists. – CuriousOne May 08 '15 at 21:29

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If the electron is supposed to have a finite size, then it has has to be a rigid body. If not, then it wouldn't be an elementary particle, it would have some inner structure that you can further investigate. So, if the electron is indeed an elementary particle and it has a finite size, then it must be a rigid body. But a rigid body can be used to create causality violations. Information can be transmitted instantaneously across the distance covered by the rigid body (if a force is applied to one side the other side will move instantly). As explained here in detail you can exploit that to send information back into your own past.

This is why in physics we assume that there are only local interactions. An effect at some point in space at some time can only have an effect at exactly the same point at the same time. Rigid bodies cannot exist, forces are transferred from one point to another via fields that only interact with each other in a local way (the interaction terms only involve the field strengths at the same points).

Count Iblis
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  • This does not disprove the idea that the electron has size. It could have size and yet be flexible, not rigid. The flexibility though I'd imagine would show up somehow once you get to conditions where it becomes important; but so too would the rigidity of a finite-extent, rigid electron. – The_Sympathizer Aug 29 '18 at 07:47
  • What about electrons as strings (or branes)? They would have a size but would not be rigid. – jinawee Jan 29 '19 at 12:13
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At school, and when studying classical physics, one assumes that particles have no size. This is done because it massively simplifies equations, and it is a good approximation if the size of the particle is believed to be significantly small than the other important length scales involved.

Actually, there are unwanted consequences when one assumes that a particle has no radius. For example, the electric field of an electron increases as $1/r^2$ as one gets closer to it. Since the electric field energy is proportional to the square of the electric field magnitude, the energy of the field of an electron in vacuum would be infinite!

In fact, in modern physics, one does not talk about the size of a given particle. Rather, this concept has been replaced by that of the probability of finding the particle at a certain place when looking for it. This uncertainty in the position of the particle can be used as a measure of its "size".

But this "size" does not behave as our normal intuition of size. For example it is variable, and generally depends on the momentum of the particle in question: the higher the momentum, the smaller the size can be. This is connected to the Heisenberg Uncertainty Principle. This is confirmed experimentally. Slow neutrons in a nuclear reactor are far more likely to collide than fast moving ones. If you want to make sure an electron passes trough a slit, you better fire it with high momentum.

In the same way, one can, by using higher and higher energies, determine the position of an electron to higher and higher precision, thus reducing its "size". But to reduce it to 0 would involve, you guessed it, infinite energy.

Andrea
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