# Are point-particles real ?

In summary, the electron is an excitation in an underlying field that has a nonzero spatial extension.

Are point-particles "real"?

I've heard a couple of times the claim that the electron (among other elementary particles) is point-like, having essentially no spatial extension.
In the framework of QFT (which forms the basis of the standard model and therefore our best current understanding of the fundamentals of physics) particles are thought of as excitations in an underlying field. These excitations, when localized, come in the form of wave packets which do have a nonzero spatial extension. The electron would not be pointlike unless its momentum was totally undetermined, right?
From what I understand, the structure of particles is determined from various scattering experiments where the momenta of the particles that are collided are quite well-defined.

Can someone explain this to me?

they would have trouble colliding otherwise :)
or only their fields collide?

The word "real" is not well defined.
It is interpretation-dependent

If you have a spatial distribution of charge, associated to its electromagnetic vertex come electromagnetic form factors. For instance, the proton with spin 1/2 has two electromagnetic form factors, F1 of Dirac and F2 of Pauli. It is an experimental observation that for the electron F1=1 (constant) and F2=0. This is measured in numerous experiments, the most stringent of which, for the purpose of testing the point-like behavior, are jets or lepton pair distributions in electron positron annihilation at high energy. For a spin 1/2, we measure for instance the expected
$$\frac{d\sigma}{d\Omega}=\frac{q^2\alpha^2}{4s}\left[1+cos^2(\theta)\right]$$

If for instance you took a simple charged scalar particle, you could show that if it has a spatial extension corresponding to a distribution of charge $\rho(x)=q f(x)$ where q is the total charge of the particle, then it would have a form factor $F(q)=\int e^{i\vec{q}\vec{x}}f(\vec{x})d^3x$ so the form factor is none other but simply the Fourier transform of the corresponding charge. The Fourier transform of a Dirac delta function is simply the constant unity, which is what we observe for electrons.

Note that, in the context of quantum field theory, creation and annihilation operators live in momentum space, so they create eigenstates of momentum, which are quantum superpositions of all positions. Please note the difference : they do not create states which have an infinite spatial extension. The idealized states created are in position space a coherent quantum superposition of all possible point-like positions. So, to answer
The electron would not be pointlike unless its momentum was totally undetermined, right?
No : the electron is still pointlike, but its position is not exactly determined unless its momentum is totally undetermined.

Thank you humanino for that answer. I'll have to think about it for a while.

Of course, you're right about the momentum-position stuff! Got a little mixed up in the old head...

I have a related question, so I won't start a new thread:

I understand that the electron degeneracy pressure arises from the Pauli exclusion principle, which forbids two electrons from occupying the same quantum state. However it has been explained to me, and I can show that the degeneracy pressure arises when the average distance between electrons approaches their de Broglie wavelength in a cold electron gas. In a hot gas, the electrons have enough energy to occupy higher states, and the degeneracy pressure declines, but couldn't this be alternatively explained as resulting from the decrease in size of the de Broglie wavelength at higher energies? This seems to indicate that the electrons do in fact have a size, given by their de Broglie wavelength.

It seems that this interpretation is incompatible with the electrons being point particles, so could someone please point out where this thought breaks down?

If I wasn't studying for an exam I would take the time to type up the math, but for now I refer you to the course-notes for my quantum class, which shows the relationship between the average spacing between electrons and the de Broglie wavelength.

http://farside.ph.utexas.edu/teaching/qmech/lectures/node65.html

I've heard a couple of times the claim that the electron (among other elementary particles) is point-like, having essentially no spatial extension.
In the framework of QFT (which forms the basis of the standard model and therefore our best current understanding of the fundamentals of physics) particles are thought of as excitations in an underlying field. These excitations, when localized, come in the form of wave packets which do have a nonzero spatial extension. The electron would not be pointlike unless its momentum was totally undetermined, right?
From what I understand, the structure of particles is determined from various scattering experiments where the momenta of the particles that are collided are quite well-defined.

Can someone explain this to me?
The expression "electron is pointlike" actually means that quantum properties of the electron can be described by a LOCAL field operator. It does not mean that the wave function of the electron is not extended in space. Of course it is.

TaylorRatliff said:
I have a related question, so I won't start a new thread:

I understand that the electron degeneracy pressure arises from the Pauli exclusion principle, which forbids two electrons from occupying the same quantum state. However it has been explained to me, and I can show that the degeneracy pressure arises when the average distance between electrons approaches their de Broglie wavelength in a cold electron gas. In a hot gas, the electrons have enough energy to occupy higher states, and the degeneracy pressure declines, but couldn't this be alternatively explained as resulting from the decrease in size of the de Broglie wavelength at higher energies? This seems to indicate that the electrons do in fact have a size, given by their de Broglie wavelength.

It seems that this interpretation is incompatible with the electrons being point particles, so could someone please point out where this thought breaks down?

If I wasn't studying for an exam I would take the time to type up the math, but for now I refer you to the course-notes for my quantum class, which shows the relationship between the average spacing between electrons and the de Broglie wavelength.

http://farside.ph.utexas.edu/teaching/qmech/lectures/node65.html

Not just electrons... Remember, we have neutron stars, and maybe quark stars? Look at degeneracy as a whole, including neutrons and the "point" issue becomes a little clearer as just one descriptive element, as Demyst and others have said in one way or another.