Elementary particles are considered to be point-like, but not point particles.
''QED, or relativistic quantum field theory in general, is not based on the notion of ''point particles'', as one sees stated so often and yet so erroneously.'' (emphasis as in the original)
This quote from p.2 of the book
O. Steinmann,
Perturbative quantum electrodynamics and axiomatic field theory,
Springer, Berlin 2000.
tells everything.
A point particle is the idealization of a real particle seen from so far away that scattering of other particles is as if the given particle were a point. Specifically, a relativistic charged particle is considered to be a point particle at the energies of interest if its interaction with an external electromagnetic field can be accurately described by the Dirac equation.
Both electrons and neutrinos are considered to be pointlike because of the way they appear in the standard model. Pointlike means that the associated bare particles are points. But these bare particles are very
strange objects. According to renormalization theory, the basis of modern QED and other relativistic field theories, bare electrons have no associated electromagnetic field although they have an infinite charge (and an infinite mass) -- something inconsistent with real physics. They do not exist.
The bare particles are points = structureless formal building blocks of the theories with which (after renormalization = dressing) the physical = real = dressed = observable particles are described. The latter have a nontrivial electromagnetic structure encoded in their form factors. (The term ''dresses'' comes from an intuitive picture form the early days of quantum field theory, where a dressed particle was viewed as
the corresponding bare particle dressed in a shirt made of infinitely many soft bare photons and bareparticle-antiparticle pairs.)
Physical, measurable particles are not points but have extension. By definition, an electron without extension would be described exactly by the 1-particle Dirac equation, which has a degenerate spectrum.
But the real electron is described by a modified Dirac equation, in which the so-called form factors figure. These are computable from QED, resulting in an anomalous magnetic moment and a nonzero Lamb shift removing the degeneracy of the spectrum. Both are measurable to high accuracy, and are not present for point particles, which by definition satisfy the Dirac equation exactly.
The size of a particle is determined by how the particle responds to scattering experiments, and therefore is (like the size of a balloon) somewhat context-dependent. (The context is given by a wave function and determines the detailed state of the particle.)
On the other hand, the deviations from being a point are usually described by means of context-independent
form factors that would be constant for a point particle but become momentum-dependent for particles in general. They characterize the particle as a state-independent entity. Together with a particle's state, the form factors contain all information that can be observed about single particles in an electromagnetic field.
An electron has two form factors, a magnetic and an electric one.
''The electric form factor is the Fourier transform of electric charge distribution in space.''
(from Wikipedia)
The electric form factor determines the charge radius of a particle, defined as the number $r$ such that
the electric form factor has an expansion of the form $F_1(q^2) = 1-(r^2/6) q^2$ if $r^2q^2\ll 1$.
(Here units are such that $c=1$ and $\hbar=1$.) This definition is motivated by the fact that the average over $e^{i q dot x}$ over a spherical shell of radius r has this asymptotic behavior. To probe the electric form factors, one usually uses scattering experiments and fits their results to phenomenological expressions for the form factor.
But the form factor contains nothing at all about interaction- or state-dependent information. The interaction-dependent information is instead coded in an external potential or a multiparticle formulation, and the state-dependent information is coded in the wave function or density matrix, which (at any given time) is independent of the Hamiltonian.
Also, the information contained in the form factor is only about the free particle in the rest system, defined by a state in which momentum and orbital angular momentum vanish identically. In an external potential, or in a state where momentum (or orbital angular momentum) doesn't vanish, the charge density (and the
resulting charge radius) can differ arbitrarily much from the charge density (and charge radius) at rest.
(The above is taken from the entry Are electrons pointlike/structureless? of my theoretical physics FAQ, where more information and references can be found.)
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