Are electrons pointlike/structureless?
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
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 everything that can be observed about single particles in an electromagnetic field.
An electron has two form factors, a magnetic and an electric one.
The mathematically most accessible derivation of the electron form
factor is in the book
The relations between form factors for spin 1/2 particles and
terms in a modified Dirac equation describing the covariant dynamics
in an electromagnetic field of a particle deviating from a point
particle are given in
The anomalous magnetic moment shows directly as a coefficient in the modified Dirac equation. On the other hand, the spectral resolution of the single-particle problem in a constant electric field leads to a spectrum from which the Lamb shift can be read off.
Foldy's paper gives a very clear analysis of the physical interpretation of the form factors, though he does not use this terminology for his coefficients. But this connection is drawn in the paper
To probe the electric form factors, one usually uses scattering
experiments and fits their results to phenomenological expressions for
the form factor.
When scattering electrons off the particle, the electrons
respond to the form factors of the particle, in a nonrelativistic
treatment more precisely to the charge distribution which can then be
computed from the electric form factor:
An intuitive argument for the extendedness of relativistic particles (which is difficult to make precise, though) is the fact that their localization to a region significantly smaller than the de Broglie wavelength would need energies larger than that needed to create particle-antiparticle pairs, which changes the nature of the system. Note, however, that even point particles (which satisfy by definition the Dirac equation exactly) have a nonzero Compton wavelength. (Localization is discussed elsewhere in this FAQ; see also Foldy's papers quoted there.)
The terminology ''particle size'' is not used in a well-defined way. But there are precise related concepts; e.g., the charge radius. it is positive even for neutral particles such as the neutron or the neutrino, but is zero for a Dirac particle. (The latter is a theoretical entity only.)
On a more formal, quantitative level, the physical, dressed particles
have nontrivial form factors, due to the renormalization necessary to
give finite results in QFT. The form factor measures the deviation
from the behavior of an ideal point particle, i.e., a particle obeying
exactly the the Dirac equation. The form factor of the electron can be
computed perturbatively from QED, and it can be indirectly determined
by experiment, e.g., through the observation of the anomalous magnetic
moment and the Lamb shift. (A point particle has no anomalous magnetic
moment and no Lamb shift since it satisfies the Dirac equation exactly.)
The "form" of an elementary particle (considered as a free particle
at rest) is described by its form factor, which is a well-defined
physical function (though at present computable only in perturbation
theory) describing how the (spin 0, 1/2, or 1) particle's response to
an external classical electromagnetic field deviates from the
Klein-Gordon, Dirac, or Maxwell equations, respectively.
The form factor contains the complete state-independent information about a free particle, since it determines the (single-particle) Hamilton operator of the free particle and everything else can be computed from it. (But it is an approximate description only since it ignores memory effects that would arise in an exact treatment of the dynamics.)
The paper
An extensive discussion of form factors of Dirac particles and their relation to the radial density function is in the paper by Yennie et al. cited above, and in
Nontrivial form factors give rise to a positive charge radius.
In his book cited above, Weinberg explicitly computes in (11.3.33)
a formula for the charge radius of a physical electron. But his
formula is not fully satisfying since it suffers from an infrared
divergence: the expression contains a fictitious photon mass,
and diverges if this goes to zero, as infrared corrections from soft
photons are not accounted for). See also Section V in the
review article
Of course, other particles also have form factors and associated
charge radii. For proton and neutron form factors, see hep-ph/0204239
and hep-ph/030305. Neutrons have a negative mean squared charge radius.
This looks strange but is not since the measure for the mean is
not positive; but it means that a classical interpretation of the
charge radius of neutrons is dubious. In the introduction of
On the numerical side,
http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1992PhDT.......130L
gives in an abstract of a 1982 thesis of Anzhi Lai
an experimental upper bound on the electron charge radius of ~ 10^{-16}
cm (But I haven't seen the thesis.)
According to (7.12) in Phys. Rev. D 62, 113012 (2000), the charge radius of neutrinos, another pointlike particle, computed from the standard model to 1 loop order, is in the range of 4...6 10^-14 cm for the three neutrino species.
hep-ph/0109138 reviews the experimental limits on the sizes of fundamental particles.
Everything discussed above assumes that the particle is isolated and at rest. The form factor tells about its response to external electromagnetic fields and about the electromagnetic field it is accompanied by.
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. For example, for a hydrogen electron in the ground state, the charge density (which must essentially cancel the proton's Coulomb field far away from the atom) is significant in a region of diameter about 10^{-11} cm (a small multiple of the Bohr radius), while the charge radius at rest is probably (in view of the above partial results) < 10^{-13} cm.
In all cases, the charge distribution is defined as the expectation of the charge density operator of the corresponding quantum field. For molecules, this charge distribution is the computational target of much of quantum chemistry, and defines the shape of a molecule. The shape of a particle determined by the form factor therefore corresponds to the equilibrium shape a molecule takes in its rest frame in the absence of forces, i.e., in its ground state, while the state-dependent shape corresponds to the much less predictable shape of a molecule interacting with its environment.
CODATA Recommended Values of the Fundamental Physical Constants: 2010 lists as ''classical electron radius'' the value 2.8179403267(27) fm. (1fm=10^{-15}m)
For nuclear radii see, e.g., p.17 of CODATA Recommended Values of the Fundamental Physical Constants: 2010
For (constituent) quark sizes see, e.g., NN Scattering and Nucleon Quark Core
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ