# What is an electron?

A narrow beam of electrons prepared by an electron source and moving in a vacuum without any external field is a reasonable preparation of free electrons in an eigenstate of the 4-momentum operator p. Free electrons are accompanied by their associated electromagnetic field (self field) since they are dressed. But this is part of the electron's state (a joint eigenstate of the 4-momentum p, with p^2=m_e^2) and the whole field moves with the particle, without having any internal motion: There are no other degrees of freedon that could be varied, apart from a single spin degree.
Dressed electrons, dressed postrons and dressed photons are the (on-shell) input and output states of the scattering processes described by the QED S-matrix, the central observable in standard treatises of QFT. Here on-shell is meant in terms of the physical (renormalized) mass, not an (infinite, ill-defined) bare mass.
Also, nothing radiates here, since the dressed electron is a stable particle. The S-matrix of QED without an external field has no contributions e --> e + k gamma, since this would violate 4-momentum conservation.

A real, free electron is described by the eigenstates psi of p with the smallest nonzero value of p^2 in the charge e sector of QED. A real, free electron can be in a pure or mixed state, since freeness specifies a definite momentum but no definite spin. Thus it can be in a mixed state with respect to spin. But this is the only freedom it has.
Just as an eigenstate psi of the Hamiltonian H of a nonrelativistic particle satisfies H psi = E psi = omega hbar psi and hence
psi(t) = e^{-i omega t} psi(0),
so an eigenstate psi of the 4-momentum operator p of a relativistic particle satisfies p psi = k hbar psi and hence
psi(x)= e^{-i x dot p} psi(0).
Every free particle in a pure state can be describe in this way since this is what it means to be free. This holds independent of any particular theory. The latter just specifies the form of psi(0).
Thus it holds for a point particle satisfying the Dirac equation, for a particle with a nontrivial form factor satisfying a phenomenological modified Dirac equation, or for a particle in the dressed 1-particle space of a quantum field theory.
The only difference in these three cases is the definition of the momentum operator. In QED it is defined quite indirectly through renormalization in Dyson's intermediate representation,

• TS Walhout, Similarity renormalization, Hamiltonian flow equations, and Dyson's intermediate representation, Phys. Rev. D 59, 065009 (1999)
while in (modified) Dirac equations it is much more tangible.

More precisely, a free relativistic electron in a pure state is described by a space-time depemndent state vector of the form
psi(x) = exp {-i p dot x/hbar} psi_0
where psi_0 is a fixed Dirac spinor with 4 components, lying in the electron subspace.
The form factor governs the interaction in case the particle is not free but coupled to an external (classical) electromagnetic field. (Coupling to a quantum field is even more complex, not discussed here.) Then psi(x) is still a Dirac spinor with 4 components, but its x-dependence is governed by a modified Dirac equation in which the form factors appear. See

• L. L. Foldy The Electromagnetic Properties of Dirac Particles Phys. Rev. 87 (1952), 688 - 693.
and the Section Are electrons pointlike/structureless? in the present FAQ.
Thus the form factors do not determine (and have nothing to do with) the state of an electron at a fixed time, but characterize how this state changes with time.
Note that the dressing of the stable asymptotic state (i.e., the free electron) and its form factors are both generated by the QED dynamics.

Compare to the much simpler quantum mechanics of an anharmonic oscillator. here the ground state is dressed - it is the lowest eigenstate of the true Hamiltonian, although it is constructed in perturbation theory from the eigenstates of the free Hamiltonian. Now an anharmonic oscillator is nothing else than a bound 2-particle system in the center of mass frame, and viewed as the latter, one sees that the ground state becomes the asymptotic unexcited in/out state of the 2-particle system (with a trivial S-matrix in this simple case). The dressing of the asymptotic state is generated by the interaction.

In QED, things are analogous, though significantly more complex. Again the dressed electron state is stable and has trivial scattering behavior since there is no way to decay into other products without violating charge or 4-momentum conservation. Again, the dressing is generated by perturbation theory from undressed point particles satisfying the free Dirac and Maxwell equations.
This is the case even in the nice, infinity-free treatment of QED in

• G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, 2nd ed. Springer, New York 1995.
The difference of the treatment there to the usual treatment lies solely in the fact that he uses point particles with the physical masses and charges to start the perturbation theory, while the standard approach begins with bare particles of infinite mass and charge that are made finite only in a mathematically questionable renormalization procedure.

Real electrons are of course not alone in the world, and hence are not strictly free. But it is commonly accepted in QFT that one considers scattering events in an asymptotic framework where free, physical particles arise in the limits t --> +-inf.
There are four levels of approximation:

• The free electron in vacuum (only with its dressing = self field):
psi(t,x) = exp {i[px - E(p)t]/h-bar} psi_0
psi_0 characterizes the state.
• The free electron in an external electromagnetic field. The state changes no longer in a harmonic fashion, but by a modified Dirac equation. The interaction is characterized by the form factor. Explicit radiation is absent or neglected, the loss of energy due to the neglected interaction with the environment is accounted for by the nonhermitian part of the form factors (which makes the Hamiltonian nonhermitian).
• The free electron as part of the asymptotic input/output of a scattering event. The transition probabilities are given by the textbook S-matrix. Here the electron is free only before and/or after scattering. Among others, it describes stimulated photon emission and radiation losses,
e + s gamma --> e + s' gamma (s,s'=1,2,...)
During the scattering process itself, the electron loses its recognizable individuallity, and all there is (meaningfully) is the quantum field with an indefinite particle content. QED gives the full observable answer, with fully dressed, free electrons entering and leaving the scattering event for t --> +-inf, but complex quantum field behavior at finite times.
• The electron in dense matter. Here the electrom loses its recognizable existence as a particle. There is additional dressing by the medium. Free is only a quasi-particle (the electron plus dressing induced by the medium), with different mass and different electromagnetic properties.
In the first three cases, the free electron is always the same object, but its context is different.
Of course, since there is currently no rigorous non-perturbative formulation of QED, everything can be calculated approximately only. But there are lots of established nonrigorous ways to approximately compute everything of interest, in principle to arbitrary order in either alpha or hbar or c^{-1}. Usually, low order results are already highly accurate.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ