The scattering behavior of a textbook Dirac particle (or ''point particle'') from an external electromagnetic field is simple and explicitly computable, but (at high accuracies) not realistic for any particle observed in Nature. Thus the Dirac particle is an idealized particle. The equation of motion of a real (as opposed to idealized) relativistic spin 1/2 particle is given by a modified Dirac equation that correctly describes the response to a classical electromagnetic field that is weak enough that pair creation and higher order terms in the field can be neglected. (This still allows motion in the central Coulomb field generated by a nucleus of charge $Z$ up to appoximately 137, and in particular the motion of an electron in the hydrogen atom.)

In the following, I derive the most general form such a modified Dirac equation may have. Special cases of this equation is actually used (and were always used) to compute the hyperfine splitting. **There is no other known way to derive the hyperfine splitting (the Lamb shift) or the anomalous magnetic moment of a spin 1/2 particle, except by using these equations.** The functions entering this modified Dirac equations (the form factors and self-energies) are either fitted to experiment (for hadrons) or derived from QED (for electrons) or QCD (for hadrons).

Since the equation must be covariant, we look for Poincare invariant field equations of Dirac type. The most general Hermitian Poincare invariant (pseudo-differential) operator on the space of a relativistic spin 1/2 particle is given in terms of $p:=i\hbar\partial$ by

\[

D_0(p):= p\!\!\!/\lambda(p^2)-m(p^2)

\]

with

\[

p\!\!\!/:=\gamma\cdot p,

\]

where $\gamma$ is the Dirac 4-vector and $\lambda(p^2)$ and $m(p^2)$ are appropriate real-valued functions.

If we allow for terms linear in an external (classical) electromagnetic field - the mean field generated by all other charged matter in the universe - with 4-vector potential $A(x)$, we find the most general first order interaction terms possible, leading to an operator of the form

\[

D(p):=D_0(p)-(j(p)\cdot A(x))

\]

with a vector

\[

j(p):=e(p^2)\gamma+i\mu(p^2)\Sigma p,

\]

where $\Sigma$ is the antisymmetric matrix with components $\Sigma_{\mu\nu}:=\tfrac12[\gamma_\mu,\gamma_\nu]$ and $e(p^2)$ and $\mu(p^2)$ are appropriate real-valued functions. (Ensuring gauge invariance would in general require additional terms nonlinear in $A$ not treated here.)

Thus Poincare invariance dictates that the single-particle dynamics of a 1-particle wave function is (to first order in the field) necessarily of the form of a modified Dirac equation

\[

D(p)\psi=0.

\]

Which functions appear must be determined for any kind of real particle by comparing with the results of spectral experiments (where the particle is confined by the field, leading to a discrete bound state spectrum) and of scattering experiments (where the particle is not confined by the field).

In the special case of a time-independent electric or magnetic field we may take a Fourier-Laplace transform, where $p$ becomes a momentum vector $p=Eu/c$, where $u$ is the 4-velocity of the mode and $E$ is a complex energy. The spectrum and the scattering matrix are then completely determined by the analytic behavior of the Greens function

\[

G(E):=D(p)^{-1}.

\]

The $u$-dependence can be removed by an appropriate Lorentz transformation. The real singularities of the $E$-dependence define the spectrum, complex singularities on the second sheet determine the resonances, and the discontinuity along branch cuts determines the scattering angles.

An external electric Coulomb field defines a hydrogen-like atom. (The special case where the particle is an idealized electron and the Coulomb field is produced by an idealized proton gives more or less idealized hydrogen atoms. One may replace the electron and/or proton by other particles to get a variety of variants.) Adding a constant electric or magnetic field induces spectral shifts. These allow to probe experimentally the structure of the functions occuring in the definition of $D(p)$.

A point particle of mass $m_0$ and charge $-e_0$ corresponds to the ordinary minimally coupled Dirac equation, which is the special case of constant functions with values

\[

\lambda(p^2)=1,~~~m(p^2)=m_0,~~~e(p^2)=e_0,~~~\mu(p^2)=0.

\]

In the absence of an external field, its spectrum consists of a single energy $E=m_0c^2$. The Coulomb field of a nucleus produces a mixed spectrum, with highly degenerate bound states due to an $SO(4)$ symmetry (from the conserved Lenz-Runge vector) and a continuous spectrum for scattering states obtained by ionization. Adding electric or magnetic fields produces a fine-splitting of degenerate levels, but there is a residual symmetry leaving some spectral degeneracy.

Real hydrogen-like atoms do not have this degeneracy but produce a hyperfine splitting defining the Lamb shift. This proves that real electrons, muons, tauons are not point particles. Indeed, all functions mentioned are now momentum-dependent and can be obtained by fitting appropriate ansatz functions to information obtained from spectral and scattering experiments. For obvious reasons one calls $m(p^2)$ the running mass and $e(p^2)$ the running charge.

Deviations from the point particle structure are traditionally characterized by an electric form factor

\[

F_1(p^2):=e(p^2)/e_0,

\]

a magnetic form factor

\[

F_2(p^2):=2m_0\mu(p^2)/e_0,

\]

and the so-called self-energy

\[

\Sigma(p\!\!\!/):=p\!\!\!/-m_0-D_0(p)

\]

involving $\lambda(p^2)$ and $m(p^2)$. The latter definition allows one to write the free Greens function in the conventional form

\[

D_0(p)^{-1}=(p\!\!\!/-m_0-\Sigma(p\!\!\!/))^{-1}

\]

suitable for simple perturbation theory.

For particles described by a quantum field theory such as the electron $e$, one can also compare with the results of renormalized perturbation theory for the scattering process $e+\gamma\to e$ at a virtual photon $\gamma$ to first order in the electromagnetic field but to arbitrary order in the number of loops. This defines the form factors and the self-energy perturbatively. For the electron, QED provides a very high accuracy description of these functions.

Since the photon is massless, QED has a nontrivial infrared behavior nottypically treated well in the standard QFT textbooks. In particular, thecontinuous spectrum starts directly at the lowest possible energy of a physical electron. This implies that the Green's function of the electron has no pole at the physical electron mass but an essential power singularity with an exponent defined within QED by an anomalous dimension. Thus the physical electron (i.e., the asymptotic QED state with electron number 1) does not have a well-defined electron mass, but its mass has a continuous spectrum, even in the absence of an external field. (Intuitively, this happens because the physical electron is dressed by a coherent state of photons that slightly change the electron momentum and hence mass; in perturbative language by a cloud consisting of an arbitrary number of soft virtual photons.) This continuous spectrum is therefore encoded in $D_0(p)$.

Thus QED tells us a lot about the nontrivial structure of the physical electron, encoded in the form factors and the self-energy, and proves that a single, nearly free electron is already a highly complex object. It is astonishing that it can be described to an exceedingly high accuracy (relative error of $10^{-12}$ in terms of two parameters only, namely the rest mass and rest charge of the electron.

References:

**Modified Dirac equations:** L.L. Foldy, The Electromagnetic Properties of Dirac Particles, Phys. Rev. 87 (1952), 688--693.

**Relativistic form factors:** D.R. Yennie, M.M. Levy and D.G. Ravenhall, Electromagnetic Structure of Nucleons, Rev. Mod. Phys. 29 (1957), 144--157.

**Popular article on particle form factors:** R. Wilson, Form factors of elementary particles, Physics Today 22 (1969), 47-53.

**1-loop calculations of electron form factors: **M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, New York, 1996. (In particular pp. 186 and 220)

**Electron branch cut and anomalous exponents: **T. Appelquist and J. Carazzone, Infrared singularities and massive fields, Phys. Rev. D11 (1975), 2856--2861.