The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order
between two charged particles in a vacuum where c is the speed of light.
It was derived before the advent of quantum mechanics and resulted from a more detailed investigation of the classical, electromagnetic interactions of the electrons in an atom.
From the Bohr model it was known that they should be moving with velocities approaching the speed of light.
[1] The full Lagrangian for two interacting particles is
where the Coulomb interaction in Gaussian units is
is the unit vector in the direction of r. The first part is the Taylor expansion of free Lagrangian of two relativistic particles to second order in v. The Darwin interaction term is due to one particle reacting to the magnetic field generated by the other particle.
If higher-order terms in v/c are retained, then the field degrees of freedom must be taken into account, and the interaction can no longer be taken to be instantaneous between the particles.
In that case retardation effects must be accounted for.
The first term on the right generates the Coulomb interaction.
The second term generates the Darwin interaction.
The vector potential in the Coulomb gauge is described by[2]: 242
which must be true if the divergence of the transverse current is zero.
is the component of the Fourier transformed current perpendicular to k. From the equation for the vector potential, the Fourier transform of the vector potential is
The inverse Fourier transform of the vector potential is
(see Common integrals in quantum field theory § Transverse potential with mass).
The Darwin interaction term in the Lagrangian is then
For interacting particles, the equation of motion becomes
The Darwin Hamiltonian for two particles in a vacuum is related to the Lagrangian by a Legendre transformation
This Hamiltonian gives the interaction energy between the two particles.
It has recently been argued that when expressed in terms of particle velocities, one should simply set
The structure of the Darwin interaction can also be clearly seen in quantum electrodynamics and due to the exchange of photons in lowest order of perturbation theory.
When the photon has four-momentum pμ = ħkμ with wave vector kμ = (ω /c, k), its propagator in the Coulomb gauge has two components.
[4] gives the Coulomb interaction between two charged particles, while describes the exchange of a transverse photon.
in this gauge, it doesn't matter if one uses the particle momentum before or after the photon couples to it.
In the exchange of the photon between the two particles one can ignore the frequency
The two parts of the propagator then give together the effective Hamiltonian for their interaction in k-space.
This is now identical with the classical result and there is no trace of the quantum effects used in this derivation.
A similar calculation can be done when the photon couples to Dirac particles with spin s = 1/2 and used for a derivation of the Breit equation.
It gives the same Darwin interaction but also additional terms involving the spin degrees of freedom and depending on the Planck constant.