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
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.[2]: 596–598
The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is[2]: 580–581
The vector potential in the Coulomb gauge is described by[2]: 242
The current generated by the second particle is
The transverse component of the current is
It is easily verified that
From the equation for the vector potential, the Fourier transform of the vector potential is
The inverse Fourier transform of the vector potential is
The Darwin interaction term in the Lagrangian is then
The equation of motion for one of the particles is
The equation of motion for a free particle neglecting interactions between the two particles is
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
The Hamiltonian becomes
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 in the last term and reverse its sign.[3]
The Hamiltonian equations of motion are
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. It has a polarization vector and couples to a particle with charge and three-momentum with a strength Since 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 compared with in the propagator working to the accuracy in that is needed here. 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.[4]