Lagrangian used in classical electrodynamics
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 free particle part is
The interaction is described by
where the Coulomb interaction in Gaussian units is
while the Darwin interaction is
Here q1 and q2 are the charges on particles 1 and 2 respectively, m1 and m2 are the masses of the particles, v1 and v2 are the velocities of the particles, c is the speed of light, r is the vector between the two particles, and
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.[2]: 596–598
Derivation in vacuum
The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is[2]: 580–581
where u is the relativistic velocity of the particle. 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
where the transverse current Jt is the solenoidal current (see Helmholtz decomposition) generated by a second particle. The divergence of the transverse current is zero.
The current generated by the second particle is
which has a Fourier transform
The transverse component of the current is
It is easily verified that
which must be true if the divergence of the transverse current is zero. We see that
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
where we have kept only the lowest order term in v/c.
The inverse Fourier transform of the vector potential is
where
(see Common integrals in quantum field theory § Transverse potential with mass).
The Darwin interaction term in the Lagrangian is then
where again we kept only the lowest order term in v/c.
Lagrangian equations of motion
The equation of motion for one of the particles is
where p1 is the momentum of the particle.
Free particle
The equation of motion for a free particle neglecting interactions between the two particles is
Interacting particles
For interacting particles, the equation of motion becomes
Hamiltonian for two particles in a vacuum
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]
Equations of motion
The Hamiltonian equations of motion are
and
which yield
and
Quantum electrodynamics
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]
![{\displaystyle D_{00}(k)={1 \over \mathbf {k} ^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4874f3df5f74da679b951018932c66adb1ddfd62)
gives the Coulomb interaction between two charged particles, while
![{\displaystyle D_{ij}(k)={1 \over \omega ^{2}-c^{2}\mathbf {k} ^{2}}\left(\delta _{ij}-{k_{i}k_{j} \over \mathbf {k} ^{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/653f2d3e18de4dee89f38131a13000ace893329a)
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
![{\displaystyle H_{int}(\mathbf {k} )={4\pi q_{1}q_{2} \over \mathbf {k} ^{2}}-{4\pi q_{1}q_{2} \over m_{1}m_{2}c^{2}\mathbf {k} ^{2}}\mathbf {p} _{1}\cdot \left(\mathbf {1} -{\hat {\mathbf {k} }}{\hat {\mathbf {k} }}\right)\cdot \mathbf {p} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/824b7f4308d342aae955cf190fc81de12db153eb)
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]
See also
References
- ^ C.G. Darwin, The Dynamical Motions of Charged Particles, Philosophical Magazine 39, 537-551 (1920).
- ^ a b c Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X.
- ^ K.T. McDonald, Darwin Energy Paradoxes, Princeton University (2019).
- ^ a b V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Relativistic Quantum Theory, Pergamon Press, Oxford (1971).