charge conservation. Here vanishes because we are considering static forces. With the current in this form the energy of interaction can be written The propagator equation for the Proca Lagrangian is The spacelike solution is which yields which evaluates to (see ) which reduces to in the limit of small . The interaction energy is the negative of the interaction Lagrangian. For two like particles traveling in the same direction, the interaction is attractive, which is the opposite of the Coulomb interaction. Darwin interaction in a plasma In a plasma, the dispersion relation for an electromagnetic wave is () which implies Here is the plasma frequency. The interaction energy is therefore Magnetic interaction between current loops in a simple plasma or electron gas The interaction energy Consider a tube of current rotating in a magnetic field embedded in a simple plasma or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as where and is the unit vector in the direction of the magnetic field. Here indicates the dimension of the material in the direction of the magnetic field. The transverse current, perpendicular to the wave vector, drives the transverse wave. The energy of interaction is where is the distance between the centers of the current loops and is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integrals and See . A current in a plasma confined to the plane perpendicular to the magnetic field generates an extraordinary wave. This wave generates Hall currents that interact and modify the electromagnetic field. The dispersion relation for extraordinary waves is which gives for the propagator where in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by the cyclotron frequency is given by (Gaussian units) and the plasma frequency (Gaussian units) Here is the electron density, is the magnitude of the electron charge, and is the electron mass. The interaction energy becomes, for like currents, Limit of small distance between current loops In the limit that the distance between current loops is small, where and and and are modified Bessel functions. we have assumed that the two currents have the same charge and speed. We have made use of the integral (see ) For small the integral becomes For large the integral becomes Relation to the quantum Hall effect The screening wavenumber can