the plasma frequency, the compression of the electron fluid is an adiabatic process and is equal to three. At low frequencies, the compression is an isothermal process and is equal to one. Retardation effects have been neglected in obtaining the plasma-wave dispersion relation. For low frequencies, the dispersion relation becomes where is the Debye number, which is the inverse of the Debye length. This suggests that the propagator is In fact, if the retardation effects are not neglected, then the dispersion relation is which does indeed yield the guessed propagator. This propagator is the same as the massive Coulomb propagator with the mass equal to the inverse Debye length. The interaction energy is therefore The Coulomb potential is screened on length scales of a Debye length. Plasmons In a quantum electron gas, plasma waves are known as plasmons. Debye screening is replaced with Thomas–Fermi screening to yield where the inverse of the Thomas–Fermi screening length is and is the Fermi energy This expression can be derived from the chemical potential for an electron gas and from Poisson's equation. The chemical potential for an electron gas near equilibrium is constant and given by where is the electric potential. Linearizing the Fermi energy to first order in the density fluctuation and combining with Poisson's equation yields the screening length. The force carrier is the quantum version of the plasma wave. Two line charges embedded in a plasma or electron gas We consider a line of charge with axis in the z direction embedded in an electron gas where is the distance in the xy-plane from the line of charge, is the width of the material in the z direction. The superscript 2 indicates that the Dirac delta function is in two dimensions. The propagator is where is either the inverse Debye–Hückel screening length or the inverse Thomas–Fermi screening length. The interaction energy is where and are Bessel functions and is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (see ) and For , we have Coulomb potential between two current loops embedded in a magnetic field Interaction energy for vortices We consider a charge density in tube with axis along a magnetic field embedded in an electron gas where is the distance from the guiding center, is the width of the material in the direction of the magnetic field where the cyclotron