Recoil Correction in the Dirac-Coulomb Problem

The expression for the first recoil correction to the Dirac-Coulomb spectrum is obtained employing the gauge invariance. Relativistic two-body problem in quantum electrodynamics has been exactly solved in the only limiting case m/M→0, α→0 at fixed Zα (here M and m are masses of the constituents, Z|e| and e are their electric charges, α = e/h̄c is the fine structure constant, h̄ = c = 1). In this limit, an infinitely heavy nucleus holds still being the source of the constant in time Coulomb field. A wavefunction of the system reduces to that of the light particle, the electron, and obeys the Dirac equation in the Coulomb field, (~α~p+ βm+ VC − E)ψ = 0. (1) Expressions for the first (linear in m/M) recoil corrections to energies of the DiracCoulomb bound states were obtained several years ago by V.M. Shabaev [1, 2]. He used the perturbation theory in Zα, summing up contributions of a given order in Zα, linear in m/M . The present note is devoted to a simple derivation of the Shabaev’s result, with only minor reference to the perturbation theory. As a guiding principle we will use the gauge invariance of QED. To begin with, let us generalize the equation (1) to an arbitrary gauge. Since VC = ZαD00 , and an infinitely heavy particle at rest can emit (or absorb) only zero component of the vector potential, we have: {αμ (pμ − ZαDμ0) + βm}ψ = 0, (2) where α0 = 1 by definition, p0 = E. The Dirac-Coulomb spectrum, that is the mutual arrangement of the Green’s function singularities at the complex E plane, is certainly gauge-invariant. Now let us take into account the motion and interaction of the nucleus to first order in 1/M . They are described by the term