Mass eigenfunction expansions for the relativistic Kepler problem and arbitrary static magnetic field in relativistic quantum theory

Abstract

We investigate the existence of orthogonality and completeness relations for the eigenvalue problem associated with the differential operator Λ=−ΠμΠμ −ieσ ⋅ (E+iB), Πμ=−i ∂μ−eAμ. The operator Λ acts on 2×1 Pauli-type spinor fields defined over all Minkowski space, and may be interpreted as the square of the mass of a charged Dirac particle moving in an external c-number electromagnetic field. We show that Λ is self-adjoint with respect to the not positive-definite inner product (φb; φa)=∫ d4x φ̄bφa, where φ̄b is defined as φ̄b=φ†b(−iΠ↙4−σ⋅Π↙). A proof is provided for the Coulomb case that the mass eigenfunctions form a complete set in spite of the indefinite metric in Hilbert space. The mass eigenfunction expansion of the propagator is worked out explicitly for the Kepler case. This mass eigenfunction expansion is expected to be quite useful for bound state calculations in quantum electrodynamics, since it involves the covariant denominators (m′)2−(m)2.