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

Journal Article · Tue Jan 01 04:00:00 EST 1985 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.526791· OSTI ID:6161117

Hostler, L C

We investigate the existence of orthogonality and completeness relations for the eigenvalue problem associated with the differential operator ..lambda.. = -Pi/sub ..mu../Pi/sub ..mu../ -iesigma x (E+iB), Pi/sub ..mu../ = -i partial/sub ..mu../-eA/sub ..mu../. The operator ..lambda.. acts on 2 x 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 ..lambda.. is self-adjoint with respect to the not positive-definite inner product (phi/sub b/; phi/sub a/) = ..integral.. d/sup 4/x phi-bar/sub b/phi/sub a/, where phi-bar/sub b/ is defined as phi-bar/sub b/ = phi/sup dagger//sub b/(-iPiX/sub 4/-sigmaxPiX). 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')/sup 2/-(m)/sup 2/.

• Research Organization: Physics Department, Wilkes College, Wilkes-Barre, Pennsylvania 18766
• OSTI ID: 6161117
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 26:1; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

645400* -- High Energy Physics-- Field Theory
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BANACH SPACE
COULOMB FIELD
EIGENFUNCTIONS
EIGENVALUES
ELECTRIC FIELDS
ELECTRODYNAMICS
ELECTROMAGNETIC FIELDS
FIELD THEORIES
FUNCTIONS
GENERAL RELATIVITY THEORY
HILBERT SPACE
INVARIANCE PRINCIPLES
LORENTZ INVARIANCE
MAGNETIC FIELDS
MASS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MINKOWSKI SPACE
POTENTIALS
QUANTUM ELECTRODYNAMICS
QUANTUM FIELD THEORY
QUANTUM OPERATORS
RELATIVITY THEORY
SPACE
SPINOR FIELDS
SPINORS