Variational derivation of relativistic fermion–antifermion wave equations in QED

Abstract

We present a variational method for deriving relativistic two-fermion wave equations in a Hamiltonian formulation of QED. A reformulation of QED is performed, in which covariant Green functions are used to solve for the electromagnetic field in terms of the fermion fields. The resulting modified Hamiltonian contains the photon propagator directly. The reformulation permits one to use a simple Fock-space variational trial state to derive relativistic fermion-antifermion wave equations from the corresponding quantum field theory. We verify that the energy eigenvalues obtained from the wave equation agree with known results for positronium.

Analysis of inter-quark interactions in classical chromodynamics

“…The formal solution of (1-19) involves the use of the symmetric Green function of that equation, and this requires a choice of gauge. We shall use the Lorentz gauge, ∂ µ A µ (a) (x) = 0, whereupon the "glue" equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) can be rewritten as an integral equation,…”

“…We have not included the free-gluon solution of the homogeneous equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) in (2-1), since free gluons do not arise and so free-gluon solutions will play no role in the present considerations.…”

“…Unfortunately, the integrals (3-12), (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)) that define the three and four point potentials cannot, in general, be evaluated explicitly, that is they cannot be expressed in terms of common analytic functions (at least we do not know how to do so). Nevertheless various general properties of these first-iterative-order non-Abelian corrections to the Coulombic inter-quark potential can be readily established, and analytical expressions can be obtained for particular situations.…”

Hyperfine structure and Zeeman splitting in two-fermion bound-state systems

“…The substitution of the partial-wave expansion (20) into the rest-frame form of Ansatz (6) leads to two categories of relations among the adjustable functions F s 1 s 2 (p):…”

“…In this work we present an analysis of the HFS of a two-fermion system in an external magnetic field based upon a reformulation of QED and the variational Hamiltonian formalism developed earlier [18][19][20]. A relativistic two-fermion wave equation for arbitrary fermion masses is, thus, derived from first principles.…”