Relativistic three-fermion wave equations (original) (raw)
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Journal of Physics B: Atomic, Molecular and Optical Physics, 1990
Coupled integral equations for a relativistic two-fermion system of masses m and M are derived variationally within the Hamiltonian formalism of quantum electrodynamics, using an improved ansatz that is sensitive to all terms in the Hamiltonian. The equations are solved approximately to determine the eigenvalues and eigenfunctions, at arbitrary coupling, for various states of the two-particle system.
Variational derivation of relativistic fermion–antifermion wave equations in QED
Journal of Mathematical Physics, 2004
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.
Journal of Physics B: Atomic, Molecular and Optical Physics, 2008
The variational method, within the Hamiltonian formalism of reformulated QED is used to determine relativistic wave equations for a system of three fermions of arbitrary mass interacting electromagnetically. The interaction kernels of the equations are, in essence, the invariant M matrices in lowest order. The equations are used to obtain relativistic O(α 2) corrections to the non-relativistic ground state energy levels of the Muonium negative ion (µ + e − e −) as well as of Ps − and H − , using approximate variational three-body wave functions. The results are compared with other calculations, where available. The relativistic correction for Mu − is found to be −1.0773×10 −4 eV.
The two-fermion relativistic wave equations of constraint theory in the Pauli–Schrödinger form
Journal of Mathematical Physics, 1994
The two-fermion relativistic wave equations of Constraint Theory are reduced, after expressing the components of the 4 × 4 matrix wave function in terms of one of the 2 × 2 components, to a single equation of the Pauli-Schrödinger type, valid for all sectors of quantum numbers. The potentials that are present belong to the general classes of scalar, pseudoscalar and vector interactions and are calculable in perturbation theory from Feynman diagrams. In the limit when one of the masses becomes infinite, the equation reduces to the two-component form of the one-particle Dirac equation with external static potentials. The Hamiltonian, to order 1/c 2 , reproduces most of the known theoretical results obtained by other methods. The gauge invariance of the wave equation is checked, to that order, in the case of QED.
2007
The variational method, within the Hamiltonian formalism of reformulated QED is used to determine relativistic wave equations for a system of three fermions of arbitrary mass interacting electromagnetically. The interaction kernels of the equations are, in essence, the invariant M matrices in lowest order. The equations are used to obtain relativistic O(α 2) corrections to the non-relativistic ground state energy levels of the Muonium negative ion (µ + e − e −) as well as of Ps − and H − , using approximate variational three-body wave functions. The results are compared with other calculations, where available. The relativistic correction for Mu − is found to be −1.0773×10 −4 eV.
3 Variational Two Fermion Wave Equations in Qed: Muonium Like Systems
2016
We consider a reformulation of QED 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. A simple Fock-state variational trial function is used to derive relativistic two-fermion equations variationally from the expectation value of the Hamiltonian of the field theory. The interaction kernel of the equation is shown to be, in essence, the invariant M matrix in lowest order. Solutions of the two-body equations are presented for muonium like system for small coupling strengths. The results compare well with the observed muonium spectrum, as well as that for hydrogen and muonic hydrogen. Anomalous magnetic moment effects are discussed.
arXiv (Cornell University), 2004
A new kind of the relativistic three-body equations for the three fermion systems are suggested. These equations are derived in the framework of the standard fieldtheoretical S-matrix approach in the time-ordered three dimensional form. Therefore corresponding relativistic covariant equations are three-dimensional from the beginning and the considered formulation is free of the ambiguities which appear due to a three dimensional reduction of the four dimensional Bethe-Salpeter equations. The solutions of the considered equations satisfy automatically the unitarity condition and for the leptons these equations are exactly gauge invariant even after the truncation over the multiparticle (n > 3) intermediate states. Moreover, the form of these three-body equations does not depend on the choice of the model Lagrangian and it is the same for the formulations with and without quark degrees of freedom. The effective potential of the suggested equations is defined by the vertex functions with two on-mass shell particles. It is emphasized that these INPUT vertex functions can be constructed from experimental data. Special attention is given to the comparison with the three-body Faddeev equations. Unlike to these equations, the suggested three-body equation have the form of the Lippmann-Schwinger-type equations with the connected potential. In addition, the microscopical potential of the suggested equations contains the contributions from the three-body forces and from the particle creation (annihilation) mechanism on the one external particles. The structure of the three-body forces, appearing in the considered field-theoretical formulation, is analyzed.
Bound-state variational wave equation for fermion systems in quantum electrodynamics
Canadian Journal of Physics, 2007
We present a formulation of the Hamiltonian variational method for QED which enables the derivation of relativistic few-fermion wave equation that can account, at least in principle, for interactions to any order of the coupling constant. We derive a relativistic two-fermion wave equation using this approach. The interaction kernel of the equation is shown to be the generalized invariant M matrix including all orders of Feynman diagrams. The result is obtained rigorously from the underlying QFT for arbitrary mass ratio of the two fermions. Our approach is based on three key points: a reformulation of QED, the variational method, and adiabatic hypothesis. As an application we calculate the one-loop contribution of radiative corrections to the two-fermion binding energy for singlet states with arbitrary principal quantum number n, and ℓ = J = 0 . Our calculations are carried out in the explicitly covariant Feynman gauge. (tr)Jms Jm J f J (p)Y ms 1 s 2 J
Variational two-fermion wave equations in quantum electrodynamics: Muoniumlike systems
Journal of Mathematical Physics, 2005
We consider a reformulation of QED 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. A simple Fock-state variational trial function is used to derive relativistic two-fermion equations variationally from the expectation value of the Hamiltonian of the field theory. The interaction kernel of the equation is shown to be, in essence, the invariant M matrix in lowest order. Solutions of the two-body equations are presented for muonium like system for small coupling strengths. The results compare well with the observed muonium spectrum, as well as that for hydrogen and muonic hydrogen. Anomalous magnetic moment effects are discussed.
Variational Two Fermion Wave Equations in QED: Muonium Like Systems
2003
We consider a reformulation of QED 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. A simple Fock-state variational trial function is used to derive relativistic two-fermion equations variationally from the expectation value of the Hamiltonian of the field theory. The interaction kernel of the equation is shown to be, in essence, the invariant M-matrix in lowest order. Solutions of the two-body equations are presented for muonium like system for small coupling strengths. The results compare well with the observed muonium spectrum, as well as that for hydrogen and muonic hydrogen. Anomalous magnetic moment effects are discussed.