The Bethe–Salpeter QED Wave Equation for Bound-State Computations of Atoms and Molecules

Abstract

Interactions in atomic and molecular systems are dominated by electromagnetic forces and the theoretical framework must be in the quantum regime. The physical theory for the combination of quantum mechanics and electromagnetism, quantum electrodynamics has been "established" by the mid-twentieth century, primarily as a scattering theory. To describe atoms and molecules, it is important to consider bound states. In the nonrelativistic quantum mechanics framework, bound states can be efficiently computed using robust and general methodologies with systematic approximations developed for solving wave equations. With the sight of the development of a computational quantum electrodynamics framework for atomic and molecular matter, the field theoretic Bethe−Salpeter wave equation expressed in space−time coordinates, its exact equal-time variant, and emergence of a relativistic wave equation, is reviewed. A computational framework, with initial applications and future challenges in relation with precision spectroscopy, is also highlighted.

Pre-Born-Oppenheimer Dirac-Coulomb-Breit computations for two-body systems

“…Evaluation of the already formulated perturbative correction with H ∆ is left for future developments, which appears to be possible along the lines reviewed in Ref. [28]. Although analytic evaluation of the energy and its corrections is not possible in this framework, the numerical results can be converged to high precision, which is demonstrated in the present work.…”

Bound-State Relativistic Quantum Electrodynamics: A Perspective for Precision Physics with Atoms and Molecules

“…( where [ ] X i 4 is the "restricted" kinetic balance for the 4-spinor of the ith particle. Finally, ) 16 1 2 16 12 16 is the antisymmetrization operator defined in ref 57, in which the permutation operator [ ] 12 16 exchanges the particle indices and the corresponding spinor structure.…”

“…, ref ) to high precision. ,− Concerning the no-pair energy (Tables and ), we will always refer to this nontrivial ++ part and highlight an explicitly correlated variational solution approach recently developed for its computation. Then, the last part of the paper outlines plans for a perturbative account of the energy-dependent term scriptV normalQ normalE normalD ( E ) = V δ + scriptV italicϵ ( E ) , , in which, of course, some of the operators will act on the +–, −+, and −– subspaces in a nontrivial manner.…”

“…In this work, a perturbative correction to the no-pair energy is considered, which may be most straightforward at the lowest orders of perturbation theory. Using a “zeroth-order” approximation of scriptV italicϵ , scriptV ϵ false( 0 false) ( E ) = ( E − h 1 − h 2 ) ∫ − ∞ + ∞ normald ε − 2 π normali s 1 s 2 scriptK normalΔ s 1 s 2 V normalI the first-order energy correction can be obtained as , normalΔ E scriptK normalΔ false( 1 , 0 false) ≈ ⟨ Φ n p | ∫ − ∞ + ∞ d ε − 2 π i false( s 1 + + s 2 + false)…”

Relativistic QED developments for precision spectroscopy

“…This approach was first proposed by Salpeter [196] to obtain recoil corrections to the energy levels of hydrogen-like atoms, applied by Fullton and Martin to arbitrary-mass two-body systems [197] and by Sucher, Duglas and Kroll and by Zhang and Drake for the helium atom [158,198,199,200]. The approach was also discussed recently in connection with the no-pair Dirac-Coulomb-Breit equation [201].…”