The quantum electrodynamic problem of two electrons

Abstract

We solve the problem of the interaction of two quasimolecular electrons at an arbitrary distance from each other, i.e., near different atoms (nuclei). We regard the interaction as a second-order effect of the quantum electrodynamic perturbation theory in the coordinate representation. Taking the natural condition of the symmetry of the retardation factor, the electron spins, and the effects of retardation of the relativistic interaction of the two quasimolecular electrons located near different nuclei into account consistently, we obtain additional terms in the interelectron interaction operator compared with the standard Breit operator.

The generalized Breit operator of the interaction between two quasimolecular electrons in the framework of the third-order effects of quantum electrodynamics

“…In view of this, in our previous paper [17], we have given the arguments that cast doubt on the possibility of using the Breit operator (1) for determination of the asymptotics of the exchange interaction, corresponding to simultaneous capture of two electrons in slow collisions of multiply-charged ions with atoms [3,4]. In essence, these arguments pertain equally to other two-electron processes with rearrangement, including radiative collisional ones, if the basic contribution to the transition probability comes from the configuration when the electrons are located near different nuclei and the approximation of independent electrons is valid as a zero approximation.…”

“…As mentioned in the introduction, the fundamental possibility of generalization of the Breit operator to the case of quasimolecular electrons has been established in papers [19,20] in the example of the problem of interaction of two bound electrons belonging to two hydrogenlike atoms, located at an arbitrary distance from each other. However, referring to these papers, we have discovered [17] that the generalized Breit operator obtained therein does not manifest the symmetry with respect to the interacting particles and, therefore, cannot be utilized in the consistent relativistic quantum theory. As mentioned in [17], the essential deficiency of the procedure of passing from the retardation factor to the corresponding operator, which was accepted in papers [19,20], consists in an unequal treatment of the pair of interacting particles.…”

“…However, referring to these papers, we have discovered [17] that the generalized Breit operator obtained therein does not manifest the symmetry with respect to the interacting particles and, therefore, cannot be utilized in the consistent relativistic quantum theory. As mentioned in [17], the essential deficiency of the procedure of passing from the retardation factor to the corresponding operator, which was accepted in papers [19,20], consists in an unequal treatment of the pair of interacting particles. The problem of two quasimolecular electrons with emission (absorption) of a real photon demands a further careful investigation for this very reason.…”

“…The method, which we shall use to solve this important quantum electrodynamic problem, represents a further development of the method applied in papers [17,19] to solving the problem of the interaction of two atomic electrons in the framework of the second-order effects of quantum electrodynamics. As shown below, consistent application of the procedure of symmetrization of the retardation factor with respect to both the particles leads to the appearance of additional terms in the relativistic operator of the interaction of the two electrons when compared to both the standard (1) and generalized [20] Breit operators.…”

“…From the form of expressions ( 15) and ( 20), it formally follows that the parameter of expansion (18) is actually not 1/c, but the small dimensionless quantity (17), and it is obvious that the expansion coefficients (18) in turn are power series in r/R. Actually, it means that if the distance between the electrons is comparable with the distance between the nuclei, then one can use formulae (20) and expand the functions f 0 , f 1 , f 2 (18) into power series with respect to r/R.…”