Massless Wave States of Two-Fermion Systems

TL;DR

This paper derives and analyzes wave equations for massless two-fermion bound states using the Bethe-Salpeter framework, revealing unique wave structures and energy quantization due to strong renormalization effects.

Contribution

It introduces a wave equation for massless two-fermion states and characterizes their stationary wave solutions and energy branches, expanding understanding of relativistic bound states.

Findings

Wave function as an infinite thread with radius exceeding the Compton wavelength

Two quantized energy branches due to renormalization effects

Stationary wave states of massless composite bosons

Abstract

It is known that in the ladder approximation the relativistic two-fermion bound-state equation of Bethe and Salpeter has solutions corresponding to the binding energy equal to the total mass of the particles. The study of these massless states has been carried out only for the bound system at rest. Of course, such composite boson can not be in the state of rest. But it is more importantly that this approach for the massless boson can not be interpreted as the limiting case of a nonzero mass system because the phase velocity of the boson wave must equal to the speed of light. Using the Bethe-Salpeter equation in the ladder approximation, we have obtained the wave equation for the massless bound states of two fermions with equal masses and the electromagnetic interaction between them. Neglecting retardation of the interaction, solutions corresponded to the stationary wave states of the composite boson, have been found. The boson wave function can be represented as an infinite, straight thread, the transverse radius of which is more than the Compton wavelength of the fermion. Two energy branches of the bosons with quantized energies have been determined. The appearance of these branches is due to the strong renormalization of the fine structure constant for the massless states.