# Sommerfeld effect as the vertex correction in three-dimensional space

## TL;DR

This paper demonstrates that the Sommerfeld effect in quantum mechanics arises from the vertex correction reduction from four to three dimensions, linking it to the wave function at the origin in two-body systems.

## Contribution

It establishes a direct connection between the Sommerfeld effect and the vertex correction as the wave function at the origin, highlighting the dimensional reduction's role.

## Key findings

- Vertex correction equals the wave function at the origin.
- Sommerfeld effect results from dimensional reduction.
- Reciprocal wave function is introduced into quantum mechanics.

## Abstract

It is shown that the correction to the vertex for fermion pair annihilation and production in the low-energy region is equal to the value of the wave function for the two-body system at the origin. The procedure also shows directly that the emergence of the Sommerfeld effect in quantum mechanics is the product of the reduction of the vertex correction from four spacetime dimensions to three-dimensional space. Meanwhile, the reciprocal wave function is introduced into quantum mechanics.