# QED for fields obeying a square root operator equation

## TL;DR

This paper explores non-local field equations with a square root operator for scalar particles, deriving scattering matrix elements in quantum electrodynamics and comparing them with traditional Klein-Gordon results, with implications for spin-1/2 particles.

## Contribution

It introduces a non-local square root operator approach to quantum electrodynamics and compares its scattering results with conventional local equations, proposing extensions to spin-1/2 particles.

## Key findings

- Derived scattering matrix elements using non-local equations
- Compared results with Klein-Gordon-based calculations
- Proposed extension to spin-1/2 particles

## Abstract

Instead of using local field equations - like the Dirac equation for spin-1/2 and the Klein-Gordon equation for spin-0 particles - one could try to use non-local field equations in order to describe scattering processes. The latter equations can be obtained by means of the relativistic energy together with the correspondence principle, resulting in equations with a square root operator. By coupling them to an electromagnetic field and expanding the square root (and taking into account terms of quadratic order in the electromagnetic coupling constant e), it is possible to calculate scattering matrix elements within the framework of quantum electrodynamics, e.g. like those for Compton scattering or for the scattering of two identical particles. This will be done here for the scalar case. These results are then compared with the corresponding ones based on the Klein-Gordon equation. A proposal of how to transfer these reflections to the spin-1/2 case is also presented.

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Source: https://tomesphere.com/paper/0704.0425