# Polar analysis of the Dirac equation through dimensions

## TL;DR

This paper investigates how the dimensionality of space-time affects the structure and degrees of freedom of the Dirac equation in polar form, revealing special cases in 4 and 8 dimensions where the equations are well-matched.

## Contribution

It provides a detailed analysis of the polar form of the Dirac equation across various dimensions, highlighting the unique cases in 4 and 8 dimensions where the degrees of freedom and equations align.

## Key findings

- In dimensions 4 and 8, degrees of freedom match the number of independent equations.
- In other dimensions, issues of under- or over-determination arise.
- Polar form effectively reveals the relationship between degrees of freedom and equations.

## Abstract

We consider the polar form of the spinor field equation in an n-dimensional space-time, studying the way in which the space-time dimension influences the number of the independent field equations and the number of the degrees of freedom of the spinor field written in the polar form: we will find that this polar form is the clearest tool to make manifest the fact that the degrees of freedom of the spinor field and the independent field equations match in dimensions 4 and 8 alone while in all the other instances there will be problems of general under-determination or over-determination.