Visualizing Imaginary Rotations and Applications in Physics

TL;DR

This paper introduces a new way to visualize complex vector rotations using imaginary angles, with applications in physics such as Lorentz transformations and quantum spin matrices.

Contribution

It proposes defining a complex vector length and visualizing rotations through imaginary angles, offering new insights into physical transformations.

Findings

Complex vector length differs from traditional Hermitian length.

Imaginary angles enable visualization of rotations in complex space.

Applications include Lorentz transformations and quantum spin matrices.

Abstract

I discuss the notions of traditional vector length, and suggest defining a complex vector length for complex vectors, as opposed to the traditional Hermitian real length. The advantages of this are shown in the development of rotations through imaginary angles. Emphasis is placed on visualizing these quantities and rotations graphically, and I show some applications in physics: Lorentz transformations, Grassmann variables, and Pauli spin matrices.