Quantum state evolution in C2 and G3+

TL;DR

This paper explores the generalization of quantum state evolution from traditional complex space to geometric algebra, revealing that the latter provides a richer, more detailed description of quantum states.

Contribution

It introduces a novel framework for quantum states using geometric algebra, extending the complex plane to 3D bivectors and analyzing the implications for state evolution.

Findings

Geometric algebra offers a more comprehensive description of quantum states.

Traditional complex space provides limited information compared to geometric algebra.

The generalized structure impacts understanding of quantum state dynamics.

Abstract

It was shown that quantum mechanical qubit states as elements of two dimensional complex space can be generalized to elements of even subalgebra of geometric (Clifford) algebra over Euclidian space. The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf fibration to maps generated by arbitrary unit value bivectors. Analysis of the new structure demonstrates that quantum state evolution in terms of two dimensional complex space gives only restricted information compared to that in even geometric algebra.