I am currently exploring an intriguing topic related to quaternion-based wave functions and have encountered a mathematical challenge that I hope to get some insights on. The concept is detailed in an article on arXiv, which presents a two-component wave function represented as a quaternion: $$\psi(\boldsymbol{x},t)=a(\boldsymbol{x},t)+b(\boldsymbol{x},t)\boldsymbol{i}+c(\boldsymbol{x},t)\boldsymbol{j}+d(\boldsymbol{x},t)\boldsymbol{k}$$
It is known that:
$$s=\bar{\psi}i\psi =(a^{2}+b^{2}-c^{2}-d^{2})\boldsymbol{i}+2(bc-ad)\boldsymbol{j}+2(ac+bd)\boldsymbol{k}$$
where the two-component wave function is defined as $ \left|\psi_{\pm}\right\rangle=\left[\psi_{+},\psi_{-}\right]^{\mathrm{T}}$ with $\psi_{+}=a+\mathrm{i}b$ and $\psi_{-}=c+\mathrm{i}d.$
I am attempting to derive the expressions: $$\begin{cases}s_1=|\psi_+|^2-|\psi_-|^2,\\s_2=\mathrm{i}(\psi_+^*\psi_--\psi_+\psi_-^*),\\s_3=\psi_+^*\psi_-+\psi_+\psi_-^*.\end{cases}$$
My questions are as follows:
1.How can one derive the above form from the known equations? I am particularly interested in understanding the steps involved in this derivation.
2.In the context of $s=\bar{\psi}i\psi$, what is the role or interpretation of the imaginary number $i$?
I would greatly appreciate any guidance or references that could help illuminate these questions. Thank you in advance for your time and assistance.
The website of the article where this issue is located is: https://arxiv.org/abs/2403.00596