Geometric Algebra Speaks Quantum Esperanto, I

The fundamental Stern-Gerlach (SG) experiments suggest that the (pure) states of a q-bit are the points of the unit sphere \(S^2\) (in some suitable system of units), with a distinguished vector corresponding to the direction of the magnetic field. The goal of this paper is to elucidate the Hermitian structure of the algebra of geometric quaternions \({\textbf {H}}=\mathcal {G}_3^+\) (that is, the even algebra of the geometric algebra of the Euclidean 3D space) which allows to regard it as the Hilbert space of the q-bit. The main results are phrased in terms of an explicit ket map \(\kappa : \textbf{H}\rightarrow E_3\) such that \(|\kappa (\mathfrak {q})|=|\mathfrak {q}|\) for all \(\mathfrak {q}\in \textbf{H}\) , and include: that \(\kappa (\mathfrak {q}')=\kappa (\mathfrak {q})\) if and only if \(\mathfrak {q}'\equiv \mathfrak {q}\) (this relation denotes that the two quaternions differ by a phase factor –a unit geometric complex number); that \(\kappa \) is onto; a check that the computed probabilities obey the statistics of the SG experiments; and a recall of the relations between the multiplicative group \(\textbf{H}^{\times }\) and the rotation group SO( \(E_3\) ). A sequel paper will explore other facets of the proposed analysis, including the study of the polarization states of electromagnetic waves and more complex spin systems. In conclusion: