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The Quaternionic Wavefunction: Spin-1/2 Particles, S3, and SU(2)

John Van Geem

Published Apr 11, 2026• 0 citations
Publisher

Abstract

This paper introduces the quaternionic wavefunction as a compact way to organize the standard spinor geometry of a spin one half particle. It shows how a two component complex state can be regrouped into quaternionic form in a way that makes the geometric relationship among spinors, the three sphere, and the transformation group SU(2) easier to see. The paper emphasizes that the mathematics is standard and that its contribution is expository. Its goal is to clarify how spin one half structure, unit quaternions, and rotational geometry fit together, while carefully distinguishing the full normalized state geometry from the projected Bloch sphere description. It is intended as a conceptual prelude to the broader RQM paper series, not as a claim of a new physical theory.

DOI
10.5281/zenodo.19502969

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This paper introduces the quaternionic wavefunction as a compact way to organize the standard spinor geometry of a spin one half particle. It shows how a two component complex state can be regrouped into quaternionic form in a way that makes the geometric relationship among spinors, the three sphere, and the transformation group SU(2) easier to see. The paper emphasizes that the mathematics is standard and that its contribution is expository. Its goal is to clarify how spin one half structure, unit quaternions, and rotational geometry fit together, while carefully distinguishing the full normalized state geometry from the projected Bloch sphere description. It is intended as a conceptual prelude to the broader RQM paper series, not as a claim of a new physical theory.
The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere rotations to be represented as quaternionic multiplications, simplifying the form of the dynamical equations. Left-multiplications generally correspond to non-unitary transformations, providing a simpler (essentially classical) analysis of time-reversal. But the quaternion viewpoint also reveals a surprisingly large broken symmetry, as well as a potential way to restore it, via a natural expansion of the state space that has parallels to second order fermions. This expansion to ‘second order qubits’ would imply either a larger gauge freedom or a natural space of hidden variables.
Abstract. The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere rotations to be represented as quaternionic multiplications, simplifying the form of the dynamical equations. Left-multiplications generally correspond to non-unitary transformations, providing a simpler (essentially classical) analysis of time-reversal. But the quaternion viewpoint also reveals a surprisingly large broken symmetry, as well as a potential way to restore it, via a natural expansion of the state space that has parallels to second order fermions. This expansion to “second order qubits ” would imply either a larger gauge freedom or a natural space of hidden variables. ‡ Present address: The Graduate Center, CUNY, New York NY 10016. ar X iv
The standard formalism treats spin-½ states as elements of a complex vector space, and the geometrical content of that description is not easy to state simply. We show that the transition probability between two spin-½ pure states can be computed using only the ordinary dot and cross products of real three-vectors. Replacing the ket |-⟩ by the quaternion unit j and applying an overall phase factor of i carries a normalised state to a real unit three-vector. Writing A∘B for the sum of the dot product and the i-component of the cross product, we prove that square of |A∘B|2 equals the standard quantum probability exactly (Theorem 2). This operation is the projection of the quaternion product of conjugate of a‾b \overline{a}b ab onto the subalgebra spanned by 1 and i (Proposition 2), and by Lagrange’s identity its four component contributions partition unit probability. The construction does not remove the complex structure from the description of a spin-½ state: it relocates it from an algebraic scalar into a distinguished spatial axis, and we state the resulting limitations precisely. The calculation requires nothing beyond vector algebra, whereas the standard route requires a complex Hilbert space.
The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits.
In this paper the geometry of two and three-qubit states under local unitary groups is discussed. We first review the one qubit geometry and its relation with Riemannian sphere under the action of group SU(2)SU(2)SU(2). We show that the quaternionic stereographic projection intertwines between local unitary group SU(2)⊗SU(2)SU(2)\otimes SU(2)SU(2)⊗SU(2) and quaternionic Möbius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining stereographic projection for much more global group Sp(2)Sp(2)Sp(2), generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic Möbius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps are entanglement sensitive.