The Quaternionic Wavefunction: Spin-1/2 Particles, S^3, and SU(2)

John G. Van Geem

Abstract

This paper introduces the quaternionic wavefunction as a compact way to organize the standard spinor geometry of a spin-1/2 particle. A normalized two-component complex state vector \((\alpha,\beta)\in \mathbb C^2\) is regrouped into one quaternion \(q_\psi = a_0 + a_1\mathbf i + b_0\mathbf j + b_1\mathbf k\). Under this regrouping, normalization becomes \(|q_\psi|=1\), so the space of normalized representatives is \(S^3\). The paper then distinguishes carefully between two roles played by that same manifold: \(S^3\) as the state-representative sphere of normalized spinors, and \(S^3\) as the manifold underlying the Lie group of unit quaternions, which models \(SU(2)\). The mathematical content is standard; the project-specific contribution is the expository framing.

1. Introduction

This prequel exists to make later quaternionic papers feel natural before the series becomes more technical. The basic teaching move is simple: a spin-1/2 state in \(\mathbb C^2\) has four real coefficients, and those same four coefficients can be grouped into one quaternion.

2. From a spinor to a quaternionic wavefunction

Write the normalized spinor as

\[\psi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}, \qquad \alpha=a_0+a_1 i, \qquad \beta=b_0+b_1 i.\]

The quaternionic wavefunction is then defined by

\[q_\psi = a_0 + a_1\mathbf i + b_0\mathbf j + b_1\mathbf k.\]

This does not add new physics. It simply regroups the same four real coefficients into one quaternion.

Normalization becomes

\[|\alpha|^2+|\beta|^2=a_0^2+a_1^2+b_0^2+b_1^2=|q_\psi|^2.\]

So normalized quaternionic wavefunctions lie on the three-sphere \(S^3\).

3. Unit quaternions and \(SU(2)\)

The same manifold \(S^3\) also appears as the set of unit quaternions, but now in a different role: as a Lie group. Using the fixed convention

\[\Phi(a+b\mathbf i+c\mathbf j+d\mathbf k)=aI-i(b\sigma_x+c\sigma_y+d\sigma_z),\]

the unit quaternions model \(SU(2)\). This is the natural transformation group for spin-1/2 states. So the quaternionic wavefunction gives one way to see, geometrically, both the state sphere and the symmetry group that acts on it.

4. Spin-1/2 and the double cover

The group \(SU(2)\) is the double cover of \(SO(3)\). That is why spin-1/2 states are not ordinary spatial vectors: a full \(2\pi\) rotation does not return the spinor to the same representative, while a \(4\pi\) rotation does.

In this paper's language, that 4 pi property lives naturally on the quaternionic-wavefunction / spinor side of the geometry, not on the projected \(SO(3)\) side.

5. Why \(S^3\) is not just the Bloch sphere

Because normalized representatives lie on \(S^3\), it is tempting to identify \(S^3\) with the physical pure-state space. That is not correct. Global phase must still be quotiented out.

After quotienting global phase, the pure-state space is \(\mathbb{CP}^1\cong S^2\), the Bloch sphere. So the paper emphasizes a key distinction:

\(S^3\) is the sphere of normalized representatives, and
\(S^2\) is the projective pure-state space.

6. Standard mathematics versus project framing

The standard mathematics here is familiar: normalized spinors in \(\mathbb C^2\), the appearance of \(S^3\), the Bloch sphere, the quaternionic model of \(SU(2)\), and the double cover \(SU(2)\to SO(3)\).

The project-specific part is the language. The term quaternionic wavefunction is used as an expository bridge because it makes the geometry of spin-1/2 structure easier to see without changing the standard theory.

7. Scope and non-claims

This paper does not propose a new physical theory.
It does not replace ordinary complex quantum mechanics.
It does not claim quaternionic Hilbert-space quantum mechanics in the Adler sense.

8. Conclusion

The quaternionic wavefunction is a compact geometric way to look at a standard spin-1/2 state. By regrouping the four real coefficients of a normalized spinor into one quaternion, the normalized-state sphere becomes visibly \(S^3\). That same manifold also underlies the unit quaternions, which model \(SU(2)\). Once those two appearances are kept conceptually distinct, the geometry of spin-1/2 particles becomes much easier to teach.

References

Adler, Stephen L. Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, 1995.
Bloch, Felix. "Nuclear Induction." Physical Review 70(7-8):460-474, 1946.
Hall, Brian C. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed. Springer, 2015.
Kuipers, Jack B. Quaternions and Rotation Sequences. Princeton University Press, 1999.
Nielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information, 10th Anniversary Edition. Cambridge University Press, 2010.
Penrose, Roger, and Wolfgang Rindler. Spinors and Space-Time. Volume 1. Cambridge University Press, 1984.