Spin-½ Probabilities from the Ordinary Dot and Cross Product: A Real Three-Vector Picture of the Spinor Inner Product

Srinivasa Mugeraya

Published Sep 24, 2026• 0 citations

Abstract

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 \( \overline{a}b \) 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.

DOI: 10.20944/preprints202308.1564.v3

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