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

1. Introduction

In quantum mechanics, spinors are used to calculate the probability amplitudes of spin-½ particles. The standard formalism classifies them as elements of a complex vector space, distinguishing them from elements of a real vector space. Because quantum state vectors are abstract objects, little is known about them beyond the fact that they predict physical outcomes exactly. The geometrical significance and physical interpretation of spinors remain difficult to state simply ([1], p. 430, [2], p. 129, [3], pp. 11–12, [29], p. 9, [30], p. 1). Given the absence of consensus on what quantum theory implies about the physical world, there is continuing interest in reconstructing its axioms from operational principles [4,37,38].

The spinor representation of spin-½ states maps directly to a unit quaternion [5]. Quaternions are isomorphic to the real algebra spanned by the identity and $\mathit{i}$ times the Pauli matrices [6,7,8], [9] (Section 13.4). The Pauli matrices are Hermitian with ${\sigma }_k^2=+1$, whereas the quaternion units are anti-Hermitian with square −1; multiplication by $\mathit{i}$ converts between the two conventions. Quaternions rotate real vectors in an ordinary sphere, whereas spin matrices rotate spinors in a Bloch sphere. Both quaternions and spinors carry four real (equivalently, two complex) components, whereas the spatial geometry of ordinary vectors involves three.

Spin, spinors and the Bloch sphere map onto the polarisation of light, Jones vectors and the Poincaré sphere respectively [10,11,12,13,14], [15] (secs. 4.7 and 22.2), [16] (Section 2.2), [17]. Spinors have been visualised geometrically using the flagpole construction [6,25] or Dirac’s belt trick. Classical analogues for spin-½ systems have been studied extensively, and the Jones vector is itself a spinor [13], so the Bloch and Poincaré spheres map onto one another. The unmeasured behaviour of an electron spin in a magnetic field can be simulated by a classical mechanism [10] (p. 2), and the polarisation of a light beam mirrors the spin characteristics of an electron beam, the differences in nomenclature arising from the historical context in which quantum mechanics was discovered [16] (p. 32). Because of this equivalence the notation developed below applies directly to the Jones calculus.

Coddens has argued that overcoming the conceptual difficulties of quantum mechanics demands a deeper comprehension of its underlying mathematics: spinors emerge naturally within the $SU\left(2\right)$ representation of the three-dimensional rotation group in ${\mathbb{R}}^3$, and there remains an enigmatic quality to that group which may stem from elementary Euclidean geometry [18] (p. 2). A quantum system is represented by a vector in an abstract Hilbert space whose geometry is not directly observable [19] (pp. 24-25), [27] (p. 1). Atiyah put the difficulty sharply: algebra is “the offer made by the devil to the mathematician”, in that passing to algebraic calculation tends to displace geometric thinking [20] (p. 7).

The conceptual gap regarding the geometric nature of spinors is often traced to the transition from Hamilton’s quaternion system to modern vector analysis. When Gibbs and Heaviside formalised the standard vector framework, they set aside the algebraic and rotational structure inherent to quaternions in order to isolate spatial geometry. That simplification advanced classical mechanics, but it removed from the vector system the tools needed to describe the continuous rotation of an oriented point. When quantum mechanics later required such transformations for spin-½ systems, those algebraic properties reappeared in the abstract form of spinors.

By re-evaluating the specific mathematical adjustments made during this historical divergence, we can re-integrate these operations and ground spinors back into observable three-dimensional geometry.

Relation to Prior Work

The material below overlaps substantially with several established bodies of work, and we state the relationship explicitly.

• The Bloch sphere. That the pure-state space of a single spin-½ system is a real two-parameter manifold has been known since Bloch [24]. The real three-vector constructed in Section 3 is a reparameterisation of that sphere, not a new geometric object.
• Quaternions. The algebra generated by the units introduced in Section 4.2 is Hamilton’s quaternion algebra $\mathbb{H}$, used throughout in the flipped (JPL) convention $\mathit{j}\mathit{i}=\mathit{k}$.
• Geometric algebra. Hestenes [21] showed that spinors can be treated as elements of the even subalgebra of a real Clifford algebra, and Doran and Lasenby [26] develop this systematically. Those treatments are more general than the present one. In their language the even subalgebra of Cl(3,0) is the quaternion algebra, so the construction below is the restriction of the geometric-algebra treatment to that even subalgebra together with a fixed projection; that restriction is what makes it elementary [39].
• Cartan’s theory and the flagpole picture. Penrose and Rindler [25] give the standard geometric realisation of a two-component spinor as a null flagpole with a flag plane.

Against that background, the specific contributions of this paper are:

(i) an explicit elementary formulation of the probability in terms of the ordinary dot and cross products of real three-vectors (Theorem 2);

(ii) the identification of that operation as a projection of the quaternion product onto a complex subalgebra (Proposition 2);

(iii) an exact probability partition following from Lagrange’s identity (Section 3.3); and

(iv) a precise statement of the limits of the construction (Section 5), in particular that it relocates the complex structure. The calculation requires nothing beyond vector algebra, while the standard route requires the machinery of complex Hilbert space.

Note on notation: The symbol $\mathit{i}$ is used for both the imaginary unit and a spatial basis vector. This is not a typographical convenience but a substantive identification, and it is legitimate precisely because the ambient algebra is $\mathbb{H}$, in which $\mathit{i}$ is a genuine element satisfying ${\mathit{i}}^2=-1$ that also serves as a basis vector of the three-dimensional vector part. The identification is made explicit as a map in Section 3.1 and its consequences are examined in Section 5. Bold $\mathit{i}$, $\mathit{j}$, $\mathit{k}$ denote algebraic units throughout; $\mathrm{\Theta }$ denotes a physical Bloch polar angle and $\alpha$, $\theta$ denote the half-angle parameters defined in Proposition 1.

Note on ‘vector’: To avoid confusion, we make it explicit that vector algebra in three-dimensional space as formulated by Gibbs–Heaviside is referred to as “real vector” or simply as ‘vector’ in this paper.

2. Probability Amplitudes in the $\mathit{j}$-Notation

2.1. Change in Polar Angle Parameter

Let $\mathrm{\Theta }$ be the polar angle of a state on the Bloch sphere and $\beta$ the phase difference. The general spin-½ superposition states, with the convention that the first coefficient is real and non-negative, are [3] (p. 41):

$${\left|+\right.⟩}_n=\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{\mathrm{\Theta }}{2}}\hspace{0.17em}\left|+\right.⟩+\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{\mathrm{\Theta }}{2}}\hspace{0.17em}{e}^{\mathit{i}\beta }\left|-\right.⟩$$

(2.1)

$${\left|-\right.⟩}_n=\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{\mathrm{\Theta }}{2}}\hspace{0.17em}\left|+\right.⟩-\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{\mathrm{\Theta }}{2}}\hspace{0.17em}{e}^{\mathit{i}\beta }\left|-\right.⟩.$$

(2.2)

The convention fixes the global phase and is a gauge choice; it degenerates at $\mathrm{\Theta }=\pi$, where the first coefficient vanishes.

Proposition 1 (Reparameterisation). Let a spin-½ pure state be written as in (2.1) with $\mathrm{\Theta }\in [0,\hspace{0.17em}\mathrm{\pi }]$, and set $\alpha \equiv \mathrm{\Theta }/2\in [0,\hspace{0.17em}\mathrm{\pi }/2]$. Then $\mathrm{\Theta }\mapsto \alpha$ is a bijection of $[0,\hspace{0.17em}\mathrm{\pi }]$ onto $[0,\hspace{0.17em}\mathrm{\pi }/2]$, and under the map of Section 3.1 every probability is a function of $\alpha ,\theta$ and the phase difference $\phi -\beta$ alone.

Proof. 

Since, $\sin \left({\displaystyle \frac{\mathrm{\Theta }}{2}}\right)=\cos \left({\displaystyle \frac{\mathrm{\Theta }}{2}}-{\displaystyle \frac{\mathrm{\pi }}{2}}\right)\mathrm{a}\mathrm{n}\mathrm{d}-\cos \left({\displaystyle \frac{\mathrm{\Theta }}{2}}\right)=\sin \left({\displaystyle \frac{\mathrm{\Theta }}{2}}-{\displaystyle \frac{\mathrm{\pi }}{2}}\right),$

Equation (2.2) may be rewritten as

$${\left|-\right.⟩}_n=\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\mathrm{\Theta }}{2}}-{\displaystyle \frac{\pi }{2}}\right)\left|+\right.⟩+\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{\mathrm{\Theta }}{2}}-{\displaystyle \frac{\pi }{2}}\right)e^{\mathit{i}\beta }\left|-\right.⟩.$$

(2.3)

Equation(2.3) shows that the spin-down state and spin-up state differ by angle of $\pi /2$. Since the Bloch polar angle is twice the half-angle parameter, the corresponding difference in Bloch sphere is $\pi$, in agreement with the standard result that spin-up and spin-down are antipodal on the Bloch sphere. The basis kets satisfy ⟨+|−⟩ = 0. It should be emphasized that this orthogonality is a statement in the state space and does not correspond to a $90°$ separation in space: as Penrose observes, “orthogonal” in Hilbert space corresponds not to “at right angles” in space but to “opposite” [22] (pp. 555–556). The spin-up and spin-down directions are antipodal on the Bloch sphere, separated by $\mathrm{\Theta }= 180°.$ Thus, in the algebra the half-angle parameter differs by $\pi /2$ between the spin-up and spin-down states. The phase $\beta$ is unchanged: it takes the same value on the Bloch sphere and in the spinor algebra. Therefore, $\mathrm{\Theta }$ is replaced by α in algebra to bring out this difference explicitly by defining $\alpha =\mathrm{\Theta }/2$. The change is only in parameters and does not change the angle in both Bloch sphere and spinor algebra. $\mathrm{\Theta }$ ↦ $\mathrm{\Theta }/2$ is injective on $[0, \pi ]$ with image $[0, \pi /2],$ on which cos $\alpha \ge 0$. The amplitude computed in Theorem 1 below depends on the parameters only through $\alpha , \theta$ and the difference $\phi - \beta$. ∎

The spinor in (2.1) can be rewritten as

$$+{⟩}_A=\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\alpha }\hspace{0.17em}\left|+\right.⟩+\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\alpha }\hspace{0.17em}{e}^{\mathit{i}\beta }\left|-\right.⟩$$

$$+{⟩}_B=\mathrm{c}\mathrm{o}\mathrm{s}\theta \hspace{0.17em}\left|+\right.⟩+\mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}{e}^{\mathit{i}\phi }\left|-\right.⟩$$

Remark 1. 

Every statement below phrased in terms of $\alpha$ or $\theta$ is a statement about reparameterisation; it is converted to angles by $\mathrm{\Theta }=2\mathrm{\alpha }$ or $\mathrm{\Theta }=2\theta$ in Bloch sphere. In particular, the $\pi /2$ separation between spin-up and spin-down noted after (2.3) is a separation in $\alpha$ or $\theta$, corresponding to $\pi$ in $\mathrm{\Theta }$.

2.2. The j-Notation

Let $S_A$ be the output-state spinor and $S_B$ the input-state spinor, corresponding to ${\left|+\right.⟩}_A$ and ${\left|+\right.⟩}_B$. Omitting the $\left|+\right.⟩$ label and replacing $\left|-\right.⟩$ by the unit $\mathit{j}$, and writing $\alpha$, $\theta$ for the half-angle parameters of the two states, we obtain

$$S_A=\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }$$

(2.4)

$$S_B=\mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\phi }.$$

(2.5)

Theorem 1. 

The probability of finding the input state $S_B=\mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\phi }$ in the output state $S_A=\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }$ is $P_{A,B}={\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +{\displaystyle \frac{1}{2}}\mathrm{s}\mathrm{i}\mathrm{n}(2\alpha )\mathrm{s}\mathrm{i}\mathrm{n}(2\theta )\mathrm{c}\mathrm{o}\mathrm{s}(\phi -\beta )+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\alpha {\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta .$

Proof. 

We define the inner product $\circ$ on states written in this notation as follows: the complex coefficients of the output state are replaced by their conjugates, the symbol $\mathit{j}$ is omitted, and corresponding terms are multiplied. Thus

$$\begin{array}{c}{S}_A\circ S_B=\left(\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }\right)\circ \left(\mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\phi }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to \left(\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}{e}^{-\mathit{i}\beta }\right)\left(\mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}{e}^{\mathit{i}\phi }\right) (\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m} \mathrm{b}\mathrm{y} \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}),\end{array}$$

so that

$$S_A\circ S_B=\mathrm{c}\mathrm{o}\mathrm{s}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}{e}^{\mathit{i}(\phi -\beta )}.$$

(2.6)

The result is the sum of a real and an imaginary part, and is therefore appropriately called a “complex dot product”. The result is equivalent to the inner product ${}_A⟨+\hspace{0.17em}|\hspace{0.17em}+{⟩}_B$.

The probability follows on multiplying by its complex conjugate. Since $e^{\mathit{i}(\beta -\phi )}$ is the conjugate of $e^{\mathit{i}(\phi -\beta )}$, their sum is twice the real part, and using $\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{s}\mathrm{i}\mathrm{n}\theta ={\displaystyle \frac{1}{2}}\mathrm{s}\mathrm{i}\mathrm{n}2\theta$,

$$P_{A,B}={\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +{\displaystyle \frac{1}{2}}\mathrm{s}\mathrm{i}\mathrm{n}(2\alpha )\mathrm{s}\mathrm{i}\mathrm{n}(2\theta )\mathrm{c}\mathrm{o}\mathrm{s}(\phi -\beta )+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\alpha {\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta ,$$

(2.7)

Here, $(\phi -\beta )$ isolates the phase difference, completing the verification of Theorem 1. $\mathrm{▫}$

Only the phase difference enters the probability; an overall phase factor has none. In particular, if the same phase shift is applied to both the input and the output spinor, the probability is unchanged.

Substituting $\alpha ={\mathrm{\Theta }}_A/2$ and $\theta ={\mathrm{\Theta }}_B/2$ recovers the standard textbook expression, confirming that (2.7) is the usual probability written in the parameters of Proposition 1.

Corollary 1. 

$P_{A,B}^{++}=P_{A,B}^{--}=P_{B,A}^{++}=P_{B,A}^{--}$

Proof. 

Symmetry under exchange of $A$ and $B$ follows because (2.7) is invariant under $(\alpha ,\beta )\leftrightarrow (\theta ,\phi )$, the phase entering only as $\mathrm{c}\mathrm{o}\mathrm{s}(\phi -\beta )$, which is even. Equality of the $(+,+)$ and $(-,-)$ cases follows by applying (2.3) to both states: each half-angle parameter is shifted by $-\pi /2$, and (2.7) is invariant under the simultaneous shift $\alpha \to \alpha -\pi /2$, $\theta \to \theta -\pi /2$. $\mathrm{▫}$

Corollary 2. 

$P_{A,B}^{+-}=P_{A,B}^{-+}=P_{B,A}^{+-}=P_{B,A}^{-+}$

Proof. 

Shift exactly one of the two parameters by $-\pi /2$ in (2.7) and apply the same symmetry argument. Together with Corollary 1 and the completeness relation $P^{++}+P^{+-}=1$, this fixes all four permutations. $\mathrm{▫}$

In both corollaries the shift is understood modulo the reparameterisation of Proposition 1: the representative with $\mathrm{c}\mathrm{o}\mathrm{s}\alpha \ge 0$ is recovered by the gauge convention of Section 2.1.

2.3. Overall Phase Factor of $e^{\mathit{i}\pi /2}$

The overall phase of a state carries no physical significance, so an arbitrary overall phase factor may be applied without altering any probability [3] (p. 18), [28] (p. 108). We therefore multiply spinors by $e^{\mathit{i}\pi /2}=\mathit{i}$.

Because the algebra is non-commutative, the side of multiplication must be specified. We apply the phase by right multiplication throughout. Right multiplication of (2.4) by $\mathit{i}$ gives $\mathrm{c}\mathrm{o}\mathrm{s}\alpha \hspace{0.17em}\mathit{i}+\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}(\beta +\pi /2)}$; left multiplication would instead give $\mathrm{c}\mathrm{o}\mathrm{s}\alpha \hspace{0.17em}\mathit{i}+\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{i}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }$, a different expression.

Right multiplying the spinors (2.4) & (2.5) by the imaginary unit $\mathit{i}$:

$${\mathit{S}}_{\mathit{A}}=cos\alpha \mathit{i}+sin\alpha \mathit{j}{e}^{\mathit{i}\left(\beta +\pi /2\right)} \mathrm{a}\mathrm{n}\mathrm{d} {\mathit{S}}_{\mathit{B}}=cos\theta \mathit{i}+sin\theta \mathit{j}{e}^{\mathit{i}\left(\phi +\pi /2\right)}$$

Let $\beta +{\displaystyle \frac{\pi }{2}}$ and $\phi +{\displaystyle \frac{\pi }{2}}$ be temporarily replaced by $\beta$ and $\phi$:

$$S_A=\mathrm{c}\mathrm{o}\mathrm{s}\alpha \hspace{0.17em}\mathit{i}+\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }$$

(2.8)

$$S_B=\mathrm{c}\mathrm{o}\mathrm{s}\theta \hspace{0.17em}\mathit{i}+\mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\phi }.$$

(2.9)

Evaluating the complex dot product of (2.8) and (2.9),

$$S_A\circ S_B=\mathrm{c}\mathrm{o}\mathrm{s}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}{e}^{-\mathit{i}\beta }{e}^{\mathit{i}\phi }$$

(2.10)

$$P_{A,B}={\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +{\displaystyle \frac{1}{2}}\mathrm{s}\mathrm{i}\mathrm{n}(2\alpha )\mathrm{s}\mathrm{i}\mathrm{n}(2\theta )\mathrm{c}\mathrm{o}\mathrm{s}(\phi -\beta )+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\alpha {\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta .$$

(2.11)

Restoring the phase shifts $\beta +{\displaystyle \frac{\pi }{2}}$ and $\phi +{\displaystyle \frac{\pi }{2}}$ in (2.10) makes no difference to the result: (2.10) and (2.11) equal (2.6) and (2.7) respectively, confirming that the overall phase factor is without physical effect in this notation as it is in the standard one. The relabelling is retained for the remainder of the paper; no probability depends on it, since only the phase difference enters.

3. The Real Three-Vector Representation

3.1. The Map and the Complex Dot Product

We now treat the states of Section 2.3 as Euclidean vectors in a real vector space. The vector dot product is written $\cdot$ and the cross product $\times$. In converting a state to a real vector, we map the imaginary unit $\mathit{i}$ to the basis vector $\mathit{i}$, and the product $\mathit{j}\mathit{i}$—rather than $ij$—to the basis vector $\mathit{k}$. We adopt the convention $\mathit{j}\mathit{i}=\mathit{k}$, the flipped (JPL) convention, rather than Hamilton’s $ij=k$. Both conventions are internally consistent; the $\mathit{j}\mathit{i}=\mathit{k}$ convention is the one consistent with the rotation convention adopted here, as discussed by Shuster for $ij=k$ rotations [23] (p. 473). The same convention is to be used in mapping spinors to quaternions [22] (p. 206).

Expanding (2.8) and (2.9) into components,

$$\begin{array}{c}{S}_A=cos\alpha \hspace{0.17em}\mathit{i}+sin\alpha cos\beta \hspace{0.17em}\mathit{j}+sin\alpha sin\beta \hspace{0.17em}\mathit{k},\\ S_B=cos\theta \hspace{0.17em}\mathit{i}+sin\theta cos\phi \hspace{0.17em}\mathit{j}+sin\theta sin\phi \hspace{0.17em}\mathit{k}.\end{array}$$

Treating $\mathit{i}$, $\mathit{j}$ and $\mathit{k}$ as spatial basis vectors, we obtain the real unit vectors

$$\mathit{a}=a_1\mathit{i}+a_2\mathit{j}+a_3\mathit{k}=(\mathrm{c}\mathrm{o}\mathrm{s}\alpha ,\hspace{0.25em}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\beta ,\hspace{0.25em}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{s}\mathrm{i}\mathrm{n}\beta )$$

(3.1)

$$\mathit{b}=b_1\mathit{i}+b_2\mathit{j}+b_3\mathit{k}=(\mathrm{c}\mathrm{o}\mathrm{s}\theta ,\hspace{0.25em}\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{c}\mathrm{o}\mathrm{s}\phi ,\hspace{0.25em}\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{s}\mathrm{i}\mathrm{n}\phi ).$$

(3.2)

The dot product of two vectors yields a real number, whereas the complex dot product of Section 2 yields the sum of a real and an imaginary number. The sum of the standard dot product $(\mathit{a}\cdot \mathit{b})$ and the $\mathit{i}$-basis component of the cross product $(\mathit{a}\times \mathit{b}{)}_{\mathit{i}}$ of vectors will be equivalent to the required complex dot product. We therefore define:

Definition 1 

(complex dot product). For real vectors $\mathit{a}$ and $\mathit{b}$,

$$A\circ B\hspace{0.33em}\equiv \hspace{0.33em}(\mathit{a}\cdot \mathit{b})+(\mathit{a}\times \mathit{b}{)}_{\mathit{i}} \mathit{i}=(a_1{b}_1+a_2{b}_2+a_3{b}_3)+(a_2{b}_3-a_3{b}_2)\hspace{0.17em}\mathit{i}.$$

(3.3)

Theorem 2. 

The probability computed from the complex dot product of the corresponding real vectors, equals the probability derived from the standard inner product of the spinors: $|A\circ B{|}^2=P_{A,B}$

Proof. 

From (3.1) and (3.2),

$$\begin{array}{c}\mathit{a}\cdot \mathit{b}=cos\alpha cos\theta +sin\alpha sin\theta \hspace{0.17em}(cos\beta cos\phi +sin\beta sin\phi )\\ =cos\alpha cos\theta +sin\alpha sin\theta cos(\phi -\beta ),\\ \left(\mathit{a}\times \mathit{b}{)}_{\mathit{i}}={(a}_2{b}_3-a_3{b}_2\right)\mathit{i}=sin\alpha sin\theta \hspace{0.17em}(cos\beta sin\phi -sin\beta cos\phi )\mathit{i}\\ =sin\alpha sin\theta sin(\phi -\beta )\mathit{i}.\end{array}$$

Adding, and combining the two trigonometric terms into a single exponential,

$$A\circ B=\mathrm{c}\mathrm{o}\mathrm{s}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{s}\mathrm{i}\mathrm{n}\theta \hspace{0.17em}{e}^{\mathit{i}(\phi -\beta )}.$$

(3.4)

Comparison with (2.6) gives $A\circ B=S_A\circ S_B$ identically. Taking the squared modulus,

$${|A\circ B|}^2={\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha {\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +{\displaystyle \frac{1}{2}}\mathrm{s}\mathrm{i}\mathrm{n}(2\alpha )\mathrm{s}\mathrm{i}\mathrm{n}(2\theta )\mathrm{c}\mathrm{o}\mathrm{s}(\phi -\beta )+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\alpha {\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta ,$$

(3.5)

$$|A\circ B{|}^2=P_{A,B},$$

(3.6)

which is (2.7). $\mathrm{▫}$

Although the computational procedure differs from the inner product of Section 2, the probabilities are identical. The calculation on the right-hand side of (3.3) requires nothing beyond the elementary dot and cross products of real vectors.

3.2. Algebraic Status of the Complex Dot Product

Definition 1 may appear ad hoc. It is not: it has a compact closed form in the quaternion algebra, which we record because it also makes the limitations of Section 5 transparent.

Proposition 2. 

Let $\mathit{a}$ and $\mathit{b}$ be pure quaternions, let $\overline{\mathit{a}}$ denote quaternionic conjugation, and let $\mathbb{S}:\mathbb{H}\to \mathbb{C}\cong \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{1,\mathit{i}\}$ be the linear projection $\mathbb{S}(q_0+q_1\mathit{i}+q_2\mathit{j}+q_3\mathit{k})=q_0+q_1\mathit{i}$. Then $A\circ B=\mathbb{S}\left(\overline{\mathit{a}}\hspace{0.17em}\mathit{b}\right).$

Proof. 

For pure quaternions, $\mathit{a}\mathit{b}=-(\mathit{a}\cdot \mathit{b})-(\mathit{a}\times \mathit{b})$, where × is the ordinary right-handed cross product and the sign of the vector part follows from the multiplication table of the flipped convention $\mathit{j}\mathit{i}=\mathit{k}$ adopted in Section 3.1 and $\overline{\mathit{a}}=-\mathit{a}$ for a pure quaternion, $\overline{\mathit{a}}\mathit{b}=(\mathit{a}\cdot \mathit{b})+(\mathit{a}\times \mathit{b})$. Applying $\mathbb{S}$ retains the scalar part and the $\mathit{i}$-component of the vector part, giving $(\mathit{a}\cdot \mathit{b})+(\mathit{a}\times \mathit{b}{)}_{\mathit{i}}\hspace{0.17em}\mathit{i}$, which is Definition 1. $\mathrm{▫}$

The complex dot product is therefore not a new algebraic operation but the image of the quaternion product under a projection onto a distinguished complex subalgebra. Two consequences follow immediately. First, no new number system is required: the construction lives entirely inside $\mathbb{H}$.

3.3. Component Decomposition and the Normalisation Identity

For unit vectors $\mathit{a}$ and $\mathit{b}$, Lagrange’s identity gives

$$(\mathit{a}\cdot \mathit{b}{)}^2+|\mathit{a}\times \mathit{b}{|}^2=\left|\mathit{a}{|}^2\right|\mathit{b}{|}^2=1,$$

(3.7)

so the four component contributions—the square of $\mathit{a}\cdot \mathit{b}$ together with the squares of the three components of $\mathit{a}\times \mathit{b}$—sum to unity identically, for every pair of states. The decomposition tabulated below is therefore an exact partition of unit probability, not a numerical coincidence of the particular example chosen.

For unit vectors the magnitude of the component of $\mathit{a}$ parallel to $\mathit{b}$ is $\mathit{a}\cdot \mathit{b}$, and the magnitude of the component orthogonal to $\mathit{b}$ is $|\mathit{a}\times \mathit{b}|$, which equals $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }(\mathit{a},\mathit{b})$. We stress that $\mathit{a}\times \mathit{b}$ and ${\mathit{a}}_{\mathrm{⟂}\mathit{b}}$ are different vectors. The decomposition below uses the components of $\mathit{a}\times \mathit{b}$ resolved along the fixed basis $\{\mathit{i},\mathit{j},\mathit{k}\}$, and the row labels of Table 1 are to be read in that sense.

3.4. Worked Example

Let the output and input states be

$$S_A=\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{2\pi }{9}}\hspace{0.17em}\mathit{i}+\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2\pi }{9}}\hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\hspace{0.17em}7\pi /18}$$

(3.8)

$$S_B=\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{5\pi }{36}}\hspace{0.17em}\mathit{i}+\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{5\pi }{36}}\hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\hspace{0.17em}7\pi /36}$$

(3.9)

that is, $\alpha ={40}^{\circ }$, $\beta ={70}^{\circ }$, $\theta ={25}^{\circ }$, $\phi ={35}^{\circ }$. Applying Section 3.1,

$$\begin{array}{c}\overrightarrow{\mathit{a}}=(0.76604,\hspace{0.25em}0.21985,\hspace{0.25em}0.60402),\overrightarrow{\mathit{b}}=(0.90631,\hspace{0.25em}0.34619,\hspace{0.25em}0.24240),\end{array}$$

$$\overrightarrow{\mathit{a}}=\left(0.76604\mathit{i}+0.21985\mathit{j}+0.60402\mathit{k}\right)$$

$$\overrightarrow{\mathit{b}}=\left(0.90631\mathit{i}+0.34619\mathit{j}+0.24240\mathit{k}\right)$$

$$\overrightarrow{\mathit{a}}\cdot \overrightarrow{\mathit{b}}=0.9168, \overrightarrow{\mathit{a}}\times \overrightarrow{\mathit{b}}=\left(-0.15581\mathit{i}+0.36174\mathit{j}+0.06595\mathit{k}\right)$$

(3.10)

Hence $P_{A,B}^{++}=86.48\mathrm{\%}$ and $P_{A,B}^{+-}=13.52\mathrm{\%}$.

As a consistency check, resolving $\mathit{a}$ into components parallel and orthogonal to $\mathit{b}$ must return $\mathit{a}$ [31] (Eq 3.8). The parallel component is $(\mathit{a}\cdot \mathit{b})\mathit{b}$ [31] (Eq. 2.23):

$${\overrightarrow{\mathit{a}}}_{\parallel \overrightarrow{\mathbf{b}}}=0.91680 \ast \left(0.90631\mathit{i}+0.34619\mathit{j}+0.2424\mathit{k}\right)$$

$$=\left(0.83090\mathit{i}+0.31739\mathit{j}+0.22223\mathit{k}\right) \left(3.11\right)$$

and the orthogonal component follows by subtraction [31] (Eq. 2.26):

$${\overrightarrow{\mathit{a}}}_{\mathrm{⟂}\mathit{b}}=-0.06486\mathit{i}-0.09754\mathit{j}+0.38179\mathit{k}.$$

(3.12)

The sum of (3.11) and (3.12) returns $\mathit{a}$, as required. Note that (3.12) and (3.10) are different vectors with different components, as observed in Section 3.3; only their magnitudes agree, both equalling $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\angle }(\mathit{a},\mathit{b})$.

3.5. Probability of the Spin-Down State

The spin-down state corresponding to (3.9) is $S_B^{-}=\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{5\pi }{36}}\hspace{0.17em}\mathit{i}-\mathrm{c}\mathrm{o}\mathrm{s}{\displaystyle \frac{5\pi }{36}}\hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\hspace{0.17em}7\pi /36}$, which maps to

$${\overrightarrow{\mathit{b}}}^{-}=0.42262\mathit{i}-0.74241\mathit{j}-0.51984\mathit{k}.$$

Projecting $\overrightarrow{\mathit{a}}$ onto ${\overrightarrow{\mathit{b}}}^{-}$ gives $\overrightarrow{\mathit{a}}\cdot {\overrightarrow{\mathit{b}}}^{-}=-0.15348$ and $\overrightarrow{\mathit{a}}\times {\overrightarrow{\mathit{b}}}^{-}=(0.33414,\hspace{0.25em}0.65349,\hspace{0.25em}-0.66163)$, from which the components below follow.

Table 2. Probability components computed against the spin-down input vector ${\overrightarrow{\mathit{b}}}^{-}$. All projections in this table are of $\overrightarrow{\mathit{a}}$ onto ${\overrightarrow{\mathit{b}}}^{-}$, not onto $\overrightarrow{\mathit{b}}$; the values therefore complement those of Table 1.

Quantity	Projection direction (onto ${\overrightarrow{\mathit{b}}}^{-}$)	Magnitude	Probability
$P_{A,B}^{+-}$	Real $+\hspace{0.25em}\mathit{i}$	0.3677	13.52%
$P_{A,B}^{++}$	$\mathit{j}+\mathit{k}$	0.9299	86.48%
Total	Real $+\hspace{0.25em}\mathit{i}+\mathit{j}+\mathit{k}$	1.0000	100%

Table 2. Probability components computed against the spin-down input vector ${\overrightarrow{\mathit{b}}}^{-}$. All projections in this table are of $\overrightarrow{\mathit{a}}$ onto ${\overrightarrow{\mathit{b}}}^{-}$, not onto $\overrightarrow{\mathit{b}}$; the values therefore complement those of Table 1.

Quantity	Projection direction (onto ${\overrightarrow{\mathit{b}}}^{-}$)	Magnitude	Probability
$P_{A,B}^{+-}$	Real $+\hspace{0.25em}\mathit{i}$	0.3677	13.52%
$P_{A,B}^{++}$	$\mathit{j}+\mathit{k}$	0.9299	86.48%
Total	Real $+\hspace{0.25em}\mathit{i}+\mathit{j}+\mathit{k}$	1.0000	100%

Table 3. Probabilities for the different permutations of $\overrightarrow{\mathit{a}}$ and $\overrightarrow{\mathit{b}}$, by Corollaries 1 and 2.

Permutation	Magnitude	Probability
$P_{A,B}^{++}=P_{A,B}^{--}=P_{B,A}^{++}=P_{B,A}^{--}$	0.9299	86.48%
$P_{A,B}^{+-}=P_{A,B}^{-+}=P_{B,A}^{+-}=P_{B,A}^{-+}$	0.3677	13.52%

Table 3. Probabilities for the different permutations of $\overrightarrow{\mathit{a}}$ and $\overrightarrow{\mathit{b}}$, by Corollaries 1 and 2.

Permutation	Magnitude	Probability
$P_{A,B}^{++}=P_{A,B}^{--}=P_{B,A}^{++}=P_{B,A}^{--}$	0.9299	86.48%
$P_{A,B}^{+-}=P_{A,B}^{-+}=P_{B,A}^{+-}=P_{B,A}^{-+}$	0.3677	13.52%

3.6. The Role of Complex Numbers

Proposition 3. 

Let ${\mathit{b}}^{-}$ be the real vector corresponding to the state orthogonal to $B$. Then ${|A\circ B^{-}|}^2=(\mathit{a}\times \mathit{b}{)}_{\mathit{j}}^2+(\mathit{a}\times \mathit{b}{)}_{\mathit{k}}^2$, and consequently ${|A\circ B|}^2+{|A\circ B^{-}|}^2=1$

Proof. 

The kets ${|+⟩}_B$ and ${|-⟩}_B$ form an orthonormal basis of the state space, so the two outcomes of the analyser exhaust unit probability. Applying Theorem 2 to each pair gives ${|A\circ B|}^2+{|A\circ B^{-}|}^2=1$. By Theorem 2 and Definition 1, ${|A\circ B|}^2=(\mathit{a}\cdot \mathit{b}{)}^2+(\mathit{a}\times \mathit{b}{)}_{\mathit{i}}^2$, and Lagrange’s identity (3.7) states that the four squared components sum to unity. Subtracting gives the stated expression. ▫

A Stern–Gerlach analyser measures only the relative orientation of the two states. Taking $\mathit{a}$ as the output state and $\mathit{b}$ as the input state, we define the spin-up channel of the analyser to correspond to the pair $(\mathit{a}\cdot \mathit{b},\hspace{0.25em}(\mathit{a}\times \mathit{b}{)}_{\mathit{i}})$ and the spin-down channel to the components $(\mathit{a}\times \mathit{b}{)}_{\mathit{j}}$ and $(\mathit{a}\times \mathit{b}{)}_{\mathit{k}}$. Proposition 3 above shows that this is not a stipulation: the two remaining components carry exactly the probability of the orthogonal outcome, so once Definition 1 is adopted the assignment is forced; Section 3.5 exhibits it numerically. Because the spin-up channel requires the sum of a real and an imaginary contribution, the corresponding probability is naturally computed using complex arithmetic. For spin-$\frac{1}{2}$ system, the complex structure of the state space can be traded for a preferred axis in a real three-dimensional space without altering any quantum probability.

This result should be placed against the known status of real-vector-space quantum theory. A single qubit can always be simulated in a real Hilbert space, and indeed any finite-dimensional quantum system can be so simulated at the cost of one additional universal “rebit” [32,33]; the construction above is a particularly economical instance of that fact for spin-½. The genuine obstruction to a real-valued formulation appears only for composite systems with independent sources, where real quantum theory makes predictions that differ from the complex theory and that experiment has excluded [34,35,36]. Nothing in the present paper bears on that question. What is shown here is narrower and geometric: for one spin-½ system the complex unit of the state space can be exchanged for a distinguished axis of real three-space, with every probability preserved.

4. Geometry

4.1. Four Components in Three Dimensions

Section 3 represents a state by a real three-vector, while a general quaternion carries four real components. The resolution is exact rather than analogical. Writing (2.4) as $S=\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathit{j}\left(\mathrm{s}\mathrm{i}\mathrm{n}\alpha e^{\mathit{i}\beta }\right)$ and expanding $\mathit{j}{e}^{\mathit{i}\beta }=\mathrm{c}\mathrm{o}\mathrm{s}\beta \mathit{j}+\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathit{k}$, the state has no $\mathit{i}$-component; right-multiplying by $\mathit{i}$ carries the scalar part onto $\mathit{i}$ and the $\mathit{j}$–$\mathit{k}$ plane onto itself, so that $S\mathit{i}$ has zero scalar part. A state in the notation of Section 2.3 is therefore a pure quaternion, and the three-dimensional subspace $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{\mathit{i},\mathit{j},\mathit{k}\}$ is where it lives: the fourth component is not hidden in a second plane, it is identically zero. Historically, quaternions were interpreted as four-dimensional for precisely this reason, and the separation of the dot and cross products arose in part from the difficulty of interpreting a fourth dimension.

The dimension count then agrees on both sides. $C^2$ has four real dimensions; imposing normalisation and quotienting by the global phase leaves ${CP}^1\cong S^2$, a real two-parameter family. On the vector side, the gauge convention of Section 2.1 fixes the phase, normalisation fixes $\left|\mathit{a}\right|=1$, and the unit vector $\mathit{a}$ of (3.1) supplies exactly the two parameters $\alpha$ and $\beta$. No information is lost or gained in the mapping.

It is worth recording which axis is distinguished and why. Setting $\alpha =0$ in (3.1) gives $\mathit{a}=\mathit{i}$, so the $\mathit{i}$-axis is the image of the basis state $|+⟩$, that is, of the quantisation axis chosen when the states were written in the form (2.1). The preferred direction of Section 5 is therefore not an artefact of the algebra: it is the measurement basis, transcribed into space.

4.2. Forms of the State and the Underlying Algebra

The state admits several equivalent algebraic forms, collected in Table 4.

Here $p$, $q$, $u$, $v$, $x$ and $y$ are real; $\theta$ is the half-angle parameter of Proposition 1, which in optics corresponds to the degree of polarisation; $\delta$ is the overall phase angle; $\phi$ is the phase difference; and $r$ is the magnitude.

The units satisfy

$${\mathit{j}}^2=-1, \mathit{j}\mathit{i}=-\mathit{i}\mathit{j}.$$

(4.1)

Together with ${\mathit{i}}^2=-1$ these relations generate the quaternion algebra $\mathbb{H}$ in the flipped (JPL) convention $\mathit{j}\mathit{i}=\mathit{k}$: from (4.1) it follows that ${\mathit{k}}^2=(\mathit{j}\mathit{i})(\mathit{j}\mathit{i})=-{\mathit{j}}^2{\mathit{i}}^2=-1$, and the full multiplication table is fixed. We emphasise that no new number system is being introduced; the units $\mathit{j}$ and $\mathit{k}=\mathit{j}\mathit{i}$ are the standard quaternion units, and the correspondence between the Pauli matrices and these units is the one recorded in [5,6,7,8,9].

4.3. Relation Between Spin-Up and Spin-Down States

From (2.4) the spin-up state is

$$S_n^{+}=\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta },$$

(4.2)

and the spin-down state is

$$S_n^{-}=\mathrm{s}\mathrm{i}\mathrm{n}\alpha -\mathrm{c}\mathrm{o}\mathrm{s}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }.$$

(4.3)

Since $\mathrm{s}\mathrm{i}\mathrm{n}\alpha =\mathrm{c}\mathrm{o}\mathrm{s}(\alpha -\pi /2)$ and $-\mathrm{c}\mathrm{o}\mathrm{s}\alpha =\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\pi /2)$,

$$S_n^{-}=\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha -{\displaystyle \frac{\pi }{2}}\right)+\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha -{\displaystyle \frac{\pi }{2}}\right)\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }.$$

(4.4)

Equation (4.3) also follows algebraically from (4.2) using the multiplication rules (4.1):

$$S_n^{-}=-\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }\left(\mathrm{c}\mathrm{o}\mathrm{s}\alpha +\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }\right)=-\mathrm{c}\mathrm{o}\mathrm{s}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }-\mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}(\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }{)}^2,$$

and since $(\mathit{j}{e}^{\mathit{i}\beta }{)}^2=-1$, this gives $S_n^{-}=\mathrm{s}\mathrm{i}\mathrm{n}\alpha -\mathrm{c}\mathrm{o}\mathrm{s}\alpha \hspace{0.17em}\mathit{j}\hspace{0.17em}{e}^{\mathit{i}\beta }$. Thus $S_n^{+}$ must be pre-multiplied by $-\mathit{j}{e}^{\mathit{i}\beta }$ to yield $S_n^{-}$, whereas in the phase-free case $S_z^{+}=\mathrm{c}\mathrm{o}\mathrm{s}0+\mathrm{s}\mathrm{i}\mathrm{n}0\hspace{0.17em}\mathit{j}$ requires only multiplication by $\mathit{j}$. The pre-multiplication is required because $\mathit{j}$ and $\mathit{i}$ anticommute. The corresponding substitution of the Pauli matrices by these units is the standard correspondence recorded in [6,7,8,9].

Equation (4.4) shows that the spin-down state is obtained from the spin-up state by a shift of −$\pi /2$ in the parameter $\alpha$. By Remark 1 this corresponds to a shift of $\pi$ in the physical Bloch angle $\mathrm{\Theta }$, so spin-up and spin-down remain antipodal on the Bloch sphere, in agreement with the standard formalism.

4.4. The Bloch Sphere and the Half-Angle

Using a Stern–Gerlach device the spin component may be measured along the $X$, $Y$ or $Z$ axis. The standard basis states are

$$\begin{array}{c}{\left|+\right.⟩}_x={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ 1\end{array}\right),{\left|-\right.⟩}_x={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ -1\end{array}\right),\\ {\left|+\right.⟩}_y={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ \mathit{i}\end{array}\right),{\left|-\right.⟩}_y={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}1\\ -\mathit{i}\end{array}\right),\\ {\left|+\right.⟩}_z=\left(\begin{array}{c}1\\ 0\end{array}\right),{\left|-\right.⟩}_z=\left(\begin{array}{c}0\\ 1\end{array}\right).\end{array}$$

Applying an overall phase factor of $\mathit{i}$ to each of these states, in the manner of Section 2.3, gives the corresponding states in the present notation, and the general states become

$${\left|+\right.⟩}_n={\displaystyle \frac{1}{\sqrt{2}}}\hspace{0.17em}\mathit{i}\left(\begin{array}{c}\mathrm{c}\mathrm{o}\mathrm{s}\alpha \\ \mathrm{s}\mathrm{i}\mathrm{n}\alpha \hspace{0.17em}{e}^{\mathit{i}\phi }\end{array}\right), {\left|-\right.⟩}_n={\displaystyle \frac{1}{\sqrt{2}}}\hspace{0.17em}\mathit{i}\left(\begin{array}{c}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \\ -\mathrm{c}\mathrm{o}\mathrm{s}\alpha \hspace{0.17em}{e}^{\mathit{i}\phi }\end{array}\right).$$

Under the map of Section 3.1, the antipodal vectors $\mathit{a}$ and $-\mathit{a}$ correspond to the same Bloch vector: writing $-\mathit{a}$ in the form $(\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }^{\prime },\hspace{0.25em}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }^{\prime }\mathrm{c}\mathrm{o}\mathrm{s}{\beta }^{\prime },\hspace{0.25em}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }^{\prime }\mathrm{s}\mathrm{i}\mathrm{n}{\beta }^{\prime })$ gives ${\alpha }^{\prime }=\pi -\alpha$ and ${\beta }^{\prime }=\beta +\pi$, and the Bloch vectors of $(\alpha ,\beta )$ and $({\alpha }^{\prime },{\beta }^{\prime })$ coincide identically. The map from the unit vector to the Bloch vector is therefore two-to-one away from the circle a₁ = 0, with a and −a identified, and the gauge convention of Section 2.1 confines pure states to the closed hemisphere $a_1\ge 0$. The bounding circle is the locus where that convention degenerates: every point of it represents the same physical state, so the hemisphere with its boundary collapsed is the Bloch sphere. The half-angle is the signature of this identification rather than an arbitrary convention. It should be distinguished from the double cover $SU\left(2\right) \to SO\left(3\right)$ of the rotation group, which is a statement about transformations rather than about states.

5. Rotation of State Vectors

Theorem 2 shows that quantum probabilities can be computed by real vector algebra. It is important to be precise about what follows: the representation is not an isometry. The complex dot product is not invariant under a simultaneous rotation of both vectors. The $\mathit{i}$-axis is a preferred direction carrying exactly the information encoded by the complex structure in the standard formalism. By Proposition 2 the operation $\circ$ is a projection onto the subalgebra $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{1,\mathit{i}\}$, and projections do not commute with rotations. The mapping of Section 3.1 therefore does not eliminate the complex unit from the description of a spin-${\displaystyle \frac{1}{2}}$ state; it relocates it, from an algebraic scalar into a distinguished spatial axis. Theorem 2 asserts that this relocation is faithful at the level of probabilities, not that the state space carries a Euclidean geometry in the ordinary sense.

The remedy is to rotate the state vectors and then map them to real vectors, rather than rotating the real vectors directly: the spinor inner product is invariant under $\mathrm{S}\mathrm{U}\left(2\right)$, so by Theorem 2 every probability computed after such a rotation is unchanged. Equivalently, one may represent the states as quaternions in the flipped convention (Section 4.2), rotate them there, map them to real vectors and then compute the probability.

6. Limitations and Outlook

Two limitations should be stated plainly. All results concern a single spin-${\displaystyle \frac{1}{2}}$ system. The construction is restricted to the sphere $\left|\mathit{a}\right|=1$. Extending it to the Bloch ball, and hence to mixed states, would require an interpretation of $r<1$ that we have not supplied.

Natural next steps are a worked application in the Jones calculus demonstrating a concrete computational saving, an extension to mixed states, and an examination of whether the projection of Proposition 2 admits a useful generalisation to higher spin.

7. Conclusions

We have shown that the probability between two spin-${\displaystyle \frac{1}{2}}$ states can be computed entirely within a real three-dimensional space. Writing states in the $\mathit{j}$-notation and applying an overall phase factor of $\mathit{i}$, each state maps to a real vector, and the squared modulus of the complex dot product—the sum of the ordinary dot product and the $\mathit{i}$-basis component of the cross product—reproduces the standard result exactly (Theorem 2). The operation is the projection of the quaternion product $\overline{\mathit{a}}\mathit{b}$ onto the subalgebra spanned by $1$ and $\mathit{i}$ (Proposition 2), and its component decomposition partitions unit probability exactly by Lagrange’s identity.

The construction is a faithful reparameterisation of the Bloch sphere in which the complex unit is relocated into a preferred spatial axis rather than removed.