Editing Spinor (section)

=== Three dimensions ===
{{Main|Spinors in three dimensions|Quaternions and spatial rotation}}

The Clifford algebra Cℓ<sub>3,0</sub>(<math>\Reals</math>) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, [[Pauli matrices|''σ''<sub>1</sub>, ''σ''<sub>2</sub> and ''σ''<sub>3</sub>]], the three unit bivectors ''σ''<sub>1</sub>''σ''<sub>2</sub>, ''σ''<sub>2</sub>''σ''<sub>3</sub>, ''σ''<sub>3</sub>''σ''<sub>1</sub> and the [[pseudoscalar]] {{math|1=''i'' = ''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>3</sub>}}. It is straightforward to show that {{math|1=(''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>2</sub>)<sup>2</sup> = (''σ''<sub>3</sub>)<sup>2</sup> = 1}}, and {{math|1=(''σ''<sub>1</sub>''σ''<sub>2</sub>)<sup>2</sup> = (''σ''<sub>2</sub>''σ''<sub>3</sub>)<sup>2</sup> = (''σ''<sub>3</sub>''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>3</sub>)<sup>2</sup> = −1}}.

The sub-algebra of even-graded elements is made up of scalar dilations,
<math display="block">u' = \rho^{\left(\frac{1}{2}\right)} u \rho^{\left(\frac{1}{2}\right)} = \rho u,</math>
and vector rotations
<math display="block">u' = \gamma u\gamma^*,</math>
where
{{NumBlk||<math display="block">\left.\begin{align}
  \gamma &= \cos\left(\frac{\theta}{2}\right) - \{a_1\sigma_2\sigma_3 + a_2\sigma_3\sigma_1 + a_3\sigma_1\sigma_2\} \sin\left(\frac{\theta}{2}\right) \\
         &= \cos\left(\frac{\theta}{2}\right) - i\{a_1\sigma_1 + a_2\sigma_2 + a_3\sigma_3\} \sin\left(\frac{\theta}{2}\right) \\
         &= \cos\left(\frac{\theta}{2}\right) - iv\sin\left(\frac{\theta}{2}\right)
\end{align}\right\}</math>|{{EquationRef|1}}}}
corresponds to a vector rotation through an angle ''θ'' about an axis defined by a unit vector {{math|1=''v'' = ''a''<sub>1</sub>''σ''<sub>1</sub> + ''a''<sub>2</sub>''σ''<sub>2</sub> + ''a''<sub>3</sub>''σ''<sub>3</sub>}}.

As a special case, it is easy to see that, if {{math|1=''v'' = ''σ''<sub>3</sub>}}, this reproduces the ''σ''<sub>1</sub>''σ''<sub>2</sub> rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the ''σ''<sub>3</sub> direction invariant, since

<math display="block">
  \left[\cos\left(\frac{\theta}{2}\right) - i\sigma_3 \sin\left(\frac{\theta}{2}\right)\right]
    \sigma_3 \left[\cos\left(\frac{\theta}{2}\right) + i \sigma_3 \sin\left(\frac{\theta}{2}\right)\right] =
  \left[\cos^2\left(\frac{\theta}{2}\right) + \sin^2\left(\frac{\theta}{2}\right)\right] \sigma_3 =
  \sigma_3.
</math>

The bivectors ''σ''<sub>2</sub>''σ''<sub>3</sub>, ''σ''<sub>3</sub>''σ''<sub>1</sub> and ''σ''<sub>1</sub>''σ''<sub>2</sub> are in fact [[William Rowan Hamilton|Hamilton's]] [[quaternion]]s '''i''', '''j''', and '''k''', discovered in 1843:

<math display="block">\begin{align}
  \mathbf{i} &= -\sigma_2 \sigma_3 = -i \sigma_1 \\
  \mathbf{j} &= -\sigma_3 \sigma_1 = -i \sigma_2 \\
  \mathbf{k} &= -\sigma_1 \sigma_2 = -i \sigma_3
\end{align}</math>

With the identification of the even-graded elements with the algebra <math>\mathbb{H}</math> of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.{{efn|Since, for a [[skew field]], the kernel of the representation must be trivial. So inequivalent representations can only arise via an [[automorphism]] of the skew-field. In this case, there are a pair of equivalent representations: {{math|1=''γ''(''ϕ'') = ''γϕ''}}, and its quaternionic conjugate {{math|1=''γ''(''ϕ'') = ''ϕ{{overline|γ}}''}}.}} Thus the (real{{efn|The complex spinors are obtained as the representations of the [[tensor product]] {{math|1=<math>\mathbb{H} \otimes_\Reals \Complex</math> = Mat<sub>2</sub>(<math>\Complex</math>)}}. These are considered in more detail in [[spinors in three dimensions]].}}) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

Note that the expression (1) for a vector rotation through an angle {{mvar|θ}}, ''the angle appearing in γ was halved''. Thus the spinor rotation {{math|1=''γ''(''ψ'') = ''γψ''}} (ordinary quaternionic multiplication) will rotate the spinor {{mvar|ψ}} through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with {{math|(180° + ''θ''/2)}} in place of ''θ''/2 will produce the same vector rotation, but the negative of the spinor rotation.

The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.