Editing Spinor (section)

== Examples ==
Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra {{math|Cℓ<sub>''p'', ''q''</sub>(<math>\Reals</math>)}}. This is an algebra built up from an orthonormal basis of {{math|1=''n'' = ''p'' + ''q''}} mutually orthogonal vectors under addition and multiplication, ''p'' of which have norm +1 and ''q'' of which have norm −1, with the product rule for the basis vectors
<math display="block">e_ie_j = \begin{cases}
+1 & i=j, \, i \in (1, \ldots, p) \\
-1 & i=j, \, i \in (p+1, \ldots, n)  \\
-e_j e_i & i \neq j.
\end{cases}</math>

=== Two dimensions ===
The Clifford algebra Cℓ<sub>2,0</sub>(<math>\Reals</math>) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, ''σ''<sub>1</sub> and ''σ''<sub>2</sub>, and one unit [[pseudoscalar]] {{math|1=''i'' = ''σ''<sub>1</sub>''σ''<sub>2</sub>}}. From the definitions above, it is evident that {{math|1=(''σ''<sub>1</sub>)<sup>2</sup> = (''σ''<sub>2</sub>)<sup>2</sup> = 1}}, and {{math|1=(''σ''<sub>1</sub>''σ''<sub>2</sub>)(''σ''<sub>1</sub>''σ''<sub>2</sub>) = −''σ''<sub>1</sub>''σ''<sub>1</sub>''σ''<sub>2</sub>''σ''<sub>2</sub> = −1}}.

The even subalgebra Cℓ<sup>0</sup><sub>2,0</sub>(<math>\Reals</math>), spanned by ''even-graded'' basis elements of Cℓ<sub>2,0</sub>(<math>\Reals</math>), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and ''σ''<sub>1</sub>''σ''<sub>2</sub>. As a real algebra, Cℓ<sup>0</sup><sub>2,0</sub>(<math>\Reals</math>) is isomorphic to the field of [[complex numbers]] {{math| <math>\Complex</math>}}. As a result, it admits a conjugation operation (analogous to [[complex conjugate|complex conjugation]]), sometimes called the ''reverse'' of a Clifford element, defined by
<math display="block">(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1</math>
which, by the Clifford relations, can be written
<math display="block">(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 = a-b\sigma_1\sigma_2.</math>

The action of an even Clifford element {{math|''γ'' ∈ Cℓ<sup>0</sup><sub>2,0</sub>(<math>\Reals</math>)}} on vectors, regarded as 1-graded elements of Cℓ<sub>2,0</sub>(<math>\Reals</math>), is determined by mapping a general vector {{math|1=''u'' = ''a''<sub>1</sub>''σ''<sub>1</sub> + ''a''<sub>2</sub>''σ''<sub>2</sub>}} to the vector
<math display="block">\gamma(u) = \gamma u \gamma^*,</math>
where <math>\gamma^*</math> is the conjugate of <math>\gamma</math>, and the product is Clifford multiplication. In this situation, a '''spinor'''{{efn|These are the right-handed Weyl spinors in two dimensions. For the left-handed Weyl spinors, the representation is via {{math|1=''γ''(''ϕ'') = ''{{overline|γ}}ϕ''}}. The Majorana spinors are the common underlying real representation for the Weyl representations.}} is an ordinary complex number. The action of <math>\gamma</math> on a spinor <math>\phi</math> is given by ordinary complex multiplication:
<math display="block"> \gamma(\phi) = \gamma\phi.</math>

An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:
<math display="block"> \gamma(u) = \gamma u \gamma^* = \gamma^2 u.</math>

On the other hand, in comparison with its action on spinors <math> \gamma(\phi) = \gamma\phi</math>, the action of <math>\gamma</math> on ordinary vectors appears as the ''square'' of its action on spinors.

Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of ''θ'' corresponds to {{math|1=''γ''<sup>2</sup> = exp(''θ σ''<sub>1</sub>''σ''<sub>2</sub>)}}, so that the corresponding action on spinors is via {{math|1=''γ'' = ± exp(''θ σ''<sub>1</sub>''σ''<sub>2</sub>/2)}}. In general, because of [[branch cut|logarithmic branching]], it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.

In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by [[abuse of notation|abuse of language]], the two are often conflated. One may then talk about "the action of a spinor on a vector". In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to [[computer graphics]]) they make sense.

====Examples====

* The even-graded element <math display="block">\gamma = \tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2)</math> corresponds to a vector rotation of 90° from ''σ''<sub>1</sub> around towards ''σ''<sub>2</sub>, which can be checked by confirming that <math display="block">\tfrac{1}{2} (1 - \sigma_1 \sigma_2) \{a_1\sigma_1+a_2\sigma_2\}(1 - \sigma_2 \sigma_1) = a_1\sigma_2 - a_2\sigma_1</math> It corresponds to a spinor rotation of only 45°, however: <math display="block">\tfrac{1}{\sqrt{2}}(1-\sigma_1 \sigma_2)\{a_1+a_2\sigma_1\sigma_2\}=\frac{a_1+a_2}{\sqrt{2}} + \frac{-a_1+a_2}{\sqrt{2}}\sigma_1\sigma_2</math>
* Similarly the even-graded element {{math|1=''γ'' = −''σ''<sub>1</sub>''σ''<sub>2</sub>}} corresponds to a vector rotation of 180°: <math display="block">(- \sigma_1 \sigma_2)\{a_1\sigma_1 + a_2\sigma_2\} (- \sigma_2 \sigma_1) = - a_1\sigma_1 -a_2\sigma_2</math> but a spinor rotation of only 90°:<math display="block">(- \sigma_1 \sigma_2) \{a_1 + a_2\sigma_1\sigma_2\} = a_2 - a_1\sigma_1\sigma_2</math>
* Continuing on further, the even-graded element {{math|1=''γ'' = −1}} corresponds to a vector rotation of 360°: <math display="block"> (-1) \{a_1\sigma_1+a_2\sigma_2\} \, (-1) = a_1\sigma_1+a_2\sigma_2</math> but a spinor rotation of 180°.

=== 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.