Editing Spinor (section)

=== 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°.