User:Jheald/sandbox/GA/Spinors in two dimensions

Per the recipes in article spinor, let us construct respresentations of spinors in two dimensions:

Component spinors: real form

1. Create an explicit representation of the Clifford algebra. (ie, a Clifford module).

Basis:

$e_{1}^{2}=e_{2}^{2}=+1;\qquad (e_{1}e_{2})^{2}=-1$

This can be achieved with the assignments (Lounesto, p. 14):

$1\simeq {\begin{pmatrix}1&0\\0&1\end{pmatrix}};\;e_{1}\simeq {\begin{pmatrix}1&0\\0&-1\end{pmatrix}};\;e_{2}\simeq {\begin{pmatrix}0&1\\1&0\end{pmatrix}};\;e_{1}e_{2}\simeq {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$

Reversion is achieved by transposing.

2. Now identify Δ, the space of spinors, as R², the space of column vectors on which the matrices act

$M{\begin{pmatrix}u\\v\end{pmatrix}}={\begin{pmatrix}u'\\v'\end{pmatrix}}$

3. These spinors can be related to elements of the algebra if we make the assignments

${\begin{pmatrix}u\\v\end{pmatrix}}\simeq {\begin{pmatrix}u&0\\v&0\end{pmatrix}}$

These elements are a sub-algebra of the original Clifford algebra, spanned by

$f_{0}={\tfrac {1}{2}}(1+e_{1})$

$f_{1}={\tfrac {1}{2}}{e_{1}-e_{1}e_{2}}={\tfrac {1}{2}}e_{2}(1+e_{1})$

A Clifford algebra on this subspace would be Cl_1,0(R)

Comments

The significance of the element ½ (1 + e₁) is clear if we consider its corresponding matrix element,

${\tfrac {1}{2}}(1+e_{1})\simeq {\begin{pmatrix}1&0\\0&0\end{pmatrix}};$

This makes clear that ½ (1 + e₁) is an idempotent

${\tfrac {1}{2}}(1+e_{1}){\tfrac {1}{2}}(1+e_{1})={\tfrac {1}{2}}(1+e_{1})$

and why it annuls elements of the Clifford algebra that correspond to the projection out of other columns

${\tfrac {1}{2}}(1-e_{1}){\tfrac {1}{2}}(1+e_{1})=0$

Nilpotent route

Alternatively, one starts by finding a nilpotent element (which will be represented by a nilpotent matrix)...

${\tfrac {1}{2}}e_{1}(1-e_{2})\simeq {\begin{pmatrix}0&0\\1&0\end{pmatrix}};$

"The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it" -- don't understand this. ... ?? maybe to do with iterating a nilpotent element to build up a flag ??

Isotropic subspace

So we get an isotropic subspace

$W=\mathrm {span} (f_{0},f_{1})\;$

is one because it contains the nilpotent above.

Why "isotropic" ? -- no clear derivation I can see yet from the more ordinary sense of "equal in all directions"

"maximal isotropic subspace" --> pull out the whole off-diagonal part of the column? No, not quite right.

Weyl spinors

The action of γ ∈ Cℓ⁰_2,0 on a spinor φ ∈ C is given by ordinary complex multiplication:

Right handed Weyl spinors: $\gamma (\phi )=\gamma \phi$

Left handed Weyl spinors: $\gamma (\phi )={\bar {\gamma }}\phi$

Can both be drived from the real spinors

More explicitly

We start with the representation of the algebra using the Weyl-Brauer matrices:

$1\simeq {\begin{pmatrix}1&0\\0&1\end{pmatrix}};\;e_{1}\simeq {\begin{pmatrix}0&1\\1&0\end{pmatrix}};\;e_{2}\simeq {\begin{pmatrix}0&i\\-i&0\end{pmatrix}};\;e_{1}e_{2}\simeq {\begin{pmatrix}-i&0\\0&i\end{pmatrix}}$

The spinors are then the space of column vectors on which these matrices act:

$M{\begin{pmatrix}u\\v\end{pmatrix}}={\begin{pmatrix}u'\\v'\end{pmatrix}}$

We now look for eigenvectors of the block-matrix $U=\left({\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}}\right)$.

This gives one eigenspace spanned by $\left({\begin{smallmatrix}1\\0\end{smallmatrix}}\right)$, which we shall call the right-handed Weyl spinor,

and one eigenspace spanned by $\left({\begin{smallmatrix}0\\1\end{smallmatrix}}\right)$, which we shall call the left-handed Weyl spinor.

In terms of elements of the algebra

• There is no element of Cl₂(R) that we can identify with M.
• Nor can we construct either Weyl spinor from elements of Cl₂(R)
• However, we can recognise M as the what could correspond to e₃ in a representation of Cl₃(R) that contained this representation of Cl₂(R)

Taking this route, the spinor space would be a space spanned by the elements

$x\;{\tfrac {1}{2}}(1+e_{3})$

a right-handed Weyl spinor corresponds to a space spanned by the elements

${\tfrac {1}{2}}(1+e_{3})\;x\;{\tfrac {1}{2}}(1+e_{3})$

and a left-handed Weyl spinor corresponds to a member of the space spanned by the elements

${\tfrac {1}{2}}(1-e_{3})\;x\;{\tfrac {1}{2}}(1+e_{3})$

where x is a general element of Cl₃(R).

• if x is a general element of Cl₂(R) -- i.e. no e₃ factors -- then the right spinor will represent the even part of x, and the left spinor the odd part.

Interpretation

?

"How the hell do I add a scalar to a vector ?"

(I know how to add a scalar to a bivector, and what it means...)