Chat with us    Hi Andrej! I will teach you how to write these wiki pages.
    
Econet / Spinors     Search:

Spinor combinatorics

Andrius Kulikauskas investigates:

What do spinors say about Bott periodicity?

I am trying to understand what are spinors. Partly, that will help me understand the physics that John Harland talks about with me, especially as regards the Dirac equation and relativity. But also, spinors manifest Bott periodicity, and I am interested to what insight they may provide.

Spin {$m$} indicates a division of everything into {$2m+1$} perspectives

In particular, I expect that spin {$m$} indicates the division of everything into {$n$} states where {$n=2m+1$}. Thus if {$m=\frac{1}{2}$} then there are two states: {$-\frac{1}{2},\frac{1}{2}$}. Whereas if {$m=1$} there are three states: {$-1,0,1$}. If {$m=0$} there is one state {$0$}.

Spinors are defined by what is done to them

Spinors are not defined by what they are. Instead, spinors are defined by what is done with them. Similarly, a "paperweight", a "piece of trash", a "sales item", a "hat" are all defined not by what they are but what is done with them.

Spinors are elements of a complex vector space

Spinors are elements of the vector spaces that are given by representations of spin groups. Currently, I am studying the Wikipedia article on Spin representation and the recipe it gives for building them.

  • I am trying to understand spinors in the simplest case, the group {$\textrm{Spin}(3)\cong SU(2)$} which is the double cover of the special orthogonal group {$SO(3)$} of the rotations of the sphere.
  • Later I should understand the spin statistics theorem

Here are my notes at Math4Wisdom: Spinors

{$\textrm{Spin}(3)\cong SU(2)$}

I am interpreting the article Wikipedia: Spin representation in terms of {$\textrm{Spin}(3)\cong SU(2)$}.

Quadratic form {$Q$} and symmetric bilinear form {$\langle\;,\;\rangle $}

{$V=\mathbb{C}^3$} with orthonormal basis {$z_1=(1,0,0),z_2=(0,1,0),z_3=(0,0,1)$} and quadratic form {$Q(c_1,c_2,c_3)=Q((c_1,c_2,c_3))=c_1^2+c_2^2+c_3^2$}.

The quadratic form {$Q$} defines a symmetric bilinear form {$\langle\;,\;\rangle $} using a polarization identity. Note that this is not sesquilinear and note also that the basis elements commute (they are not anticommmuting). The polarization identities allow us to characterize an angle between vectors in terms of the squared lengths of the two vectors, their sum and their difference. Specifically:

{$2\langle u, v\rangle = Q(u+v)-Q(u)-Q(v)$}

{$2\langle u, v\rangle = Q(u)+Q(v)-Q(u+v)$}

{$4\langle u, v\rangle = Q(u+v)-Q(u+v)$}

Isotropic spaces {$W$} and {$W^*$}

{$v\in V$} is isotropic if {$Q(v)=0$}. Subspace {$W$} of {$V$} is totally isotropic if all of its elements are isotropic.

We have isotropic elements {$z_1+iz_2$} and {$z_1-iz_2$} and isotropic spaces {$W=\{\lambda(z_1+iz_2) | \lambda\in\mathbb{C}\}$} and {$W^*=\{\gamma(z_1-iz_2) | \gamma\in\mathbb{C}\}$}. In this way, we can pair up the original basis elements.

{$V$} is odd-dimensional so we also define {$U=\{\beta x_3 | \beta\in\mathbb{C}\}$}

Note that {$\langle z_1 + iz_2, z_1 - iz_2\rangle = \langle z_1, z_1 \rangle - i\langle z_1, z_2 \rangle + i\langle z_2, z_1 \rangle - i^2\langle z_2, z_2\rangle = Q(z_1) + Q(z_2) = 1 + 1 = 2$} where two terms cancelled out because of symmetry.

Thus we have two isotropic elements {$z_1+iz_2$} and {$z_1-iz_2$} for which the bilinear form {$\langle z_1 + iz_2, z_1 - iz_2\rangle$} is not zero.

{$V=W\oplus U \oplus W^*$} where the three latter spaces are all one-dimensional.

{$V$} acts on the exterior algera {$S=\wedge^\bullet W$}

{$V$} acts on the exterior algebra {$S=\wedge^\bullet W$}, which is likewise one-dimensional. This action has {$v=w+w^*\in W\oplus W^*$} act on {$\psi\in S$} by

{$v\cdot\psi=\sqrt{2}(w\wedge\psi + \iota(w^*)\psi)$}

{$\iota(w^*)$} is defined by the bilinear form {$\langle x_1+ix_2, x_1-ix_2\rangle $} which relates {$W$} and {$W^*$}. We have {$w^*=\gamma (x_1-ix_2)$} thus {$\iota(w^*)=\iota(\gamma (x_1-ix_2))=\langle x_1+ix_2, \gamma (x_1-ix_2)\rangle $}

Real spin representations

A real spin representation is understood as embedded within a complex spin representation in one of several possible ways. Some of these may be thought of as the removal of one generator (yielding a real structure) or two generators (yielding a quaternionic structure). This may be related to defining a codimension 1 (defining a reflection) or a codimension 2 (defining a rotor) although I think here these are complex codimensions.

Spinors and zero

An isotropic vector has length zero but can have an orientation.

Odd orthogonal group has an outer zero and even orthogonal group has an inner zero (this latter case yields chiral spinors)

{$(1+i)^n$}

{$i+1$} is proportional to the eight-root of {$1$}. It generates numbers expressing Bott periodicity. They are given by the sequences A146559 for the exponential generating function of {$\sin x e^x$} and A009545 for the exponential generating function of {$\sin x e^x$}.

Geometric product

Math Stack Exchange. Visualizing the area described by the dot product?

{$(\textbf{v}\cdot\textbf{w})^2 + \parallel \textbf{v}\wedge\textbf{w} \parallel^2 = \parallel \textbf{v} \parallel^2 \parallel \textbf{w} \parallel^2 $}

Dimensions of embedded Lie groups

I should relate the combinatorics of spin representations with the dimensions of embedded Lie groups, notably the role of the diagonal of the matrix, what it contributes to the dimension.

Literature