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Spin representation combinatorics, Spin representation background, Spinors

Andrius Kulikauskas investigates:

What do spin representations 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.

Divisions of everything

Rotations are shifts in perspective

The mathematics of spin representations breaks up n-dimensional space into a maximal number of independent rotations. Each rotation pairs two dimensions, rotating from ont to the other, and analogously, in a division of everything, a shift in perspective pairs two perspectives, shifting from one to the other. In each case, there is an initial state and a final state.

In particular, consider free will and fate. Free will can be modeled with {$x\pm iy$} where there is a choice to be made between interpreting {$i$} as clockwise or counterclockwise. Fate is what comes after the choice is made, so that there is no longer a choice to be made. If we chose {$x+iy$}, then we are left with {$x-iy$}.

Rotations are planar numbers, complex numbers (roots of unity).

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 {$0$}.

Spin representations are the maximal representations

Spin representations are maximal in various senses. They run up against the symmetry of the totality, the widget that relates counting forwards and counting backwards, the difference between odd and even dimensions as regards how everything fits together.

A division of everything is a maximal structuring of perspectives in terms of structural pairs. So there is a general structural principle that structure wants to be maximized.

In the study of the tensor product {$S\otimes S$}, there is a breakdown in terms of independent pairs. And when signs are alternating, then the objects that survive have a maximum number of independent pairs. Note also that this breakdown into independent pairs codes the Hermite polynomials which are fundamental for the quantum harmonic oscillator.

A rotation of 90 degrees is two reflections separated by 45 degrees

Note that a rotation of 45 degrees is an eighth root of unity {$\sqrt{2}(1+i)$}. How does such a rotation relate to the two reflections it separates? The rotation includes the 45 degrees plus two angles (by which the reflection occurs) which add up to another 45 degrees.

Implicit covering group (spin group) and explicit (special orthogonal group)

We can think of the spin group as formalizing the unreflective answering mind and the special orthogonal group as formalizing the reflective questioning mind.

{$(1+i)^n$}

{$i+1$} 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 {$\sin x e^x$}.

Understanding the sequence of steps

The Wikipedia article Spin representation describes the construction of spin representations.

Here are the steps that I need to understand.

  • Define a quadratic form {$Q$} on {$V=\mathbb{C}^n$}.
  • Polarization identities define a symmetric bilinear form {$\langle \;,\; \rangle$}.
  • This allows us to break down {$V=W\oplus W^* \oplus U$} where {$W, W^*, U$} are totally isotropic subspaces, as follows. The dimension of {$V$} is {$n$} and we can define {$m$} such that {$n=2m$} if {$n$} is even, and {$n=2m+1$} is {$n$} is odd. Then {$V$} has an orthonormal basis {$x_1,\dots,x_m, y_1,\dots,y_m$} with an additional element {$u$} if {$n$} is odd. We pair up basis elements to construct {$a_j=x_j+iy_j$} as the basis elements for {$W$} and to construct {$\alpha_j=x_j-iy_j$} as the basis elements for {$W^*$}. Note that {$Q(x_j+iy_j)=0$} and likewise {$Q(x_j-iy_j)=0$}. Note that {$\langle \alpha_j, a_j \rangle = \delta_{ij}$}.
  • Note that we can write a bilinear form as {$\langle u, v \rangle = u^TBv = \sum_{i,j=1}^n u_iB_{ij}v_j$} and in our case we have

{$$B=\begin{pmatrix} 0 & I \\ I & 0 \\ \end{pmatrix}$$}

  • In general, for any {$n\times n$} matrix {$M$} and any {$u,v\in V$} we have {$\langle Mu,v \rangle = u^TM^Tv = \langle u, M^Tv \rangle$}. We can, in our case, consider {$v=w+w^*, w\in W, w^*\in W^*$} and we can understand {$w$} and {$w^*$} as in {$V$}. Similarly, we can understand any {$m\times m$} matrix {$A$} as an {$n\times n$} matrix {$A$} where the extra entries are all zero. Then we can observe {$\langle Aw,w^*\rangle = w^TA^Tw^*=\langle w,A^Tw^*\rangle$}. Recall that {$W$} and {$W^*$} are totally isotropic and so {$\langle Aw_1,w_2 \rangle = 0$} for all {$w_1,w_2\in W$} and likewise {$\langle Aw^*_1,w^*_2 \rangle = 0$} for all {$w^*_1,w^*_2\in W^*$}.

{$$\rho_A=\begin{pmatrix} A & 0 \\ 0 & -A^T \\ \end{pmatrix}$$}

  • Given {$m\times m$} matrix {$A$}, we define {$\rho_A$} as above. We observe that {$\langle\rho_A u,v\rangle = -\langle u,\rho_A v\rangle$} for all {$u,v\in V$}. To see this, decompose {$u=u_w+u_{w^*}, v=v_w+v_{w^*}$} where {$u_w,v_w\in W, u_{w^*},v_{w^*}\in W^*$}. We have {$\langle \rho A (u_w+u_{w*}),v_w+v_{w^*} \rangle = \langle Au_w -A^Tu_{w^*},v_w+v_{w^*}\rangle = \langle Au_w,v_{w^*}\rangle - \langle A^Tu_{w^*},v_w\rangle = \langle u_w,A^Tv_{w^*}\rangle - \langle u_{w^*}, Av_w \rangle$}{$ = - \langle u_{w}, Av_w \rangle - \langle u_{w^*}, Av_w \rangle + \langle u_w,A^Tv_{w^*}\rangle + \langle u_{w^*},A^Tv_{w^*}\rangle$}{$ = \langle u, -Av_w \rangle + \langle u, A^Tv_{w^*}\rangle = \langle u, -\rho_A v_w \rangle + \langle u, -\rho_A v_{w^*}\rangle = -\langle u, \rho_A v\rangle$}. This means that {$\rho_A$} is skew and {$\rho_A \in \frak{so}$}{$(n)$}.
  • Understand how this relates to the Lie algebra {$\frak{so}$}{$(n)$}! How is it that the diagonal matrices of the form {$\rho_A$} are elements of the Cartan subalgebra? Don't these matrices have zeroes on the diagonal?

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.

Ideas to investigate

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.

Three dimensions

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

Literature