Lorentz spinors of $SO(n,1)$ and conformal spinors of $SO(n,2)$

Question

• It would be great if someone can give me a reference (short enough!) which explains the (spinor) representation theory of the groups $SO(n,1)$ and $SO(n,2)$.

I have searched through a few standard representation theory books and I couldn't find any.

• More specifically I would like to know how a Lorentz spinor of $SO(n-1,1)$ (say $Q$) is "completed" to a conformal spinor of $SO(n,2)$ (say $V$) by saying,

$V = (Q, C\bar{S})$

where $C$ is a "charge conjugation operator" and $S$ is probably another $SO(n-1,1)$ spinor.

Is there some natural Clifford algebra representation ($\Gamma$) lurking around here with respect to which I can define the "charge conjugation operator" as $C$ such that $C^{-1}\Gamma C = - \Gamma ^T$? (...in general a representation of the Clifford algebra also gives a representation of $SO(n,1)$..I would like to know as to how this general idea might be working here...)

• Some of the other aspects of this group theory that I want to know are an explanation for facts like,

  • $Sp(4)$ is the same as $SO(3,2)$, and the fundamental of $Sp(4)$ is the spinor of $SO(3,2)$
  • $SU(2,2)$ is the same as SO(4,2), and the fundamental of $SU(2,2)$ is the spinor of $SO(4,2)$

  (...just two "facts" hoping that people can point me to some literature (hopefully short!) which explains the systematics of which the above are probably two examples...)

Answer 1

Albert Crumeyrolle, "Orthogonal and Symplectic Clifford Algebras", Kluwer, Dordrecht, 1990.

One finds for the Clifford algebra $C(m,n)$ with basis vectors $\gamma^\mu$ that the elements $\frac{1}{2}(\gamma^\mu\gamma^\nu-\gamma^\nu\gamma^\mu)$ generate a Lie algebra that is isomorphic to $so(m,n)$. With two new basis vectors in $C(m+1,n+1)$ that are orthogonal to the $\gamma^\mu$, $\gamma_P$ and $\gamma_M$, say,

• $\frac{1}{2}(\gamma_P+\gamma_M)\gamma^\mu$ generate what can be taken as translations,
• $\frac{1}{2}\gamma_P\gamma_M$ generates what can be taken as dilatations,
• and $\frac{1}{2}(\gamma_P-\gamma_M)\gamma^\mu$ generate what can be taken as special conformal transformations.

The spinors are ideals of the Clifford algebras, for which the precise structure ---real, complex, or quaternionic--- depends on the dimensionality and on whether one is working over the reals or over the complex field.

The Crumeyrolle is not written for Physicists, but the underlying structure is the same.