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When doing calculations with Weyl spinors, terms like $\theta\sigma^\mu\theta^\dagger$ appear. I know that for 3+1 spacetime dimensions, $\sigma^\mu = (\textbf{1}, \sigma^i)$ with $i=1,2,3$ the usual Pauli matrices. But what if we consider 1+1 or even $D$+1 dimensions?

Qmechanic
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ersbygre1
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  • What do you know about Clifford groups and algebras? – DanielC Sep 27 '17 at 07:58
  • @DanielC I know the elements have to fulfill anti commutator relations ${ \gamma^\mu,\gamma^\nu } = 2\eta^{\mu\nu}\textbf{1}$. – ersbygre1 Sep 27 '17 at 08:07
  • You asked about the generalization to D spatial dimensions. I asked you for your mathematical background because I found a possible source to answer your question https://arxiv.org/abs/hep-th/0506011 – DanielC Sep 27 '17 at 08:29
  • That's great, thank you very much! I guess I should rather say, my knowledge of Clifford algebras is somewhat limited. I can't find the answer to $\sigma^\mu$ in 1+1 dimensions in this paper.. – ersbygre1 Sep 27 '17 at 08:37
  • I will try to look this up. Namely, if there are spinors for SO(1,1). – DanielC Sep 27 '17 at 08:46
  • there are spinors for SO(1,1), they are representations of its double cover Spin(1,1) – Mozibur Ullah Sep 27 '17 at 13:59

1 Answers1

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A covariant Pauli matrix $\sigma^\mu$ is defined as $$\sigma^\mu=e^\mu_a\,\sigma^a\, \tag{1}$$ where $e^\mu_a$ is a vielbein with a Lorentz index $\mu$, and a flat spacetime (tangent space) index $a$. $\sigma^a$ is a Pauli matrix in flat spacetime.

$$\sigma^a=({\bf{1}},\,\sigma^i)\,\tag{2}$$ where ${\bf 1}$ is an identity matrix and $\sigma^i=(\sigma^1,\,\sigma^2,\,\sigma^3)$ are the usual Pauli matrices.

The above definitions do not make any reference to the number of dimensions of the spacetime. Hence they are true in all spacetime dimensions. You just need to use the tangent space Pauli matrices $\sigma^a$ shown in $(2)$ as per the number of spacetime dimensions you are working in.

This discussion might be helpful in getting $\sigma^a$ in higher dimensions.

vyali
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