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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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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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