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I am struggling to understand spinors in 1+1 spacetime. I know in this case the Clifford algebra is realized by two by two matrices so the spinors have two components. Then what do we mean by spin or spin components in 1+1 dimensions? There is not any $su(2)$ subalgebra in the algebra $so(1,1)$ so I do not see things such as $\pm \frac 12$ eigenvalues showing up. Do the two degrees of freedom of this spinor represent particles/antiparticles with no mention to spin components?

Any reference (for physicists) to this subject or to representations of the group $SO(1,1)$ would be welcome.

Diracology
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1 Answers1

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  • In 1+1D the restricted Lorentz group $SO^+(1,1)\cong \mathbb{R}_+$ is simply connected and only contains a boost $B$.

  • In light-cone coordinates $x^{\pm}=\frac{t\pm x}{\sqrt{2}}$, the Minkowski metric becomes off-diagonal $$ds^2~=~dt^2-dx^2~=~2dx^+dx^-, \qquad \eta_{\pm\mp}~=~1,\qquad \eta_{\pm\pm}~=~0,$$ while a restricted Lorentz matrix becomes diagonal: $$\Lambda~=~\begin{pmatrix}e^{\eta} & 0 \cr 0 & e^{-\eta} \end{pmatrix} ~=~e^{\eta B},\qquad B~=~\begin{pmatrix}1 & 0 \cr 0 & -1 \end{pmatrix},$$ where $\eta$ is the rapidity.

  • A Majorana-Weyl spinor $\psi\in\mathbb{R}$ of weight/"spin" $w\in\mathbb{R}$ is 1-dimensional, and transforms as $\psi^{\prime}=e^{w\eta}\psi$ under restricted Lorentz transformations.

  • A Dirac/Clifford representation in 1+1D is 2-dimensional. $$\{\sigma_{\mu},\sigma_{\nu}\}~=~\eta_{\mu\nu}{\bf 1}_{2\times 2}, \qquad \mu,\nu=\pm.$$

  • $Spin^+(1,1)\cong \mathbb{R}^{\times}:=\mathbb{R}\backslash\{0\}$ is the double cover of the restricted Lorentz group, cf. Ref. 1. It has 2 connected components.

References:

  1. V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, 2004; p. 193.
Qmechanic
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