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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.

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    $\begingroup$ There are no rotations in 1+1, only boosts. Have you solved the Dirac equation? $\endgroup$ Jun 1, 2018 at 14:12
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    $\begingroup$ meaning... $\endgroup$ Jun 1, 2018 at 14:13
  • $\begingroup$ @CosmasZachos Is $P^2$ the only Casimir of the Poincare group in $1+1$? In other words, is there any other quantity that (along with momentum) labels irreducible representations of the Poincare group? $\endgroup$
    – Diracology
    Jun 1, 2018 at 20:50
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    $\begingroup$ Yes, writing down the trivial Poincare algebra you may confirm $P^2$ is the only Casimir. $\endgroup$ Jun 1, 2018 at 22:31
  • $\begingroup$ related: physics.stackexchange.com/q/470148/226902 $\endgroup$
    – Quillo
    Jun 30, 2022 at 13:09

1 Answer 1

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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.
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