2
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ There are no rotations in 1+1, only boosts. Have you solved the Dirac equation? $\endgroup$ – Cosmas Zachos Jun 1 '18 at 14:12
  • $\begingroup$ meaning... $\endgroup$ – Cosmas Zachos Jun 1 '18 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 '18 at 20:50
  • 1
    $\begingroup$ Yes, writing down the trivial Poincare algebra you may confirm $P^2$ is the only Casimir. $\endgroup$ – Cosmas Zachos Jun 1 '18 at 22:31
5
$\begingroup$

In 1+1D the restricted Lorentz group $SO^+(1,1)\cong \mathbb{R}_+$ 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. $$\{\sigma_{\mu},\sigma_{\nu}\}~=~\eta_{\mu\nu}{\bf 1}_{2\times 2}, \qquad \mu,\nu=\pm.$$

$\endgroup$
  • $\begingroup$ What are the possible values of $w$? Does it have any physical meaning? $\endgroup$ – Diracology Jun 1 '18 at 20:22
  • $\begingroup$ I updated the answer. $\endgroup$ – Qmechanic Jun 2 '18 at 13:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.