# How do spinors arise in systems of less than 3 effective spatial dimensions as representations of the Lorentz group?

In 3+1 Minkowski spacetime, we can use the fact that the Lie algebra of the Lorentz group decomposes (I am omitting some details here to keep it short) into $$su(2) \oplus su(2)$$ and then use the fact that spinors are the fundamental representations of $$su(2)$$ to start building representations of the Lorentz group$$^\dagger$$ of the form $$(j_+, j_-)$$ where $$j_\pm$$ label representations of $$su(2)$$.

My question is: since there are systems that include spinors and live in 2+1 or 1+1 dimensions being studied at the moment, how do spinors come into play? How do we get them as representations of the (universal cover of the) Lorentz group in 2+1 and 1+1 dimensions? Can their Lie algebras be decomposed in similar ways or maybe there is a different definition of spinors in these dimensions (i.e. not as the fundamental representation of $$su(2)$$)?

$$^\dagger$$or rather its universal cover since we're interested in projective reps of the Lorentz group

There is a systematic way to define spinor representations in any number of dimensions. The key idea is the following. The SO$$(d-1,1)$$ algebra can be written

$$i[ \Sigma^{\mu\nu}, \Sigma^{\sigma\rho}]=\eta^{\nu\sigma} \Sigma^{\mu\rho} + \eta^{\mu\rho}\Sigma^{\nu\sigma} - \eta^{\nu\rho} \Sigma^{\mu\sigma} -\eta^{\mu\sigma} \Sigma^{\nu\rho}$$ where $$\eta^{\mu\nu}=$$diag(-1,+1,...,+1) and $$\Sigma^{\mu\nu}$$ is an antisymmetric matrix which packages the rotation and boost generators $$J_i,K_i$$. For instance in $$d=3+1$$ we have $$\Sigma^{\mu\nu}=\begin{pmatrix}0 &K_1 & K_2 & K_3\\-K_1 & 0 & J_3 & -J_2\\-K_2 & -J_3 & 0 & J_1 \\-K_3 & J_2 & -J_1 & 0\end{pmatrix}.$$

Moreover, whenever we have $$d$$ Dirac matrices $$\Gamma^{\mu}$$ satisfying

$$\{ \Gamma^\mu,\Gamma^\nu\} = 2\eta^{\mu\nu}$$

we automatically have a representation of SO($$d-1,1$$) using

$$\Sigma^{\mu\nu} = -\frac{i}{4} [\Gamma^\mu,\Gamma^\nu]$$ as one can check. So all we need are suitable $$\Gamma$$s and we are done; the spinors can be identified by building raising/ lowering operators out of the $$\Gamma$$s as usual. In $$d=1+1$$ we can use

$$\Gamma^0 = \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix}\quad\Gamma^1 = \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}$$ and in dimension 2+1 we can tack on $$\Gamma^2 = \Gamma^0\Gamma^1$$. There's a systematic way to get higher dimensions, too, and to find out which representations are irreducible. (Note the number of components of the spinor goes like $$2^{d/2}$$ for $$d$$ even or $$2^{(d-1)/2}$$ for $$d$$ odd).

• Great! Just a question: this seems pretty straightforward: introduce gamma matrices such that their commutator satisfies the appropriate Lie algebra and you're basically done; so why do textbooks on QFT usually get into the trouble of decomposing the Lorentz algebra into $su(2)\oplus su(2)$ in order to explain spin states? Maybe because in this way we can clearly see the motivation behind labelling reprenestations of the Lorentz group by the notation $(j_+ , j_-)$? Through what was shown in the answer, the labelling $(j_+ , j_-)$ due to the decomposition $su(2)\oplus su(2)$ is not apparent. Nov 14, 2020 at 18:11
• Not sure, but if I were to speculate, I'd guess it's a pedagogical tool. Students learn QFT after learning the quantum mechanics of SU(2), so it's hard to pass this up. Such a splitting can't work for $d=6,7,10,11$, for instance, where the Lorentz group has odd dimension. Nov 14, 2020 at 19:22