# Weyl and Majorana-Weyl spinors why need commutation?

Let $\psi$ denote a Dirac spinor then Weyl spinors are defined by: $$\psi_{L,R}=\frac{1}{2} (I\pm \gamma)\psi$$ on even dimensions $\gamma$ commutes with $\sigma_{\mu \nu}$ (generators used to define spinor rep) and as such this is clearly a subrep of the dirac spinors. But I don't understand why it is then claimed what we can't define Weyl spinors for odd dimensions. Might it also be true that $\psi_{L,R}$ form a subrep in this case - even if less explicitly?

On a related note Majorana-Weyl spinors are said to only exist in $2 \mod 8$ dimensions which is derived on the condition that: $$[\gamma, C]=0$$ but I cannot see why this needs to be true. Since surely we only need them to share a single eigenvector rather then all of them (c.f. this post).

Please can someone explain these two points?

Note on Notation: $\gamma=i^{-k}\gamma_0 \ldots \gamma_{d-1}$ and $C$ is the operator for which Majorana spinors are defined: $\psi =C\psi$

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• I think I have found the answer to the first of these, i.e. in the odd case $\gamma\propto 1$ so the projector simply projects back onto the Dirac spinors or $0$ and in either case a trivial subrep. I am still confused about the second part. – Quantum spaghettification Dec 26 '17 at 13:18
• I have found H. Murayama’s notes: hitoshi.berkeley.edu/230A/clifford.pdf to be a good and clear intro to the eightfold periodicty of Majorana and Weyl fermions. You want to look at, and understand the origin of, his table 4 for Minkowski signature. – mike stone Dec 26 '17 at 21:44
• Appendix E of de Wit, B. and Smith, J. (1986), Field Theory in Particle Physics (North-Holland Personal Library), V1, ISBN 978-0444869999 should do the trick. – Cosmas Zachos Dec 27 '17 at 14:24
• If mathematically inclined, peruse the Clifford algebra clock. – Cosmas Zachos Dec 27 '17 at 16:38
• You seem to have taken an alarming and mysterious left turn with your oracular "surely we only need them to share a single eigenvector rather than all of them". I doubt your question is meaningful, unless you explain this preposterous misconception. – Cosmas Zachos Dec 27 '17 at 21:29