# SUSY and van der Waerden spinor notation

I am currently trying to read into SUSY and I am running into trouble with the van der Waerden spinor notation for Weyl spinors.

I am looking for resources that construct and justify the index notation given to the Weyl spinors, especially van der Waerden spinor notation.

The literature seems to be all over the place in terms of notation, also lecture notes that just introduce notation rather than develop it seem to have contradictions.

For example, in the lecture notes https://www.sissa.it/tpp/phdsection/OnlineResources/6/susycourse.pdf (1) (bottom of) page 26/ (top of) page 27 it states that we can link the inequivalent reps (that is the fundamental and anti-fundamental reps) via the definition $$(\bar \sigma^0)^{\dot \alpha \alpha}(\psi_\alpha)^*=:\bar\psi^\dot\alpha$$. This is different going from source to source. Immediately after, in the lecture notes (1), it states we can set $$(\psi_\alpha)^\dagger =: \bar\psi_\dot\alpha$$. This seems like too many definitions and makes it look like the notation will be inconsistent or trivial as we also have $$\epsilon_{ \dot \alpha \dot \beta}\bar\psi^\dot\beta =: \bar\psi_\dot\alpha$$. We get different answers when travelling around the maps between the 4 reps in different orders.

Another example of this is in Ben Allanchs lecture notes, http://www.damtp.cam.ac.uk/user/examples/3P7.pdf, stating both definitions, which in mind render the notation inconsistent.

• Welcome to the jungle. Unfortunately, one has different conventions (just as the metric tensor in Minkowski spacetime is either diag (+---) or diag (-+++)). Use a single book/source for basic SuSy and stick to its notation, after checking its consistency througout the text. Jul 22 '21 at 23:04
• Wess and Bagger's textbook 'Supersymmetry and Supergravity' is the authority on this notation. Check it out if you can find a free version. Jul 22 '21 at 23:25
• One problem is that interesting SUSY occurs in more than 4d. I've seen various confilcting extensions. I've just spend the morning try to grok Weinberg vol 3 chapter 32 for his take. Any recommendations for the best extended version? Jul 22 '21 at 23:32
• @mikestone Unfortunately the SUSY algebra depends greatly on the spacetime dimension (more specifically the dimension modulo 8). I'm not sure if there's a standard notation for each dimension modulo 8, but I am sure there's a standard notation for $2d$, $10d$ and $11d$, since those are the most widely studied dimensions outside of $4d$. I don't know what they are; maybe someone else knows? Jul 23 '21 at 2:12
• @fewfew4 Thanks for the comment. It's exactly how the gamma algebra leads to the allowed SUSY R symmetries displaying 8-fold Bott sequence $\ldots {\rm O}(16r)\supset {\rm U}(8r)\supset {\rm Sp}(4r)\supset {\rm Sp}(2r)\times {\rm Sp}(2r)\supset {\rm Sp}(2r) \supset {\rm U}(2r) \supset {\rm O}(2r) \supset {\rm O}(r)\times {\rm O}(r)\supset {\rm O}(r)\ldots$ that interests me. Jul 23 '21 at 13:36

Coincidentally I have also been catching up with SUSY in the last couple of days. I suggest you look at "Introduction to SUSY - H. Muller-Kirsten, A. Wiedemann (World Scientific 2010)". The first chapter should be what you are looking for, especially Section 1.3 on $$SL(2,C)$$ spinor representations. And almost all the identities are worked out in great detail in component notation.