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In quantum field theory class, we are doing spinors. The representations of the Lorentz algebra $\mathfrak{so}(3,1) = \mathfrak{su}(2) + i \mathfrak{su}(2)$ are indexed by two integers. Then (1,2) and (2,1) are the left and right spinor representations, $\phi_a, \phi_{\dot{a}}$.

In the homework, we're asked to prove identities using spinor notation like $\psi\chi = \chi\psi$ and $(\psi\sigma^\mu\chi^\dagger)^\dagger = \chi\sigma^\mu\psi^\dagger$ by unsuppressing the indices (so $\psi \chi = \psi_a \chi^a$ and $\psi^\dagger \chi^\dagger = \psi^{\dagger\dot{a}} \chi_{\dagger\dot{a}}$.

I started developing "box" notation to keep track of which index is being matched/contracted with which other index in expressions like $ \epsilon^{ab}\epsilon^{\dot{a}\dot{b}}\sigma^\mu_{a \dot{a}}\sigma^{\nu}_{b\dot{b}} $. Since these are anticommuting variables, such notation would have to remember order of multiplication.

I started coming up with nice ways to handle upper and lower indices with upward and downwoard pointing boxes, tricks for handing Dirac and Majorana spinors and Hermitian conjugates. Has diagrammatic calculus for spinors already been built somewhere?? I think Penrose did such a thing for twistors.

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    $\begingroup$ You should look in "Birdtracks" by Cvitanovic if you have not done so already. If you are doing something different then I would be interested in seeing what you come up with. $\endgroup$ Jan 16, 2011 at 13:00
  • $\begingroup$ I don't know the answer to this question, but I would like to see a convincing argument that these diagrammatic calculi are actually useful. I'm serious: I'd love to be convinced that it makes sense to go down this path. $\endgroup$ Jan 16, 2011 at 13:13
  • $\begingroup$ @Jose: have you seen mathoverflow.net/questions/25187/… ? $\endgroup$ Jan 16, 2011 at 14:37
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    $\begingroup$ I think I agree with José. If one can write Wess and Bagger[1] without use of such diagrammatic calculi, then I think it's likely of little use. What I've found of most use is superfields. [1] press.princeton.edu/titles/2149.html $\endgroup$ Jan 16, 2011 at 16:48
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    $\begingroup$ @Everyone: I was not "dissing" diagrammatic techniques in general. I'm well aware of their usefulness in quantum field theory and also in proving identities in braided categories,... I was merely expressing a desire to be convinced that in the restricted context of representation theory their use is advantageous. $\endgroup$ Jan 16, 2011 at 18:36

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