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In normal vector representation, vectors can be contracted as follows: $$v^\mu v_\mu$$

with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of indices:

$$\psi^{a} $$ $$ \psi^{\dot{a}} $$ $$ \psi_a $$ $$ \psi_{\dot{a}}$$

How do I contract them?

Also, a vector is a rank-2 spinor as the following: $v^{a \dot{b}}$, so is a covariant vector like this: $u_{\dot{a}b}$? What about $w^{a}_{\dot{b}}$ and other countless ways I can construct a rank-2 spinor?

Qmechanic
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Habouz
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1 Answers1

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  1. Undotted upper and lower indices can be contracted. Undotted indices are raised and lowered with the Levi-Civita symbol/tensor $\epsilon_{ab}$.

  2. Dotted indices work similarly. Dotted indices are raised and lowered with $\epsilon_{\dot{a}\dot{b}}$.

  3. For more information, see e.g. this & this Phys.SE posts.

Qmechanic
  • 201,751
  • Hmm. So if I have a vector that can be represented by 2 spinors: $v^{a\dot{b}}$, it transforms like this: $N_a^c v^{a\dot{b}} N_{\dot{b}}^{\dot{d}}=N_a^c \psi^{a}\psi^{\dot{b}} N_{\dot{b}}^{\dot{d}}=\psi ' ^{c} \psi ' ^ {\dot{d}} = v ' ^ {c \dot{d}}$ – Habouz Jul 14 '22 at 10:56