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For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as $$ T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}} $$ with local coordinate $x=(x^{1},\dots,x^{n})$ of $M$. The transformation rule of tensors is nothing but the change of local coordinates of $M$, i.e. if one changes the local coordinate $x\mapsto y$ with Jacobian matrix $g$, then the tensor above transforms like $$ T_{a,b}^{\ \ \ \ c} \mapsto T_{A,B}^{\ \ \ \ C}=T_{a,b}^{\ \ \ \ c}g_{A}^{\ \ a}g^{\ \ b}_{B}g^{\ \ C}_{c}. $$ How should one think of the transformation rule of spinor fields? What local change reflects the transformation rule?

My question is a bit ambiguous, but I hope everyone can understand what I try to say. Thank you.

M. K.
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    My understanding is that they transform like spin representations (http://en.wikipedia.org/wiki/Spin_representation) of orthogonal groups. But I don't think this explicit description in local coordinates is a good way to think about tensors and it's probably even less of a good way to think about spinors. – Qiaochu Yuan Sep 10 '12 at 20:05
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    The proper answer to this question should address both Weyl and Majorana spinors as well as the methods to construct scalars from said quantities. – James S. Cook Sep 11 '12 at 04:30
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    See my answer here. – Moishe Kohan Apr 05 '23 at 16:20

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