Timeline for Classical geometric interpretation of spinors
Current License: CC BY-SA 3.0
4 events
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Jun 2, 2011 at 19:43 | history | edited | Igor Khavkine | CC BY-SA 3.0 |
Fixed typo (V should have been C^3).
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Jun 2, 2011 at 14:31 | comment | added | Paul Siegel | This extends to a representation of the Clifford algebra of $V$ on $S$ which in turn restricts to the spin representation of $Spin(V) \subseteq Cl(V)$. Some or all of this might only be valid for even dimensional $V$; I can't quite remember. But in the even dimensional case the spin representation is graded, and the direct summands each carry an irreducible representation of the spin group in one dimension lower. | |
Jun 2, 2011 at 14:26 | comment | added | Paul Siegel | The following observation should show how to generalize your answer to higher dimensions. Let $V$ be a real vector space and let $P$ be a polarization of $V \otimes \mathbb{C}$, i.e. a maximal isotropic subspace, i.e. an isotropic subspace such that $V \otimes \mathbb{C} = P \oplus \overline{P}$. Let $S$ denote the exterior algebra of $P$, and define an action of $V$ on $S$ by allowing a vector $v$ to act as exterior product with $v$ minus interior product with $v$ (up to a constant - the square root of 2, I think). | |
Jun 2, 2011 at 13:04 | history | answered | Igor Khavkine | CC BY-SA 3.0 |