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started something at twistor space
added a section twistors for 4d Minkowski spacetime with basics on the actual original definition and motivation for twistors.
(from looking around I gather this is now the only discussion on the web that comes out right away with admitting what a twistor actually is, conceptually :)
added an actual section twistor space with discussion of how $\mathbb{K}P^3$ encodes light-like geodesics in Minkowski spacetime. Did this in the generality that $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$, hence for Minkowski spacetime of total dimensions 3,4, and 6, in order to amplify the algebraic pattern.
To discuss twistor space for Minkowski spacetime, it is useful to work more generally with $d$-dimensional Minkowski spacetime for $d \in \{3,4,6,8\}$.
Do you mean 10 rather than 8?
Then
…vectors in Minkowski spacetime in $d = \{2,3,4,10\}$…
why those dimensions?
For $d = 10$ there is no elegant statement like this, due to the non-associativity of the octonions
There’s plenty of discussion about why there can’t be an $\mathbb{O} P^3$ here. Enough associativity for $\mathbb{O} P^2$, but not projective space. Does the lack of octonion spinors tell us anything, or is it just an inconvenience?
Sorry for the silly typos, fixed now. Thanks for catching them. I’ll try to catch the next train home to get some sleep.
Something is wrong in the part of the entry. The entry incorrectly says
The original twistor correspondence (Penrose 67) is the correspondence
$\left( \array{ && Gr_{1,2}(\mathbb{C}^4) \\ & \swarrow && \searrow \\ \mathbb{C}P^3 && && Gr_2(\mathbb{C}^4) } \right) \;\;\;\;\; \simeq \;\;\;\;\; \left( \array{ && SL_\mathbb{C}(4)/SL_{\mathbb{C}}(2) \\ & \swarrow && \searrow \\ SL_{\mathbb{C}}(4)/SL_{\mathbb{C}}(3) && && SL_{\mathbb{C}}(4)/(SL_{\mathbb{C}}(2)\times SL_{\mathbb{C}}(2)) } \right) \,,$The right-hand side is plainly wrong entrywise: all the quotients should be quotients of SL-s by the appropriate parabolics (block-triangular), which are therefore bigger than the diagonal block SL-s written. This way even the dimensions do not fit.
Isn’t it the standard expression for the Grassmannians? But there is much room to expand on this. Feel free to remove that line and start afresh.
This article appeared recently on the arxiv about the twistor $\mathbf{P}^1$ and its role in number theory, algebraic geometry, and mathematical physics
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