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nLab > Latest Changes: twistor space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2013

    started something at twistor space

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2013
    • (edited Oct 31st 2013)

    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 :)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2016

    added an actual section twistor space with discussion of how 𝕂P 3\mathbb{K}P^3 encodes light-like geodesics in Minkowski spacetime. Did this in the generality that 𝕂∈{ℝ,ℂ,ℍ}\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.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 18th 2016

    To discuss twistor space for Minkowski spacetime, it is useful to work more generally with dd-dimensional Minkowski spacetime for d∈{3,4,6,8}d \in \{3,4,6,8\}.

    Do you mean 10 rather than 8?

    Then

    …vectors in Minkowski spacetime in d={2,3,4,10}d = \{2,3,4,10\}…

    why those dimensions?

    For d=10d = 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 𝕆P 3\mathbb{O} P^3 here. Enough associativity for 𝕆P 2\mathbb{O} P^2, but not projective space. Does the lack of octonion spinors tell us anything, or is it just an inconvenience?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2016
    • (edited Oct 18th 2016)

    Sorry for the silly typos, fixed now. Thanks for catching them. I’ll try to catch the next train home to get some sleep.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 14th 2018

    Corrected a subscript.

    diff, v36, current

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeSep 18th 2019

    Added monographs

    • L. J. Mason, N. M. J. Woodhouse, Integrability, self-duality and twistor theory, Oxford Univ. Press 1996
    • R. S. Ward, R. O. Wells, Jr. Twistor geometry and field theory, Cambridge Univ. Press 1990

    diff, v37, current

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeSep 18th 2019

    Added also two references on palatial twistor theory, a new hope of Penrose, related to quantization.

    diff, v37, current

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeSep 18th 2019

    Added also two references on palatial twistor theory, a new hope of Penrose, related to quantization.

    diff, v37, current

    • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeSep 22nd 2019
    In my humble opinion one can find the actual twistor space existing as the bond between Dimer molecules and one can find the four momentum particle actually existing as the caged electron in the Dimer cavity in the microtubules in our neurons. The caged electron is in a state of partial decay and is also a Hilbert space. The lattice forming along magnetic lines between the opposing molecules is the larger twistor space but twistor space and the caged electron together form an energy tensor. frowningbuddha at g mail dot com
    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeFeb 11th 2021

    Something is wrong in the part of the entry. The entry incorrectly says

    The original twistor correspondence (Penrose 67) is the correspondence

    ( Gr 1,2(ℂ 4) ↙ ↘ ℂP 3 Gr 2(ℂ 4))≃( SL ℂ(4)/SL ℂ(2) ↙ ↘ SL ℂ(4)/SL ℂ(3) SL ℂ(4)/(SL ℂ(2)×SL ℂ(2))), \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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2021

    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.

    • CommentRowNumber13.
    • CommentAuthorGuest
    • CommentTimeFeb 12th 2022

    This article appeared recently on the arxiv about the twistor P 1\mathbf{P}^1 and its role in number theory, algebraic geometry, and mathematical physics

1 to 13 of 13

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