Talk:Twistor theory

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Unless I am missing something, I thought complex spaces don't have metric signatures, unlike real spaces. Phys 06:28, 1 Dec 2004 (UTC)

I'm a physics student at UCLA, I've only read about Twistors but from what Roger Penrose describes it as it is "Higher dimensional spinors" (from his tome "Road to Reality"). So, rather than the conventional 2 dimensional complex vector space, there is instead a 4 dimensional complex vector space. This is, of course, from an undergrad's understanding, so I may have missed the boat completely!

Further the twistor depends on the spinor momentum \pi\,_{A'} where it satisfies

p^{ \alpha\, } = \bar{ \pi\,^{ A } }\pi\,^{ A' }

the noncontracted multiplication of it and its complex conjugate is equal to the four-momentum. These, however, are the last two coordinates of the Twistor.

The first two are related to the angular momentum, but can be calculated from the momentum components of the twistor.

Twistors are concerned, first of all, with incidence (in the geometric sense of the word). That is to say, with (say) a light cone, if an event lies within the light cone it is in incidence with the light cone. The coordinates of the event in incidence with the twistor is rAA'. That is

r^{ A A' } = \begin{bmatrix} t+z & x-iy \\ x+iy & t-z \end{bmatrix}

in spinor form. The relation of the angular momentum components \omega\,^{ A } to the linear momentum components \pi\,_{A'} are

\omega\,^{ A } = i r^{ A A' } \pi\,_{A'}

Thus a spinor Z^{\alpha \,} = (\omega \,^{A}, \pi \,^{A'} ). The norm of the twistor has a unique relation to the helicity of the particle in question with the equation

s = \frac{ 1 }{ 2 } \bar{Z_{ \alpha }}Z^{ \alpha }\,

where s is helicity. That's the primer. Cheers! - Pablo.


Can someone tell me why this is "For some time there was hope that the twistor theory may be the right approach towards solving quantum gravity, but this is now considered unlikely."?

Thanks

Alan ( alan_stafford@btinternet.com )

  • I think he means why is twistor theory no longer perceived as a great hope for approaching quantum gravity...not why that section is in the text. Well, for one it works primarily with special relativity rather than general relativity...twistors could be seen more as a quantization of the light cone rather than a quantization of gravity itself. It does, nonetheless, give insight into making events "fuzzy" rather than null vectors (the paths light takes). I recall reading from a monograph from the late 80s or early 90s that Christopher Isham thought Twistors would be more of a "last resort" rather than a "frontal assault" for quantum gravity, so "when all else fails" we can still fall back on it! The reason for his stance, I think, was the promise of Ashtekar variables in the canonical approach to quantum gravity appeared to be superior to the promise of the twistor programme. That is not to say however that twistors are useless, they are actually rather interesting things...it's just that there are other approaches which appear to be superior at the moment.

How does the link "Music based on twistor theory" have anything to do with Twistor theory - besides the band's founder (Mr. Hogan) studying Twistor Theory at a university and adopting the name "TWISTOR" for his DJing business? There may be some valid relationship between Twistor Theory and Mr. Hogan, but there seems to be no basis in Twistor theory within the music itself. Davidl9999 19:38, 29 October 2007 (UTC)