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 )

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