# A Penrose twistor theory correctly predicts 4 dimensions

1. Mar 10, 2017

### Urs Schreiber

The following is well known but keeps being underappreciated:

1) Twistor theory exists not just in dimension 4, but also in dimensions 3 and 6 and to some extent in dimension 10.

Ingemar Bengtsson, Martin Cederwall,
"Particles, Twistors and the Division Algebras", Nucl.Phys. B302 (1988) 81-103
http://inspirehep.net/record/247269

Edward Witten,
"Twistor-like transform in ten dimensions",
Nuclear Physics,
Section B, Volume 266, Issue 2, p. 245-264. (1986) 10.
http://dx.doi.org/10.1016/0550-3213(86)90090-8 [Broken]

2) The reason "twistors work" is the fact that in these dimensions there is a magical coincidence by which a) Minkowski spacetime is identified with 2x2 hermitian matrices with entries in the complex numbers (for 4d) or real numbers (for 3d) or quaternions (for 6d), or octonions (for 10d): the generalized Pauli matrices. Moreover, in these dimensions the spin group happens to be isomorphic to the special linear group on two entries with coefficients in this number system.

For more on how this work see at "Twistor space" here

3) These algebraic facts that make twistors work in dimensions 3,4 6 and to some extent in dimension 10 are precisely the same algebraic facts that make the Green-Schwarz super-string work in these dimensions. See at division algebras and supersymmetry for more on this.

So the mathematics that makes twistors work, is just the same mathematics that makes the Green-Schwarz superstring work. Both theories agree on which spacetime dimensions are possible.

Last edited by a moderator: May 8, 2017