- Twistor theory
The twistor theory, originally developed by
Roger Penrose in1967 , is the mathematical theory which maps thegeometric objects of the four dimensional space-time (Minkowski space ) into the geometric objects in the 4-dimensional complex space with themetric signature (2,2). The coordinates in such a space are called "twistors."The twistor theory was stimulated by a rationale indicating its particular usefulness in emergent theories of
quantum gravity .The twistor approach appears to be especially natural for solving the equations of motion of
massless fields of arbitrary spin.In
2003 Edward Witten used twistor theory to understand certainYang-Mills amplitudes, by relating them to a certainstring theory , thetopological B model, embedded in twistor space. This field has come to be known astwistor string theory .Details
Twistor theory is unique to 4D
Minkowski space and does not generalize to other dimensions ormetric signature s. At the heart of twistor theory lies theisomorphism between theconformal group Spin(4,2) and SU(2,2), which is the group of linear transformationsof determinant 1 over a four dimensional complex vector space leaving a Hermitian norm of signature (2,2) invariant.* mathbb{R}^6 is the real 6D vector space corresponding to the
vector representation of Spin(4,2).
* mathbf{R}mathbb{P}^5 is the real 5Dprojective representation corresponding to the equivalence class of nonzero points in mathbb{R}^6 under scalar multiplication.
* mathbb{M}^c corresponds to the subspace of mathbf{R}mathbb{P}^5 corresponding to vectors of zero norm. This is conformally compactified Minkowski space.
* mathbb{T} is the 4D complex Weyl spinor representation and is called twistor space. It has an invariant Hermitian sesquilinear norm of signature (2,2).
* mathbb{PT} is a 3D complex manifold corresponding to projective twistor space.
* mathbb{PT}^+ is the subspace of mathbb{PT} corresponding to projective twistors with positive norm (the sign of the norm, but not its absolute value is projectively invariant). This is a 3D complex manifold.
* mathbb{PN} is the subspace of mathbb{PT} consisting of null projective twistors (zero norm). This is areal-complex manifold (i.e. it has 5 real dimensions, with four of the real dimensions having a complex structure making them two complex dimensions).
* mathbb{PT}^- is the subspace of mathbb{PT} of projective twistors with negative norm.mathbb{M}^c, mathbb{PT}^+, mathbb{PN} and mathbb{PT}^- are all
homogeneous spaces of theconformal group .mathbb{M}^c admits a conformal metric (i.e. an equivalence class of metric tensors under
Weyl rescaling s) with signature (+++-). Straight null rays map to straight null rays undera conformal transformation and there is a unique canonical isomorphism between null rays in mathbb{M}^c and points in mathbb{PN} respecting the conformal group.One thing about mathbb{M}^c is that it is not possible to separate positive and negative frequency solutions locally. However, this is possible in twistor space.
mathbb{PT}^+ simeq SU(2,2)/left [ SU(2,1) imes U(1) ight]
ee also
*
Twistor space
*twistor string theory
*Invariance mechanics References
External links
* [http://www.twistordiagrams.org.uk/ Twistor Theory and the Twistor Programme]
* [http://mathworld.wolfram.com/Twistor.html MathWorld - Twistors]
* [http://users.ox.ac.uk/~tweb/00001/index.shtml Roger Penrose - "On the Origins of Twistor Theory"]
* [http://users.ox.ac.uk/~tweb/00002/index.shtml Roger Penrose - "The Central Programme of Twistor Theory"]
* [http://users.ox.ac.uk/~tweb/00003/index.shtml Richard Jozsa - "Applications of Sheaf Cohomology in Twistor Theory"]
* [http://users.ox.ac.uk/~tweb/00004/ Fedja Hadrovich - "Twistor primer"]
* [http://users.ox.ac.uk/~tweb/00006/index.shtml Roger Penrose and Fedja Hadrovich - "Twistor Theory"]
* [http://homepage.mac.com/stephen_huggett/Elements.pdf Stephen Huggett - "The Elements of Twistor Theory"]
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