335. What is twistor

Good Saturday evening, True Geeks!

Since everybody currently seems to be a bit crazy about twistors – see for example, the Witten’s paper “Perturbative gauge theory as a string theory in twistor space” and the buzz it started – I decided that the time has come for me to learn what it is and write minireview post about it.

So, what is twistor? Twistor is a straight line in the complex projective space , which is use to realize 4-dimensional Minkowski space, correspondingly, twistor space is a linear space of all such lines. Twistors were first introduced by Roger Penrose (on the left) in the late 1960s.

The set of all twistors – lines in the complex projective space – depends on 4 complex parameters. Minkowski space is realized if we choose a subset in the twistor space that depends on 4 real parameters. The idea to use complex geometry in order to work with real spacetime is highly non-trivial and powerful as we will see. The (twistor) space of all lines in can be interpreted as a complexified and conformally compactified Minkowski spacetime. If you consider two causally connected events (for example, connected by a ray of light) in Minkowski space, they correspond to lines in the twistor space which are crossing each other. Euclidean 4-dimensional space can also be naturally realized as a subset of twistor space, so Wick rotation is a very natural operation in the twistor language.

Congruence of null lines

The fundamental idea of Penrose is that the primary physical structure is not 4-dimensional Minkowski space, but complex twistor 3-dimensional space: twistor equivalents of various physical quantities should be described easier than the 4-dimensional quantities defined in Minkowski space. According to Penrose, many physical field equations just follow from the analyticity conditions in the twistor space. After you deal with a quantity in the twistor space, you reduce it to its equivalent in Minkowski space by an integration over twistors. In other words, you introduce some kind of integral transformation from the complex 3-dimensional twistor space to the real 4-dimensional Minkowski space.

This program is easily realized in the simplest case of free massless fields of various spins: scalar field, spinor, vector field and linearized Einstein equations. These massless fields propagating in 4-dimensional flat spacetime correspond to solutions of Cauchy-Riemann conditions in the twistor space. This fact is not of a physical interest, though, since what you really get is new representations of solutions of linear differential equations.

Next time I am going to discuss a bit how twistors help dealing with solutions of non-linear differential equations and present some examples.