# Wavefunction in twistor theory

#### dimension10

I was reading Penrose's 1968 paper on twistor theory and it seems interesting.

Even upon reading around half the paper, I did not find the answer to one of the questions that came to my mind while reading the first few pages.

Somewhere in the first few pages (Section 0, 1 and 2), it was written that (from my memory)

"This should not be taken as simply making points on spacetime discrete (although this has already been attempted) but rather to smear the discrete points out, in a manner similar to the one found in Quantum theory for several state variables."

So, I presume there must be some wavefunction describing the points on spacetime. I am guessing something like

$$\Psi=\cos(2\pi x_0)\cos(2\pi x_1)\cos(2\pi x_2)\cos(2\pi x_3)$$

in 4 dimensions using natural units:

$$\ell_p=ic_0t_p=1$$

Am I right or is the wavefunction something else? Or is this smearing out described without using a wavefunction.

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#### mitchell porter

Penrose's transform maps space-time to twistor space, and it also maps wavefunctions on space-time to wavefunctions on twistor space. Maybe it's the idea of functions on twistor space itself that you're missing?

#### dimension10

Penrose's transform maps space-time to twistor space, and it also maps wavefunctions on space-time to wavefunctions on twistor space. Maybe it's the idea of functions on twistor space itself that you're missing?
Thanks!

So can the Penrose's transform map anything on ordinary spacetime to twistor spacetime?

Thanks again!

#### dimension10

Also, how do you actually take the penrose transform of a function?. Many sources say that it is the complex analog of the radon transform but I don't see any way to extend the radon transform to a complex analog.

#### sheaf

Thanks!

So can the Penrose's transform map anything on ordinary spacetime to twistor spacetime?

Thanks again!
Firstly, a minor point: it's "twistor space" rather than "twistor spacetime". Twistor space isn't a spacetime, but rather a complex manifold constructed from objects which live in spacetime.

There isn't a transform for *anything* between spacetime and twistor space. An example of what you *can* transform to twistor space is the set of solutions of the zero rest mass field equations (e.g. Maxwell equations). Think of twistor space as a space whose points represent null rays in Minkowski space (It's a bit more general than that, however, the full twistor space actually corresponds to the space of alpha planes in complexified Minkowski space). Take a point in Minkowski space - you can think of this as being defined by the set of all null rays through that point (i.e. the null rays which make up the light cone of the point)- there is a two-sphere's worth of such things. Hence a point in Minkowski space is represented by a subset of twistor space which is topologically a two sphere.

In fact it's more than that - it has the structure of a projective line. Now the magic of the Penrose transform is that if I take a certain type of function on twistor space, then I can construct a solution of the zero rest mass equations on spacetime as follows: to find the value of the field at a point x in Minkowski space, take a function on twistor space, restrict it to the projective line Lx corresponding to x, and perform a contour integral over a contour on Lx. The function must have a singularity structure on twistor space such that the intersection of Lx with the singular points is such that the contour can't be deformed to zero without hitting the singularities. If you do this then (1) the field you get on Minkowski space is a solution of the appropriate zero rest mass equations (the helicity of the solution depends on the homogeneity of the function on twistor space you took). (2) the field is independent of the choice of contour.

It gets even more involved - you can have many functions on twistor space representing the same field on Minkowski space. In fact more rigorously, the function classes are really elements of sheaf cohomology <cough> groups on twistor space.

There are generalizations of this correspondence:
* to curved spacetimes (self dual and anti self dual solutions of Einstein's equations) and deformed twistor spaces,
* to spacetimes with Euclidean signature, where instanton solutions of the Yang Mills equations can be generated from twistor structures,
* to scattering amplitudes on Minkowski space etc.

Twistor inspired methods have seen a bit of a recent resurgence, mainly in the supersymmetric context, with Witten's twistor string theory and the N=4 SYM scattering amplitude work of Arkani-Hamed et al.

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