Posted: Oct 23, 2012 7:29 am
Post 14 More Scattering

In the last post, I described the situation depicted in this diagram

twistorScatt.jpg (27.29 KiB) Viewed 2416 times

namely, that in a tree-level MHV scattering process, the gluons are represented in twistor space by points which must lie on a line (by which we mean a complex projective line, which has one complex = 2 real dimensions, and the topology of a sphere! Welcome to algebraic geometry).

What we are aiming to do is to give the twistor picture of the usual Feynman diagram-style treatment of scattering. Here the incoming particles are represented by a set of wavefunctions, as are the outgoing particles, and the scattering amplitude is given by sandwiching a scattering operator between the in-state and the out-state.

In post #4 I gave the twistor picture of the solutions of the free Maxwell equations. In twistor space, they corresponded to certain functions which
1) Had certain homogeneity properties
where
n = 0 for helicity +1
n=-4 for helicity -1

2) Had certain singularity properties, such as poles

There are certain freedoms in the choice of function used to represent a Maxwell field - you can add various local functions on twistor space and not change the result. The rigorous way to phrase this correspondence, conventional in twistor circles, is to use the formalism of Čech sheaf cohomology, but that's quite involved to describe. There is a rigorous isomorphism between Čech and Dolbeault cohomology.

There is an easier way, which is slightly(!) easier to understand, and will be easier to apply to our scattering problems, namely that the fields correspond to Dolbeault cohmology classes. The definition of Dolbeault cohomolgy is given in the appendix. Although we gave the example of Maxwell fields, the correspondence with twistor space Čech/Dolbeault cohomology classes applies to all zero-rest-mass fields, regardless of their spin, so we'll keep it general.

Twistor Wavefunctions as Dolbeault Cohomology Classes

A helicity +h solution of the zero rest mass field equations satisfies

and a helicity -h solution satisfies

So, in Dolbeault terms, the correspondence, for positive helicity h, is

where

i.e. the first Dolbeault cohomology group (see the appendix for a description).

For negative helicity the correspondence is

The means just that we're dealing with forms which are homogenous of degree n in the twistor coordinates i.e. .

The Dolbeault correspondence is made explicit by the integral formulas:
For the helicity h solution:

and for the helicity -h solution:

The integrals are over the two plane X (projective line Lx when we go to projective twistor space) in twistor space corresponding to the space-time point x. Note that these are distinct from the integral formulas in post #4 – these were integrals of functions holomorphic on certain subsets of twistor space, whereas the integrals here are of smooth one-forms (wedged up to make two-forms). More rigorously stated, the integral formulas in post 4 are realizations of Čech cohomology, whereas the formulas here are realizations of Dolbeault cohomology. These concepts are properly described using the formalism of sheaves (which I’m not going to do here since this is just a sketch). The Čech and Dolbeault descriptions are identical (Čech-Dolbeault isomorphism).

Homogeneities of Scattering Amplitudes

In theory, a scattering amplitude should be a function of the helicities, momenta and polarization vectors of the n incoming gluons:

However, the momenta and polarization vectors are fully encoded in the spinors . The polarization vectors, written in terms of spinors, are things like

Since the scattering amplitudes are linear in the polarization vectors, they must obey relations

Then, in performing the transformation to twistor space using the “half Fourier transform” outlined in the previous post, if we just make the naïve replacements

then the condition on the scattering amplitudes becomes:

The label i labels the gluons. Then after the transform to twistor coordinates, the scattering amplitude is now a function on several copies of twistor space, each copy labelled with an i. Since is just the twistor homogeneity operator , this tells us that the scattering amplitude is homogeneous of degree in the twistor coordinates .

Scattering Amplitudes as Dolbeault Cohomology Classes

Now in the last post, I showed that a gluon scattering amplitude could be transformed to twistor space by a Fourier integral in one of the spinor variables:

There was a bit of a gloss-over there, namely an assumption that the spinors are real in order to perform an ordinary Fourier transform. This is the case with spacetime signature ++--. For other signatures, strictly speaking you would have, instead, to do a contour integral.
Anyway, the integrand is a two-form in the lambda spinor variable which is closed, i.e. maps to zero under the application of the operator. It is thus a representative of a Dolbeault cohomology class which lives in . The H2 means we’re dealing with a second cohomology class, which in Dolbeault terms means two forms (see appendix below) on (some subset of . The means that we’re dealing with functions homogeneous of degree (-2h-2) in the coordinates on non projective twistor space . is more accurately described as the sheaf of germs of holomorphic functions, homogeneous of degree (-2h-2).

Twistor Interpretation of Scattering of External particles

In momentum space, we have an n-particle scattering amplitude that’s a function of the momenta

However, in general, the incoming particles won’t be momentum eigenstates, but rather will be described by wavefunctions which (in momentum space) are n functions of the momenta . These correspond to n solutions of the zero rest mass equations. Scattering of particles in these states is given by an amplitude (where is the momentum space version of the wavefunction ).

In twistor terms, the scattering particles are represented by elements of the cohomology groups (+ve helicity), (-ve helicity)and, as we’ve discovered, the momentum space scattering amplitudes are also represented by cohomology elements, namely of (+ve helicity), resp of (-ve helicity).
So if we have n incoming gluons with helicities , each gluon will pair up an element of with an element of of . Combination of such entities is via the cup product. If, in the Dolbeault case, we’re representing the cohomology classes by differential forms, this is just the wedge product, and we end up with a 3-form. This is a good thing, because the subset U of projective twistor space is 3 (complex) dimensional, so we can integrate it. (The homogeneity (-4) is just right because the three form will be a function multiplied by the twistor volume form

which is homogeneous of degree 4.
With a careful handling of indices etc. all these considerations transfer over to supertwistor space, and gluons of theory.

Appendix: Dolbeault Cohomology

Back in post #7 I briefly described the concept of differential forms. These were built out of antisymmetric tensor products of dual vectors. You can extend the concept of differential forms to complex manifolds.

A function on a complex manifold is written as where is the dimension of the manifold and are local coordinates. We need both and coordinates to describe general functions - for example suppose I wanted where , then *must* make an appearance because .

So when we consider differential forms on a complex manifold M, they will have both and terms. For example the one-form obtained by differentiating a function is

In general we can have differential forms with different numbers of barred and unbarred indices, e.g. a (p,q) form

The space of such objects is denoted . The usual exterior derivative has holomorphic and antiholomorphic parts

given by

given by

Concentrating on the 's, if I take a (p,q) form and it, and then the result, I always get zero. Piss easy to see. When things have this property, mathematicians feel the irresistable urge to draw a little diagram showing the spaces which the mappings operate between:

Then, because they're so rock'n roll, they say, "let's look at the elements of which maps to zero, and let's consider two of them to be the same if the difference between them is the image of some element of ". The set of such objects has a group structure under addition and is called the Dolbeault cohomology group . If we're talking about differential form *fields* on a complex manifold M, we denote the Dolbeault cohomology group by .

And that, ladies and gentlemen, is why mathematicians have NO TROUBLE in getting laid.