What is a Twistor: Explaining Geometric Applications

[mnb96]

Hello,
I am familiar with Clifford algebras, the concept of spinor and the Hopf fibration of the S^3 sphere.
I know that all of these somehow relate to the concept of twistor.

Does anyone know how to explain in simple words what is a twistor and what it is useful for in geometry (please do not mention direct real-world applications in physics, let's just stick with pure geometry).

Thanks.

 

[arkajad]

There are several definitions. For instance you can define twistor as a vector of the natural representation space of SU(2,2). A more general definition is given by Crumeyrolle in "Ortogonal and Symplectic Clifford Algebras", Kluwer 1990. In Chapter 12.3 Twistors he has Definition 12.3.1:

Every irreducible representation for a Clifford algebra being called a spinor space, we will call twistor space all direct sums of spinor spaces.

Then he moves to direct sums of just two irreducible representations of the complexified Clifford algebra.

I think it is safe to say that "a twistor for CL(p,q) is nothing else but a spinor for Cl(p+1,q+1)".

 

[mnb96]

Thanks arkajad for your reply!
sorry for the trivial question: I understand the notations SU(n) and CL(p,q), but what is usually meant by SU(p,q)? Let's first clarify this point.

 

[arkajad]

Let G be nxn diagonal matrix G=(1,...,1,-1,...,-1) with p entries +1 and q entries -1, p+q=n.
Then SU(p,q) i the group of (complex) nxn matrices U such that U*GU=G and det(U)=1.

Sometimes G is defined differently, differing from the above diagonal form by similarity transformation.

 

[mnb96]

Ah! I see...
So basically G is used to mimic a non-euclidean metric.

Isn't this the same as considering an element of SU(2,2) as an http://books.google.fi/books?id=0Nj...CB4Q6AEwAg#v=onepage&q=outermorphism&f=false" f on CL(2,2) such that f(\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3\mathbf{e}_4) = det(f)\mathbf{e}_1 \mathbf{e}_2 \mathbf{e}_3 \mathbf{e}_4 = \mathbf{e}_1 \mathbf{e}_2 \mathbf{e}_3 \mathbf{e}_4[/tex] ?<br /> <br /> Back to the discussion of twistors, were you essentially saying that one definition of twistor is:<br /> ...an element of the <u>natural representation space</u> of SU(2,2)...<br /> <br /> Could you point out what definition you used for &quot;natural representation space&quot;?<br /> Thanks a lot!

 

[arkajad]

1. Probably whe you write CL(2,2) - you mean real Clifford algebra. Element of SU(2,2) are complex matrices.

Natural representation is the representation on C^4 endowed with a pseudo-hermitian form of signature (2,2):

\langle Z,Z&#039;\rangle ={\bar{Z}}_1 Z_1+{\bar{Z}}_2 Z_2-{\bar{Z}}_3 Z_3-{\bar{Z}}_4 Z_4

with the standard matrix action. Then U\in U(2,2) is equivalent to \langle UZ,UZ&#039;\rangle=\langle Z,Z&#039;\rangle

 

[mnb96]

Thanks arkajad for your help!
I see the mistake...SU(p,q) are unitary matrices, hence they have complex entries, so my poor attempt of trying to understand the intuition behind the definition of twistors was vain.

It seems there is some "higher knowledge" out of my range involved in this, because I can't yet see how twistors relate to other simple concepts (spinors, Hopf fibrations...)

*** EDIT: Perhaps I found an useful http://arxiv.org/abs/math-ph/0603037" written in a formalism I am more familiar with. Let's see if I can get something out of it.
It looks promising, because the authors claim to be able to represent twistors as "4-d spinors with a position dependence", which to me sounds a bit more accessible than the other introductions I found on the net.

 

[arkajad]

In my opinion the article is unnecessarily complicated. It could have been all said on two pages at most. Anyway - if you will have questions or problems - I will try to help you as much as I can.

 

[mnb96]

Yes, I just realized that that article seems to be an extended version of another paper. It indeed contains a long introduction on some well-known concepts of Clifford algebra.

In the next days (or weeks) I will go more carefully through that paper and I will probably resume this post as soon as I find something difficult.

By the way, if you say that article is unnecessarily long, I suppose you have already spotted the most important sections that contain the heart of the matter. This might be easy for you, but difficult for me, since I don't know the topic.

Could you please tell me which are the sections which in your opinion are the most critical?

Thanks again for your endless effort to help people.

 

[arkajad]

I do not know which sections are critical. The group SU(2,2) is mentioned only once. You can try to read the paper, but I don't think you will learn from it what are twistors. So just read what is new to you and fits you needs. After that you will probably have to look for a different paper. I suggest, just for change, to have some other perspective, get and read this paper (replace xx with tt in the URL):

hxxp://www.fuw.edu.pl/~slworono/Twistory.html

 

[mnb96]

thanks!
I downloaded the article you mentioned.
I will try first to get something useful out from that paper I found, and if everything goes well I will switch to that source you gave me.
I'll eventually let you know how things go.