Sunday, January 30, 2011

Motives, Twistors and Amplitudes













Nima Arkani-Hamed gave a recent talk on 01/26 entitled "Space-Time, Quantum Mechanics and Scattering Amplitudes". He essentially covers all the recent progress in the study of scattering amplitudes in dual twistor variables. He ends with hints at an underlying theory that gives rise to AdS/CFT and QFT, which might be based on the mathematical theory of motives.

For those unfamiliar the theory of motives, the goal within the mathematical community is to define a unified cohomology theory, from which all others (de Rham, Čech, singular, etc.) are special cases. It is interesting that a unified theory of physics would coincide with this platonic goal of mathematicians. Perhaps Edward Witten foresaw such a convergence and the 'M' of M-theory stood for motive all along. Either way, category theorists saw this coming a few years ago.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

— Albert Einstein

6 comments:

Kea said...

Oh, I recognise that window from Waterloo! And I said that the M should stand for Motives some years ago now ... sigh.

kneemo said...

Yes, of course, who could forget 'Monday Motives' back in 2007?

http://kea-monad.blogspot.com/2007/03/monday-motives.html

Kea said...

Of course, with my health deteriorating due to long term stress, isolation and poverty, no one need bother giving me any credit.

CarlBrannen said...

I'm enjoying the talk. Not yet over my head. The computation <24>^4/<12><23><34><45><56> I'm finding interesting because this is in the form of a Berry-Pancharatnam phase. That is, you can write the denominator as the trace of a product of pure density matrices and so does not depend on complex phase choices. Not so with the <24>^4.

So one of the things I'm trying to improve in myself is understanding of BP phase.

CarlBrannen said...

One of the most interesting parts of the lecture was how he used the Pauli matrices to encapsulate spin-1 photons.

The result was that twistors are 2x2 matrices with 0 determinant. Of course pure density matrices have zero determinant so this seems like its related to the way I prefer to work on things. I think generalizing should not be that hard. Course I haven't sunk time in it yet...

And of course I liked the commentary on the evilness of gauge symmetry.

kneemo said...

Yup, the mapping to 2x2 Hermitian matrices with zero determinant is just a mapping to rank one matrices in a Jordan algebra, i.e., a projective line. For 2x2 Hermitian matrices over the complex field, the determinant gives the norm squared of a vector in (3,1)-spacetime. Hence, rank one 2x2 Hermitian matrices describe lightlike objects. This sets up a correspondence between the points of the projective line CP^1~S^2 and rays on the lightcone. The corresponding Hopf fibration is S^3->S^2 with fiber S^1. Penrose knows this well.

To generalize to twistors in C^4, Donaldson and many others find it easier to represent twistors as elements of H^2 and map to rank one 2x2 Hermitian matrices over the quaternions. This maps twistors to the the projective line HP^1~S^4. This case corresponds to the Hopf map S^7->S^4 with fiber S^3. I suspect the scattering amplitudes can also be computed in this quaternionic formalism.