Friday, September 17, 2010

Exact Yangian symmetry, non-trivial scattering amplitudes, no IR singularities: only a dream?

I have work hardly to understand in more detail the formulation of the scattering amplidudes in terms of Yangian invariants defined by Grasmannian integrals from the recent article of Nima Arkani-Hamed and collaborators and its predecessor The S-matrix in Twistor Space.

The exact super Yangian invariance would be extremely attractive constraint on the theory and I proposed in the earlier posting a generalization of this symmetry to TGD framework obtained by replacing finite-dimensional conformal algebra with various infinite-dimensional super-conformal appearing in TGD. It seems that in momentum degrees of freedom this symmetry gives the super-conformal Yangian symmetry of N=4 SYM so that an appropriate generalization of the Grassmannian approach should work also in TGD context.

The problem is that this symmetry applied to scattering amplitudes allows only the trivial Yangian invariant which is constant (this does not however mean physical triviality, only the triviality of loop contributions). The argument demonstrating this is very simple and can be found at the end of the recent article by Nima and others. Non-triviality of the scattering amplitudes is due to the infrared singularities spoiling the Yangian invariance. The basic idea of the approach of Nima and others was indeed the idea that the leading IR singularities code for the scattering amplitudes. As a romantic soul I cannot avoid the feeling that something is wrong in QFT approach. Note that the IR singularities of the scattering amplitudes are also physically problematic and one must develop a procedure for eliminating them. This however leads difficulties with Yangian aesthetics.

The question is whether one can circumvent this objection in TGD framework, where the physical particles appearing as incoming and outgoing states of particle reactions are bound states of fundamental massless fermions assigned with ligh-like wormhole throats whereas virtual particles are massive on mass shell particles with both positive and negative energies so that loop integrals reduce to sums subject to very powerful constraints from on mass shell property (zero energy ontology). The first good news is that the study of the simple special case shows that at least in this case the algebraic discretization of virtual masses and even of virtual four-momenta suggested by the p-adicization (number theoretical universality) and modified Dirac equation does not spoil the applicability of the Grassmannian approach involving residue integrals along complex contours: a discrete version of loop integral in momentum space is obtained. The number theoretical beauty of the residue integrals is that they make sense also p-adically unlike Riemannian integral.

It might be possible to achieve Yangian invariance and non-trivial scattering amplitudes in TGD framework and the bound state masses bring in also the natural infared cutoff as a simple modification of the killer argument of Nima and others to a less lethal form implies.

The basic point is that the action of super-generators on bound states with constraints between momentum components and thus between components of components of super-twistors (in particular their super parts) associated with different wormhole throats carrying fermion quantum numbers can annihilate the amplitude without it being constant. The non-locality of the bound states is in accordance with the non-locality of Yangian symmetry and bound state mass scale brings in naturally a physical IR cutoff.

This argument and a vision about how the Grassmannian integral approach might generalize to TGD framework can be found in the article What could be the generalization of Yangian symmetry of N=4 SUSY in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

The reader interested in other articles written during this year about quantum TGD can find them here.


donkerheid said...

This sounds very elegant. I wonder if it can be learnt by ordinary humans :).

Matti Pitkänen said...

Not easy but I hope possible;-). In this optimistic mood I am making attempts to learn at least some of the background at deeper level and I feel myself very very stupid most of the time. There are so many parallel developments and the mathematics is so sophisticated at the technical level. The outcome, Yangian symmetry, is however something crystal clear, allows an obvious generalization in TGD framework.

donkerheid said...

Isn't the last sentence enough? :))

Matti Pitkänen said...

Better to adopt diplomatic attitude and leave some backdoors open;-). I do also my best to avoid a role of guru.

donkerheid said...

:D I see.

Ulla said...

You know I hate gurus :) I want to think myself.

I did not try to make you a guru, then there would be no need for falsifications :) I did my best to explain things in other ways, as you know. You was angry also at that.

I just tried to give you the honour you deserved. Show the human. Was it too much? I can do a new trial.

Ulla said...

To your knowledge, (light cone):
We present a nonlinear realization of E_8 on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined ``light cone'' in 57 dimensions. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times...