Nimas 'new' talk (from 2.5.) with many new ideas. "You can identify everything in 3-D space" and he says infinities are important.

Spacetime, Quantum Mechanics and Scattering Amplitudes

sun 22.5. Matti has written a new chap. Motives and Infinite Primes.

Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory.

May 16, 2011: The Kinematic Algebra From the Self-Dual Sector.

Ricardo Monteiro and Donal O'Connell, arXiv:1105.2565

KITP Informal Discussion, So I must look what it could be.

We identify a di ffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed o ff-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, nding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.

Introduction: A deeper study reveals concrete relationships between gauge and gravity scattering amplitudes. Early in the first heyday of string theory, Kawai, Lewellen and Tye (KLT) derived a relation, valid in any number of dimensions, expressing any closed string tree amplitude in terms of a sum of products of two open string tree amplitudes [1]. Taking the field theory limit of the string amplitudes, these KLT relations imply in particular that any graviton scattering amplitude can be expressed in terms of a sum of products of colour ordered gluon scattering amplitudes.

A more insightful \squaring" relationship between gauge theory and gravity amplitudes was conjectured recently by Bern, Carrasco and Johansson (BCJ) [2], and then proven by Bjerrum-Bohr, Damgaard, and Vanhove [3] and by Stieberger [4]. These authors have shown that scattering amplitudes in gravity can be obtained from Yang-Mills scattering amplitudes, expressed in a suitable form, by a remarkably simple squaring procedure.

[1] H. Kawai, D. C. Lewellen, S. H. H. Tye, Nucl. Phys. B269, 1 (1986).

[2] Z. Bern, J. J. M. Carrasco, H. Johansson, Phys. Rev. D78 (2008) 085011. [arXiv:0805.3993 [hep-ph]].

[3] N. E. J. Bjerrum-Bohr, P. H. Damgaard, P. Vanhove, Phys. Rev. Lett. 103, 161602 (2009). [arXiv:0907.1425 [hep-th]].

[4] S. Stieberger, [arXiv:0907.2211 [hep-th]].

[5] T. Sondergaard, Nucl. Phys. B821, 417-430 (2009). [arXiv:0903.5453 [hep-th]].

[6] N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard and P. Vanhove, JHEP 1006, 003 (2010) [arXiv:1003.2403 [hep-th]].

KITP talks, many of them.

"Cubic Feynman diagrams appear quite naturally in the self-dual sectors." Fits nicely with Wilzcek too. In fact TGD has also these.

"On the gravitational side, we have seen how to compute MHV amplitudes from cubic diagrams in light-cone gauge" they say, up to a mild assumption about the behaviour of the higher terms in the gravitational light-cone Lagrangian. The fact that the BCJ relations hold (at least on-shell) for all graviton amplitudes suggests that there is some cubic theory which can be used to compute these amplitudes."

They don't say quaternion?

They talk of "the self-dual Yang-Mills (SDYM) equations in Minkowski spacetime". Is that ZEO?

Seems the fantastic castle is crushing down? All the many different dimensions were simply illusions?

Why not look at TGD? What is so disgusting with it?

## söndag 22 maj 2011

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Ulla,

SvaraRaderaInterestingly I followed a lead last night may 25 which did not get me very far and there was no better method than drawing things in the flat two dimensional case (with the understanding of course higher dimensions were logically represented).

So there was a posting not on this suggestion I had made earlier for it seems that beyond say fourspace we really do not gain much by viewing thing in multidimensional or natural dimensions of a matrix. It does raise the issue of what is clearly communitive or not when we take the roots of things.

Lubos has a post today on the dipole and shape of an electron. The may as well be toroidial in shape for what we can see and interpret. Anyway, I smashed an ellipse into a circle in a symbol on my post of kepler's second law- this is the case when we reduce things from the outside into ideal spheres or taking infinity into account as important we have the same old problem of viewing stretches where the congruence of such ellipses intersect in areas.

Thus, at some stretch of infinity as if seeing cosmic strings we are limited in our notions and perceptions to a universe centered on ourselves and it is not clear that either the big bang view is helpful or any sort of inflation model.

Such strings can only be understood and comprehensive of a more general space if we used the practical methods of Kea, and the numerical insights of TGD. Other than that I am not sure what else may be there save that I remain more advanced only from the standpoint of longtime familiarity with notions of the metaphysics and what I imagine from Aristotle's lost book on three space and the thought of everything in it, his Stereonometry.

The PeSla