Sunday, September 11, 2011

Could TGD be an integrable theory?

During years evidence supporting the idea that TGD could be an integrable theory in some sense has accumulated. The challenge is to show that various ideas about what integrability means form pieces of a bigger coherent picture. Of course, some of the ideas are doomed to be only partially correct or simply wrong. Since it is not possible to know beforehand what ideas are wrong and what are right the situation is very much like in experimental physics and it is easy to claim (and has been and will be claimed) that all this argumentation is useless speculation. This is the price that must be paid for the luxury of genuine thinking.

Integrable theories allow to solve nonlinear classical dynamics in terms of scattering data for a linear system. In TGD framework this translates to quantum classical correspondence. The solutions of modified Dirac equation define the scattering data. The conjecture is that octonionic real-analyticity with space-time surfaces identified as surfaces for which the imaginary part of the biquaternion representing the octonion vanishes solves the field equations. This conjecture generalizes the conformal invariance to its octonionic analog. If this conjecture is correct, the scattering data should define a real analytic function whose octonionic extension defines the space-time surface as a surface for which its imaginary part in the representation as bi-quaternion vanishes. There are excellent hopes about this thanks to the reduction of the modified Dirac equation to geometric optics.

I do not bother to type 10 pages of text here but refer to the article An attempt to understand preferred extremals of Kähler action and to the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts of "Physics as Generalized Number Theory".


L. Edgar Otto said...


Of course (even in the back of my mind I can imagine an ultimate chaos and indeterminacy beyond such statistics if these are solved.

Clearly, the fractal filling of planes would be differentiable too in such a setting as you consider (so in what sense are these terms to be compared and generalized?)

In my post today I use the lattice of Coxeter as planes or membranes and these form a cycle of five (or ten) as the duality projects from one level to another. These are very hard thoughts to grasp and keep a model in my head and digest it and evaluate it. But all is flatness here.

So, beyond Hamiltonian and phase space and so on, is there some restrictions (none of these things vanish by the way to hide the paradoxes of nature) on your idea of membranes and can your wormholes loop as if a three and greater model of Riemanns excluding the zero and negative in one axis while it is still a soloton of motion in itself or out to the world.

So the question I asked of space expansion relates to the one of these other notions in the topic context of my last comments here.

iota is a ray like idea, a scaless omnium if we want to connect it to cosmology. In my notation ironically I used Greek iota and nu (and chi eventually) as roman numbers for the cube notation. Vaguely I imagine (as the conformal idea of complex numbers involved I certainly have assumed without the questioning as some like Rowlands suggest cautiously) is that of four of the eight cubes- the electric and magnetic flow of things.

ThePeSla said...

Sorry for not responding. I spent a week in hospital and am gradually returning to work.

Ulla said...

I thought you were on a sunny beach in a better weather. Nothing serious, I hope, not the first warning? I become worried.

Entanglement and the Higgs mechanism?

L. Edgar Otto said...


That was a great pdf article from 2010. It begins to see the higher picture beyond quantum formalism in the way we can understand information and the process proposed of the changes locally or globally in such information as qubits etc. In fact the issues are part of the general questions of Penrose's quantanglement and Majorana information transmission as well as Bell et al.

If Matti is proposing the higher membranes as also like orbitals I think this is right on and compliments my more detailed speculations.

I can see why in a vague intuitive way one can imagine the teleportation like DNA imprinting on say water- and also why TGD will answer some of our questions some have not yet decided to ask.

One thing for sure it is a little naieve to just apply string like or wormhole like or dark matter like theories now conceived to the deeper functioning of matter and life. It requires a more unified theory but is a good start.

Matti should also consider more than the manifolds but at least of all three curvatures as to what is the nature of the physical and finite.

Interesting how many of his and my formula values show up- as if these authors are almost at the end of the quest. Still, we need such work to light the way.

The PeSla