Friday, March 25, 2005

Dualities as wave-particle dualities in infinite-dimensional context?

I finally got the opportunity to start musing again about TGD dualities and to see what new understanding has emerged during the last two weeks.

1. HO-H duality as p-q duality in the configuration space of 3-surfaces?

HO-H duality means that space-times can be regarded either as surfaces of H=M4xCP_2 or of its tangent space HO=M^8. This reminds strongly of co-tangent bundle and the familiar p-q (wave particle) duality of quantum mechanics. Indeed, in the geometric quantization you select Lagrange manifold which means assigning canonical momenta p(q) to position coordinates q. This choice corresponds in very general sense to Bohr quantization although it is not unique without further conditions coming in practice from symmetries. The idea is obvious. p-q duality in the infinite-dimensional context would mean following. a) In q-represention you represent the quantum theory using configuration space spinor fields in the space defined by the positions in the configuration space CH of 3-surfaces X^3 in H=M^4xCP_2. Each X^3 corresponds to a four-surface X^4(X^3) ("Bohr orbit going through X^3"). These surfaces are unique apart from physically very important complications caused by classical non-determinism, which can be overcome by appropriately generalizing X^3. That is allowing X^3 to consist of sequence of space-like disjoint components with time like separations, a snapshot of space-like disjoint components with time like separations, a snapshot about non deterministic time evolution: this would be the first guess. b) In p-representations related to q-representation by generalized Fourier transform you represent the quantum theory in the fiber of cotangent bundle of CH. What is the new and marvellous element that the canonical momenta are represented as 4-surfaces in the tangent space of HO=M^8, tangent space of H! Duality means Bohr orbit representation assigning to 4-surface in H a 4-surface in HO. Here the power of geometric thinking becomes obvious. Armed with this generalization of q-p duality you can go through the various aspects of HO-H duality, make it much more precise, and derive a lot of highly non-trivial looking new predictions. One of the most beautiful implications of strong form of this duality would be that the 4-surfaces in CH representing positions q and the 4-surfaces CHO representing canonical momenta p are set-theoretically identical! CH 4-surface has the metric, Kähler structure, and Kähler action induced from H and CHO 4-surface has all this induced from HO! Duality is indeed in question! This picture generalizes also to the level of HQ-coHQ duality stating that the theory can be represented either in terms of hyper-quaternionic 4-surfaces or co- hyper-quaternionic 4-surfaces of H (HO). Now cotangent bundle is replaced by the 4+4-dimensional normal bundle and duality of course makes only for 4-D surfaces in 8-D imbedding space.

What about M-theory dualities?

The generalization of configuration space of 3-surfaces to that of configuration space of 1-surfaces with Kähler metric (which need not exist mathematically except in very rare cases, sorry!) allows also a deeper understanding of spontaneous compactification and the dualities of M-theory. All spontaneous compactifications, that is theories in given target space would have same cotangent space, whose points are representable as strings in flat target space. The non-compactified theory corresponds to the perturbative theory which is always the same whereas compactifications correspond to non-perturbative theories, which are the "real" theories. The duality mapping strings in two representations to each other would be analogous to Bohr quantization assigning to a given position q canonical momentum p(q), and Lagrange manifolds of cotangent bundle of compactified theory would determine a huge variety of different dualities unless there are some constraints from symmetries. Probably these constraints are very important, at least in TGD. Of course, this formal picture need not make sense since the existence of infinite-dimensional Kähler geometry poses so strong constraints that the notion of dynamical target space must be given up. For instance, the Kähler geometry of loop groups is unique and has Kac-Moody algebras as symmetries. In the more general case this geometry need simply not exist at all. This would of course mean getting rid of the landscape problem. Unfortunately, it would very probably mean that the physics predicted by string models and M-theory does not have much to do with what we see in laboratory. During these 16 years of configuration space geometry in TGD framework I have been wondering why string theorists refuse to take the notion of infinite-dimensional geometry seriously and why they have spent a lot of time with matrix models and alike. It might be a good idea to see whether the theory allows a formulation in terms of Kähler geometry of the configuration space of strings and what the existence of this formulation predicts. My strong conviction is that this would finally pave the way to TGD and we could start doing physics again. For the first draft of these ideas see the last section of TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts Matti Pitkänen


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