Bert Schroer Says:I added the bold face at the end. My comment is here.October 8th, 2006 at 4:15 pm
I repeat, there is the maldacena conjecture about a relation between a duality relation between a N–> infinity gauge theory and some form of 5-dim. gravity and there is a mathematical theorem about a AdS–CFT holography (or better correspondenc) and both have, according to our best knowledge nothing to do with each other. From a conceptual point of view the correspondence is rather trivial because the same substrate of matter is only changing its spacetime encoding (for more details see may essay) and olthough the physical interpretation changes it does not change miraculously (e.g. a spin 2 particle must have been there already on the CFT side). If the more than 400 people who worked on this problem would in addition to their computations have leaned back a while and looked at the published theorem may be we would have known by now what the Maldacena conjecture really mean conceptually. But I would predict that nobody at this late time will do this, the right time has passed. Certainly I would not loose time on such a physically fruitless project, but on the other hand as mathematical physicist I find this change of spacetime encoding in holographic projections very interesting; most interesting if the smaller spacetime is not a brane but rather a null-surface (see my last section and the cited literature).
Some comments to Bert Schroer about null surfaces, or lightlike surfaces, as I have used to call them.3-D lightlike surfaces in 4-D space-time, which itself is a surface in 8-D space-time H=M4×CP2, can be seen as fundamental quantum dynamical objects in Topological Geometrdynamics. They are identified as parton orbits. The effective metric 2-dimensionality with ensuing super-conformal symmetries makes D=4 as a space-time dimension unique.
The infinitesimal transformations respecting null surface property form a Kac-Moody type algebra of conformal transformations of H localized with respect to X3 decomposing to representations of 1-D Kac Moody algebra.
The cones H+/-= δ M4+/-×CP2 are also crucial for the formulation of theory and the 3-D lightlikeness of δ M4+/- makes possible super-conformal symmetries of new kind based on canonical algebra of H+/- and its super-counterpart. General Coordinate Invariance predicts quantum holography at level of H+/-apart from effects implied by the failure of the complete classical determinism of the classical theory.
The resulting theory at fundamental parton is almost (absolutely important physically!) topological CFT defined by Chern-Simons action for Kähler gauge potential of CP2 projected to X3. The second quantized fermionic counterpart of C-S action is fixed by the requirement of super-conformal symmetry. The theory allows N=4 super-conformal symmetries of various kinds broken for lightlike 3-surfaces which are not extremals of C-S action (have CP2 projection with dimension D>3). No space-time (Poincare) super-symmetries and thus no sparticles are predicted.
Super-symmetrization of super-canonical algebra is possible for a sub-algebra of superconformal symmetries for which Noether charges defined as 2-D integrals over partonic 2-surface reduces to 1-D integrals as duals of closed 2-forms. The (super-)Hamiltonians of this sub-algebra have vanishing spin and color quantum numbers and thus leave invariant the choice of various quantization axis.
The vertices of the theory are described by almost topological having stringy character. Correlations between partons (propagators) involve interior dynamics determined by a vacuum functional defined as a determinant of the Dirac operator and assumed to reduce to an exponent of Kähler action for absolute extrema playing the role of Bohr orbits for particles identified as 3-surfaces: this would be quantum holography at the level of space-time surface. Interior dynamics of space-time surface codes for non-quantum fluctuating classical observables allowing to realize quantum measurement theory at fundamental level. There would be thus a direct connection between quantum holography and quantum measurement theory.
The mathematical methods of string theory can be applied to TGD and one can see the target space of string theories as a fictive concept associated with the vertex operator construction assigning to the Cartan algebra of Kac-Moody algebra a target space. In TGD framework spontaneous compactification can be seen only as an ad hoc attempt to give physical content to the theory.
For more details see my blog and homepage, in particular What's New sections to get view about the recent rapidly evolving situation in TGD.
With Best Regards,
Matti Pitkanen
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