Sunday, June 29, 2008

Is N=8 super-gravity finite and why this would be the case?

There is an interesting discussion in the blog of Peter Woit about the recent evidence for the finiteness of N=8 super-gravity and perhaps even N<8 super-gravities. I have nothing to say about horrible technicalities of these calculations and I can only admire from distance what the people involved are able to do. Nothing however prevents me to make some uneducated comments from TGD point of view.

Theoretical physicist as a thinker or as a computational virtuoso?

Certainly the cancelation of infinities would be a proof for the magic mathematical power of deep principles like General Coordinate Invariance, Equivalence Principle, and Super Symmetry. That Einstein with his not so fantastic calculational skills discovered the first two principles should make clear that it is conceptual thinking that matters most even in theoretical physics. The return to the study of N=8 super-gravity after 24 years of super-string models and the possible reduction of super string models to a mere auxiliary calculational tool would be a clear-cut answer to the question "Theoretical physics as the art of conceptualization or as mere computational methodology?".

What could be the symmetry behind cancellations of infinities in N=8 supergravities?

One question represented in discussions concerns the underlying symmetry causing the cancelation of the perturbative infinities.

  1. In the latest postings (see this, this,and this) I have told about how the notion symplectic quantum field theory emerges from TGD. These theories are highly analogous to conformal field theories. Fusion rules generalize and N-point functions are proportional to symplectic invariants assignable to the set of points defined by N arguments. The basic symplectic invariants are the symplectic areas of the geodesic triangles defined by 3-point subsets of the arguments of N-point function. Under very general assumptions these invariants vanish when two of N arguments co-incide because the areas of the triangles containing these two points vanish. The behavior of N-point functions is smoothed out so that the artificial UV cutoff is replaced with a dynamical one. Infrared cutoff comes in turn from the size of the causal diamond serving as imbedding space correlate for the zero energy state. One can thus get rid of local divergences as very general arguments based on the absence of local interaction vertices and Kähler geometry of the "world of classical worlds" indeed predicted long time ago.

  2. Very naively, the quantum field theory limit of TGD obtained by replacing the light-like 3-surfaces appearing as "lines" and 2-D partonic surfaces appearing as the vertices of generalized Feynman diagrams are taken to lines and points. The limiting theory could still have symplectic invariance as a hidden symmetry of N-point functions.

The obvious question is whether the N-point functions of N≤8 super-gravities have this kind of symplectic symmetry as a hidden symmetry? Note that N=4 super-conformal symmetry is the most natural candidate for the conformal super-symmetry of TGD and that TGD does not predict super-symmetry at the level of Poincare algebra so that no sparticles are predicted.

Why TGD is needed?

It is very difficult to understand how N=8 super-gravity could be consistent with standard model and my strong conviction is that one cannot avoid TGD if one is interested in physics.

  1. My belief is that it is the generalization of super-conformal invariance provided by the replacement of strings with light-like 3-surfaces as fundamental objects in H=M4×CP2 which is needed. The maximal conformal symmetries are obtained only for 4-D space-time and 4-D Minkowski space.

  2. "Number theoretical compactification" -or more precisely, the duality of pictures based on the identification of 8-D imbedding space as hyper-octonionic space-time HO= M8 or as H=M4×CP2 is absolutely essential for the recent formulation of quantum TGD and most "must-be-trues" of quantum TGD follow from the basic prerequisite of this duality. More precisely: space-time surfaces identified as hyper-quaternionic surfaces of HO contain preferred hyper-complex (commutative) plane HC=M2 defining the plane of non-physical polarizations in their hyper-quaternionic tangent space M4: this kind of tangent spaces are parametrized by CP2 which implies the duality.

  3. One implication of the existence of preferred plane M2 is justification for what I call generalized coset construction (see this) providing an elegant generalization of Einstein's equations replacing them with the condition that the super Virasoro algebras associated with the symplectic analog of Kac-Moody algebra and the ordinary Kac-Moody algebra defined by a symplectic sub-group of isometries of the imbedding space act in the same manner on physical states (the isometric Super Virasoro corresponds to the gravitational four-momentum and symplectic Super Virasoro to the inertial four-momentum). This provides the long sought-for explicit realization of Equivalence Principle in TGD framework.

I dare safely say that there is only a single possible choice of the imbedding space and this choice is consistent with the standard model. The geometry of classical spinor fields in the world of classical worlds and thus also physics is unique just from its mathematical existence, and physics as a generalized number theory vision allows to identify this unique and very special world of classical worlds (as does also the inspection of Particle Data Tables;-)). This motivates my humble suggestion for colleagues: why not to accept the facts and start to transform TGD to mathematical physics. A good place to start would be construction and classification of symplectic quantum field theories.

For a summary of quantum TGD see the article Topological Geometrodynamics: What Might Be the First Principles?.

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