Riemann Revived II
S duality in the guise of the Langlands program is a truly awe inspiring component of stringy triality.Recall that the complex number form of S duality has a modular group symmetry. This group appears all over the place in M theory. For example, we looked at the Banach-Tarski paradox in terms of a ternary tiling of hyperbolic space as a Poincare disc. This tiling marks the boundary of the circle with a nice triple pattern of accumulating points.
Alain Connes and Matilde Marcolli say that the Riemann zeta function is related to the problem of mass, which in turn we have seen is related to three stranded braids and M theory triality, of which S duality is a piece. It appears that no part of mathematics is left untouched by gravity.
Update: a new paper by Witten et al on 3D gravity, prominently featuring the modular group, has appeared on the arxiv. See the picture on page 49, and the J invariant on page 52. The paper shows that the partition function cannot be given a conventional Hilbert space interpretation. Holomorphic factorisation is suggested as a possible mechanism for extending the degrees of freedom. For instance, the complexified Einstein equations are considered. They say:
we think another possibility is that the non-perturbative framework of quantum gravity really involves a sum not over ordinary geometries in the usual sense, but over some more abstract structures that can be defined independently for holomorphic and antiholomorphic variables. Only when the two structures coincide can the result be interpreted in terms of a classical geometry.I confess to finding this statement a little ill-phrased, since some more abstract structure presumably does not begin with traditional complex analysis.
posted by Kea | 8:11 AM
5 Comments:
Funny. One of my heretical beliefs about QM is that Hilbert space is just a mathematical convenience; nature should instead be modeled at a fundamental level with a Banach space. The difference is that a Hilbert space has an inner product.
One can obtain a Hilbert space from a Banach space if the parallelogram law is satisfied.
Mathematically, it's natural that the parallelogram law should be satisfied in the limit of very small vectors. Similarly our quantum mechanical physics experiments are performed in the limit of very small energies and momenta compared to the Planck scale. But from that's it's hardly possible to conclude that nature satisfies the parallelogram law "all the way down".
Yes, those ubiquitous turtles can be annoying.
Indeed it is ill-phrased. Not having read the Witten paper yet, I would think that his statement about "more abstract structures" may be coming around to the structures that Kea has described. His talk shows that he has largely abandoned strings in favour of Representations.
This sum-over-geometries-dogmatics remains to me one of the strangest mammoth bone in theoretical physics landscape filled by ancient stuff. Sum-over-histories is nothing but an over-abstraction from Hamiltonian time evolution which has remained ill-defined even in single particle quantum mechanics with non-trivial interactions. The only justification was quantum classical correspondence working in non-perturbative situation.
Path integrals lived a high time in my youth and I of course tried also to construct quantum TGD using path integral although the failure was obvious from beginning.
Finally came the realization that the world of classical worlds must be geometrized and the idea that Einstein's geometrization program applies also to quantum theory. An idea which for some very strange reason is still waiting to be discovered by the mainstream. Also the geometrization of fermion statistics followed automatically via anti-communications of space gamma matrices and the connection with HFFs of type II_1 gives for the approach its real technical power. Despite the work of mathematicians like Connes, people simply refuse to realize the power of these algebras and try to understand their physical meaning.
The attempts to construct geometry of world for the classical worlds led also to the idea that Kahler function of world of classical assigns a unique space-time surface to the 3-surface: this just from 4-D general coordinate invariance for basic objects which are 3-D. Bohr orbitology becomes exact part of quantum theory. One could hardly dream of anything more fascinating form of quantum magic. Quantum classical correspondence in this new sense has been the strongest conceptual tool of TGD since then.
After having worked for two decades with this program and learned its amazing power as an idea generator, I cannot but just wonder how irrational a process the evolution of mainstream theoretical physics is. Some authority decides to follow the wrong path just because he does not of anything better and everyone follows. "We do not know of anything better!": isn't this the most often heard justification for string models! This kind of statements from the mouth of theoretical physicist should be criminalized;-)!
Hi Kea,
I am impressed by this Witten paper, particularly figure 2, page 13.
This reminds me of the virtual cylinder formed by the geodesic helical trajectory of a single electron around a single proton while both are in motion.
It may also represent a single planet around a single star.
If there were 8 concentric virtual cylinders around the "orbifold singularity", then this might represent our solar system [omitting dwarf planets].
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