Wednesday, December 05, 2007

Witten and 3-D quantum gravity

Witten has written together Alex Maloney a new paper about 3-D quantum gravity. They calculate the partition function of pure gravity on an AdS3 space by summing the contributions from classical geometries, including quantum corrections, finding that “the result is not physically sensible”. They propose as a possible cure a sum over complex geometries or sum over some new kind of objects.

In a posting to Kea's blog I considered the basic problems resulting from sticking to sum over histories dogma and discussed TGD approach as an alternative. Just to enjoy the experience of becoming quoted in italics by a theoretician whose ideas I take very seriously, I glue my comments below before continuing.

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;-)!

After this little bit sour self-quotation in the style of an older statesman who feels of having not quite received the attention he deserves, I continue in a similar self-referential tone by recalling my old postings commenting Witten's ideas. See this and this.

TGD indeed has strong resemblances with Witten's theory. The basic difference is that 3-geometries are replaced by light-like 3-surfaces. They are solutions of vacuum Einstein's equations in a generalized sense. Cosmological constant is however vanishing whereas Witten's geometries have negative cosmological constant which might be responsible for unphysical nature of the resulting theory: conformal field theory should not contain explicit length parameters.

The metric 2-dimensionality of the basic objects implies generalized conformal invariance and almost topological QFT property. Chern-Simons action and its fermionic counterpart are dictated by super-conformal symmetry as a definition of basic dynamics. Light-likeness is a metric notion and makes topological QFT almost topological so that one can speak about masses, etc...

Again I find myself stumbling with these mammoth bones. Colleagues refuse to take seriously a technically completely trivial but physically extremely profound generalization of 2-D conformal invariance although it explains space-time dimension among other things. Perhaps this is about insulted egos: perhaps it is too painful to admit that something went badly wrong for 24 years ago.

A further bonus is that 4-D quantum gravity emerges via quantum classical correspondence as a (not 1-1) mapping of quantum physics at 3-D lightlike surfaces to the classical physics in the interior of space-time sheets. This replaces gravitational holography in TGD framework. The construction of space-time sheet assignable to given 3-D lightlike 3-surfaces can be done in quite detailed manner now and an unexpected connection with Higgs fieds emerges.


At 11:58 AM, Blogger Kea said...

Thanks for the comment, Matti.

Conformal field theory should not contain explicit length parameters.

Yes, I agree that it really is this simple. I can't understand how in a mere 10 years (since the discovery of apparent cosmic acceleration) physicists can have become so fond of a fudge factor.


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