The String Coffee Table

The following was my summary of the ill reasoning of Bogdanov & Bogdanov as originally posted here, which is reproduced in a distorted fashion in their book:

I think this is their line of reasoning:

They look at the general form of any partition function $Z(\beta )=\mathrm{Tr}(\exp (-\beta H))$. They set beta equal to zero and find, lo and behold, $Z(0)=\mathrm{Tr}(1)$. They notice that the Hamiltonian has disappeared in this expression! They conclude that $\mathrm{Tr}(1)$ must be the partition function of a topological field theory, because they think you obtain the partition function of a topological field theory by setting the Hamiltonian in $\exp (-\beta H)$ equal to zero. Let me call this ‘result’ A.

Next they want to apply this insight to something and search for a setup that justifies setting $\beta \to 0$, thereby arriving at the FRW cosmology, where $\beta \to 0$ as the scale factor $R\to 0$. (At this point they mention the word, just the word, ‘Hagedorn temperature’, not noticing that, considering the role the Hagedorn temperature plays in string cosmology, this is bordering on self-parody.)

They reason as follows: ‘At the initial singularity we have $\beta =0$, therefore physics ‘at the initial singularity’, by result A, is described by topological field theory.’ This is ‘result’ B.

(By the generality of ‘result’ A it does not matter which field theory they are considering. But they are thinking of their $H$’s as the Hamiltonians of field theories on fixed FRW backgrounds, not of the Hamiltonian constraint of some theory of gravity.)

The next step is to assert, C, that a topological field theory is a field theory defined on a Riemannian manifold. Since, by result B, ‘every field theory is a topological field theory at the initial singularity’ it thereby follows that the metric of spacetime ‘at the initial singularity’ must be Riemannian, which is ‘result’ D.

Next, they realize that D is in contradiction to the original assumption of an FRW cosmology with pseudo-Riemannian metric! Being confronted with a paradox they invoke quantum mechanics and postulate that the signature of the metric must be subject to quantum fluctuations ‘at the initial singularity’. That’s ‘result’ E.

It remains to be understood how the Foucault pendulum comes into play now. Even more so, since this doesn’t fit the pattern of using modern termionology.

Just to make sure: I do not think that any of the above is valid reasoning. I am writing this just to point out what I think are the central ‘ideas’ the authors had when writing their articles and how this led them to their conclusions.