Why? Because the Bogdanoff brothers are proposing something that has, speculatively, the potential to be an alternative story about quantum gravity. As a string theory believer, I would say "a new dual description of quantum gravity i.e. string/M-theory". What they are proposing is a potential new calculational framework for gravity. I find it unlikely that these things will work - but it is probably more likely than loop quantum gravity and other discrete approaches whose lethal problems have already been identified in detail.
In the previous article I focused on the creation of myths and their idea about the fluctuating spacetime signature. But let me now ask you about the following proposal that is included among their refreshing speculative ideas.
Usually we assume that the geometry completely breaks down at the sub-Planckian distances, together with the spacetime topology and all other things. It's because the excited string states and other states appear together with infinitely many higher-derivative corrections. But let us now believe that geometry is a useful picture despite all these effects.
Imagine that you start with a generic gravity action whose pure gravitational part has terms like
- L = R / (16.pi.G) + alpha. R (wedge) R + beta . R^2 + gamma . R^3 ...
and so on. The topological term "R (wedge) R" would have to be replaced by something else in higher dimensions. At long distances, the first, Einstein-Hilbert term is important (after the vacuum energy, of course). At short distances, we usually assume that the infinite tower of higher-derivative terms takes over and we can't say anything; the metric is not a relevant degree of freedom anymore because we must really add a whole tower of new, equally important states predicted by string theory.
But try to assume, together with the Bogdanoff brothers, that it is not the case. Your task is nothing less :-) than to finish their work e.g.
- find an appropriate theory of gravity in which the steps below can be justified
- explain that the higher-derivative terms disappear or decouple in this theory at very short distances, or that they can be ignored for other reasons
- isolate a topological term that does not disappear in this limit
- write down the path integral as a summation over the actual quantum foam - different topologies of the Universe - in analogy with the quantum foam in topological string theory - where the contributions are labeled by "exp(-S)" where "S" is a combination of topological invariants of the d-manifold such as the Euler character
- this will be your dual, sub-Planckian expansion of the partition sum or maybe just some important quantities in quantum gravity
Alternatively, prove that this picture by Bogdanoffs is impossible in any quantum theory of gravity. I can't do either.