Lorentzian quantum gravity

They say, “gravity is non-renormalizable”. That conclusion is derived for a theory that is not even Euclidean.

“Expand the metric \(g_{ab}=\eta_{ab}+h_{ab}\) against a fixed background \(\eta_{ab}\) and integrate \(h_{ab}\) over all real values”. What ensures that \(g_{ab}\) is Euclidean or Lorentzian? Nothing. What do we learn from the result? You draw your own conclusions.

I’m interested in studying genuinely Lorentzian theories of quantum gravity, which integrates over genuinely Lorentzian configurations. One example is Lorentzian simplicial quantum gravity. A finding that is trivial to make is that such integrals avoid singularities, but this may have some non-trivial implications to our understandings of singularity. Another finding is that the semiclassical approximations of the quantum theories require a nonsmooth extension of general relativity. Another finding is that the conventional boundary conditions for the universe are not applicable, so new ideas are needed.

{to do: elaborate on Lorentzian quantum cosmology}

In the future, I hope to understand better:

• What do the theory spaces look like for Lorentzian theories of quantum gravity? Which actions lead to well-defined path integrals with continuum limits, which don’t?

• Are there some tangible phenomemological implications? Perhaps associated with bouncing cosmology and black holes?