Dear Vesselin,
I very much enjoyed reading your essay, which is right on topic and warns against grave metatheoretical misconceptions. Indeed, many research directions are plagued with the misconception that "mathematics in physics is merely a description and therefore even fundamental mathematical entities (such as a manifold) do not represent counterparts in the physical world." And I fully agree that an instance of this misconception is that related to the reality of spacetime, with the most negative implication being in the misguided idea that gravity is a force that should be put in the same Procrustean bed as the other forces. I like how you explain the reality and the necessity of the Minkowski spacetime, in the special relativistic limit, and that of Lorentzian manifolds and gravity as an inertial effect of curved spacetime.
I agree that such misconceptions lead to the idea that gravitation should be quantized on equal footing with the other forces, and the failure of the perturbative method (which is just a perturbative method) leads physicists too quickly to the conclusion that it is general relativity's fault. From this viewpoint, I find semi-classical gravity more compelling than any approach to quantum gravity known so far. This doesn't exclude the need to quantize gravity and even spacetime (and by this I don't mean to discretize it, as it is often equated). Einstein's equation relates the stress-energy, which should be quantum, with curvature, which is spacetime geometry. Given that metric, Levi-Civita connection and various curvature tensors are fields, maybe they should be quantized too, in order to make Einstein's equation work for quantum fields. But I think that our current procedures of quantization are artificial and ad-hoc, in the sense that they worked in a domain, and were extrapolated in the absence of the true understanding of quantization, to other domains. Related to the perturbative approach to quantum gravity, I think perturbative methods are approximations made outside their range of validity, and fixed with various renormalization techniques, and perhaps non-perturbative methods should be found. But they work as approximations, and it would be great if the infinities in quantum gravity would somehow disappear. Most approaches lead to the conclusion that some dimensional reduction effects are needed at high energy scales. I worked several years at singularities in general relativity, to show that although they are different than usual spacetime events, they still can be treated without infinities and without modifying general relativity. As it happens, I realized that the dimensional reduction effects introduced ad-hoc by many researchers to remove the infinities, appear naturally in the presence of singularities. So general relativity is able to take care of its own singularities (rather that these being evidence of its breakdown), and as a bonus, singularities may heal the infinities in perturbative quantum gravity. Of course, I think that we would rather find a non-perturbative solution, in which gravity preserves its status of inertial effect of curvature, and not a force.
Thank you for the wonderful and insightful reading!
Best wishes,
Cristi Stoica
