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I am not asking about why or how gravity should be quantized, or what the problem with renormalization is, or what the discrepancy is between QM and GR is. Those are beautifully described in other questions.

Spacetime is widely accepted as being continuous and showing no discreteness.

And it safely rules out all hypotheses that the spacetime may be built out of discrete, LEGO-like or any qualitatively similar building blocks.

Does the Planck scale imply that spacetime is discrete?

Gravity is accepted to be spacetime curvature.

the Einstein-Hilbert action, or "gravity is the curvature of spacetime"

Does the curvature of spacetime theory assume gravity?

Why do we say "Spacetime Curvature is Gravity"?

If spacetime doesn't show any discreteness, then naively I would think that it's curvature doesn't either. Its curvature is gravity itself, so that would mean that gravity doesn't show any discreteness either. But if it doesn't show any discreteness, then it can't be quantized?

The only thing I can think about is the EM field but that does show some discreteness. We do know that Em energy can be transferred in quanta, based on experiments (like the photoelectric effect), and we do know that it can be stored in quantized energy levels in atoms. There is just no example like that for gravity.

So basically the question is whether the gravitational field needs to show some discreteness to be quantized or not.

Question:

  1. If gravity is spacetime curvature, and spacetime doesn't show any discreteness, then gravity doesn't show any discreteness and can't be quantized?
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    $\begingroup$ I don't understand what you think discreteness has to do with quantization (other than the origin of the word "quantization" perhaps). Position and momentum of a particle in free space aren't discrete either but have perfectly fine quantum versions. $\endgroup$
    – ACuriousMind
    Apr 3 at 16:28
  • $\begingroup$ @ACuriousMind I am just talking about the way for example the EM field shows discreteness in some experiments, like the photelectric effect. EM energy can be transferred in quanta or be stored in atoms in quantized levels. I believe this has contributed to how the EM field got quantized and how we knew it could be. I did not find any similarity in the case of the gravitational field though. $\endgroup$ Apr 3 at 16:39
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    $\begingroup$ @ÁrpádSzendrei But when we quantize something like EM, the classical analogue is classical field theory (which is continuous). The problems with quantizing GR don't arise from the smoothness of classical spacetime (though people do explore discretised spacetimes as an approach to QG too) $\endgroup$
    – Eletie
    Apr 3 at 16:51
  • $\begingroup$ "Gravity is accepted to be spacetime curvature." I think the correct statement is : the Newtonian gravitational field is functionally dependent on the curvature of spacetime", so easy conclusions are out. $\endgroup$
    – anna v
    Apr 3 at 17:56
  • $\begingroup$ look at loop quantum gravity en.wikipedia.org/wiki/Loop_quantum_gravity "LQG postulates that the structure of space is composed of finite loops woven into an extremely fine fabric or network". The theory would have been thrown out if it does not end in the limit in Newtonian gravitational fields. $\endgroup$
    – anna v
    Apr 3 at 18:02
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But if it doesn't show any discreteness, then it can't be quantized?

This is incorrect. Energy is a counter example. It is not discrete but is quantized. Quantization comes naturally from the axioms of QM even with continuous wavefunctions and non-discrete operators.

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  • $\begingroup$ Thank you. I do get what you say. Maybe I wasn't clear enough. Energy is a beautiful example for me, because it can be stored in atoms (in discrete levels), and can be transformed into the EM field's energy (in quanta) and vica versa. But can we transform it into the gravitational field, at least, can we have a thought experiment for this? I cannot think of one. This is why I am asking, if the gravitational field does not show any kind of example for this, then maybe this should mean that quantization is not as we think for the gravitational field.. $\endgroup$ Apr 3 at 21:34
  • $\begingroup$ I am not claiming any specific relationship between energy and gravity. My point is that “if it doesn't show any discreteness, then it can't be quantized” is false reasoning. There exist known quantities that do not show any discreetness and can be quantized. That is the point I am making. $\endgroup$
    – Dale
    Apr 4 at 1:24
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I think the right way of thinking about this may be: General Relativity is formulated with the idea of continuous manifolds. This would, indeed, be inconsistent with a discretized spacetime.

I don't think that ends up being the problem with quantising gravity, though. Since, as somebody has already pointed out, something being continuous doesn't mean it can't be quantized.

I'm no expert on this, though. Please disregard my input in favour of a more reliable source.

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