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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?
Qmechanic
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Árpád Szendrei
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    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. – ACuriousMind Apr 03 '21 at 16:28
  • @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. – Árpád Szendrei Apr 03 '21 at 16:39
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    @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) – Eletie Apr 03 '21 at 16:51
  • "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. – anna v Apr 03 '21 at 17:56
  • look at loop quantum gravity https://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. – anna v Apr 03 '21 at 18:02
  • How is this question not asking what it claims to be not asking? This is indistinguishable to me from "what the discrepancy is between QM and GR" with a particular focus on a proposal for what that discrepancy might be (based on a wrong idea that quantization has to do with discreteness). In any case, the EM field is not discrete in any way that the gravitational field isn't -- as pointed out by others. –  Apr 03 '21 at 20:58
  • @Eletie thank you, yes, right to the point. I do get that on the one hand, quantization might not need an obvious way of discreteness, but on the other hand, people are still trying to quantize gravity by hypothesizing as you say discretized spacetimes. – Árpád Szendrei Apr 03 '21 at 21:27
  • @DvijD.C. I do get what you say about that an obvious way of discreteness is not necessary for quantization, but, the EM field does show different (from the gravitational field) things in experiments in my opinion, because I cannot think of any experiment like the photoelectric effect for gravity. Thus, we do not have a clear idea for an experiment that would proof some sort of (like for the EM field) discreteness for the gravitational field. – Árpád Szendrei Apr 03 '21 at 21:31

2 Answers2

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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.

Dale
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  • 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.. – Árpád Szendrei Apr 03 '21 at 21:34
  • 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. – Dale Apr 04 '21 at 01:24
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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.

V.L. Proud
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