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The past decades of research in perturbative quantum gravity has shown that calculations are possible, that results are unique, and that corrections to classical results are extremely small. (Donoghue's work is an example, but that of many others is as well.)

What are the reasons that people are unsatisfied with this combination of general relativity and quantum theory, especially if one uses the Hilbert Lagrangian as a starting point for pQG?

Why do people explore loop quantum gravity, or causal sets, or spin foams, or several other such theories, when perturbative quantum gravity works well?

Non-perturbative effects seem to arise only at the Planck scale, which does not seem accessible anyway. Doesn't this mean that pQG is perfect for all practical purposes?

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  • $\begingroup$ maybe this reference will help arxiv.org/abs/0907.4238v1 $\endgroup$
    – anna v
    Feb 21 at 13:28
  • $\begingroup$ Why do you think people are unsatisfied with perturbative quantum gravity? $\endgroup$
    – Andrew
    Feb 21 at 13:38
  • $\begingroup$ Well, people seem to develop ideas like string theory, loop quantum gravity, causal sets etc just because they dislike perturbative quantum gravity. $\endgroup$
    – user85598
    Feb 21 at 15:36
  • $\begingroup$ @Christian , perturbations only involve calculation on background Minkowski space, while the theory is supposed to be background independent. I think the issue here is that we are simply applying usual qft on classical gravity and looking for possible modifications, which is kind of uneven tbh. I feel that necessary changes are needed in both quantum theory and classical gravity using a deeper foundation, which should inherently be non perturbative in nature $\endgroup$
    – KP99
    Feb 22 at 8:28
  • $\begingroup$ But the question is: if perturbative QFT works well, why should a non-perturbative theory be needed at all? $\endgroup$
    – user85598
    Feb 23 at 1:03

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Let me draw a historical parallel (here is some context you can read afterwards).

"Vanilla" quantum mechanics is Galilean-invariant, but in view of special relativity we need something more sophisticated. To make allowances for Lorentz transformations, relativistic quantum mechanics approximates quantities at leading order in a small variation around the Galilean case. But this isn't enough to model all conditions. When particles are created or destroyed, we need a fully-fledged Lorentz-invariant quantum field theory, which took a fair bit of work. First, we had a "free" (non-interacting) version, and later we worked out how electromagnetism looks from that perspective, and eventually understood the nuclear interactions the same way. Unfortunately, gravity has proven harder.

So far, I've talked about QFT in flat spacetime. Its curved counterpart, QFTCS, is a step short of full QG, but a step beyond perturbative QG. PQG should be compared to RQM, approximating quantities at leading order in a small variation around special relativity's metric tensor. This doesn't address particle number subtleties any better than RQM does. In fact, such issues are even more complicated in QFTCS than its flat counterpart, partly because the definition of particle number, or even a vacuum state, runs into very geometry-dependent difficulties. As a further problem, PQG is nonrenormalizable.

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There is nothing wrong with perturbative quantum gravity, in its regime of validity. When describing processes with energies less than the Planck scale, perturbative quantum gravity can be considered as a low-energy effective theory, and quantum corrections can be computed in a systematic way.

However, there are situations which lay outside the regime of validity of the effective field theory of gravity.

  • What happens near the singularity of a black hole?
  • What happens near the Big Bang singularity?

And, there are conceptual questions that the effective field theory of gravity cannot answer. Unitarity of perturbative quantum gravity breaks down at the Planck scale, which is an indication that something more is needed to have a consistent quantum theory. How can we consistently treat gravity in a quantum mechanical way?

The history of physics is a balance of building successful effective theories that cover a range of phenomena, and digging deeper to find a more complete understanding that underlies the current best theories. You could call this the reductionist paradigm. Quantum gravity theories try to find what deeper theory replaces perturbative quantum gravity outside its regime of validity.

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From Wikipedia:

${S={1 \over 2\kappa }\int R{\sqrt {-g}}\,\mathrm {d} ^{4}x,}$ where ${g=\det(g_{\mu \nu }})$ is the determinant of the metric tensor matrix, $R$ is the Ricci scalar, and $ \kappa =8\pi Gc^{-4}$ is the Einstein gravitational constant ($G$ is the gravitational constant and $c$ is the speed of light in vacuum).

So, the action is calculated in a spacetime that is assumed to be curved in advance. For a sufficiently flat spacetime this is gives no problems. But if you want to calculate interactions of masses by means of virtual gravitons, the virtual condensate surrounding mass (mass is supposed to couple to the virtual gravitons in the vacuum) distorts spacetime itself, contrary, for example, the condensate of a virtual photon condensate surrounding electric charge. And that's where the trouble lies. The flat spacetime approximation is, well, just an approximation.

So how do masses interact? By coupling to a virtual graviton field (in flat spacetime), or by moving in a distorted spacetime? How can a graviton condensate curve spacetime around a massive object? That's the problem with describing gravity by masses coupling to virtual gravitons in the vacuum. A mass that couples to them, changes the geometry of spacetime. Unlike the gauge particles of basic interactions.

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