Results

The dynamical laws for gravity may be captured by a theory of quantum gravity, or something else. My current (as of October 8, 2023, based on Section 1.2 of the PhD thesis) credence for the possibilities are:

1. Geometric-variable path integrals
   Credence: ∼ 50%.
   In General Relativity, gravity is captured by spacetime geometry. Straightforwardly, path integrating over spacetime geometries yield quantum theories of gravity. Examples include versions of simplicial quantum gravity (quantum Regge calculus) and dynamical triangula- tion.I assign the highest credence to this possibility, because it is hardest to argue against. In con- trast to most other theories discussed below, I do not know any way to rule out Lorentzian simplicial quantum gravity (see Chapter 2 of PhD) by comparing with known facts. To know if the extant theories in this category are true, the challenge is to develop techniques to evaluate the path integrals efficiently, draw predictions, gather data, and test against yet unknown facts.
2. Gauge-variable path integrals
   Credence: ∼ 20%.
   General relativity can be reformulated in terms of frame fields. This leads to alternative gravitational path integrals based on gauge variables. Examples include Ponzano-Regge models, and various spin foam models.For geometric variables, the path integral sum is constrained by inequalities to Lorentzian or Euclidean geometries. For gauge variables, it is far less clear which configurations should be included in the path integral. The many different answers give rise to a large variety of gauge-variable path integrals, of which only a small portion have been studied in detail. The pool is large enough for one to be optimistic that at least one member could be true, but due to limited understanding of the models, there lacks sufficient reason to single out any candidate as particularly promising. Therefore I assign a modest credence.
3. Causal set path integrals
   Credence: ∼ 3%.
   The causal set approach captures gravity by fundamentally discrete sets, and provides ex- amples for gravitational path integrals based on neither geometric nor gauge variables. Causal set path integrals face some long-standing issues. A causal set does not carry a spacetime dimension, nor a tangent space structure to couple to fermions. Consequently, no extant causal set path integral can be viewed as a viable candidate for the true theory of quantum gravity, and the faint hope has to be set on future inventions.

4. Holography
   Credence: ∼ 5%.
   As far as our universe goes, AdS/CFT correspondence seems to only supply false theories, because our universe is not AdS.There is the hope that some dS holography theory applicable to our universe will be in- vented in the future. So far so bad, after many years, so my credence is low.
5. Wheeler-DeWitt equation
   Credence: ∼ 1%.
   Defining a theory of quantum gravity (e.g., Canonical loop quantum gravity; quantum ge- ometrodynamics) by the Wheeler-DeWitt equation Hψ = 0 seems to me to be misguided from the beginning, due to its focus on ψ, instead of empirical predictions. Unlike in un- dergraduate quantum mechanics, where p = |ψ|2, there is no clear way to reach empirical probabilistic predictions p from ψ in the present context.To derive empirical probabilities, one could adopt a functional integral approach (see Chapter 6 of PhD). If one insists on a differential equation approach, one could parameterize the family of mathematical quantities which do yield empirical probabilities, and try to derive an equa- tion of how these quantities change as the parameters vary. This equation may have little to do with the Wheeler-DeWitt equation, which means one should not focus on the Wheeler- DeWitt equation in the first place.
6. Other known approaches
   Credence: ∼ 5%.
   The other extant approaches I am know of are all quite premature. For instance, functional renormalization group asymptotic safety, developed in the Euclidean setting, is inapplicable in the Lorentzian setting. Perturbative approaches, limited to the perturbative setting, are capable of neither offering a full theory of quantum gravity, nor addressing questions about black hole interior and quantum cosmology, where quantum gravity is actually in need. String theory turns to holography or the distant dream of M-theory to address the challenge for a non-perturbative formulation. The former case does not seem promising, as discussed above, while the latter dream seems to remain distant.Since these are all long-standing issues, unsolved not due to the lack of trying, my credence is low.
7. Something else
   Credence: ?
   It could be that the true theory is not a quantum theory, or a quantum theory in some brand new approach. I find it difficult to estimate the chance for this possibility, so leave it unspecified.

By the above Bayesian analysis, my current best bet for the correct theory of gravity is a non-perturbatively defined path integral in the metric variable.

In particular, I find most promising the Lorentzian theory developed here:

• Ding Jia, 2022, “Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects”. [article, arxiv]

Currently, we do not understand the lattice refinement limit of the theory well. Some partial progress are reported in:

• DJ, 2022, “Light ray fluctuations in simplicial quantum gravity”. [article, arxiv]
• DJ, 2023, “Light ray fluctuation and lattice refinement of simplicial quantum gravity”. [article, arxiv]

What happens to black hole and cosmological singularities in quantum gravity?

One possibility is that they are avoided trivially:

• DJ, “Is singularity resolution trivial?”. [arxiv]

The idea is that for some Lorentzian gravitational path integrals, singular configurations are excluded from the sum by definition.

What phenomenological implications does the above mechanism of singularity avoidance have?

• DJ, “Semiclassical singularity is compatible with quantum singularity avoidance” [ttoe]

The interesting finding is that quantum singularity avoidance sometimes come hand-in-hand with singular semiclassical spacetimes. The result suggests one to consider singular bounce effective spacetimes for phenomenological studies.