A Which is the quantized variable in quantum gravity?

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In Quantum Mechanics, the position (or momentum) variable is quantized. I define "quantization" as promoting a variable into a probability distribution.

For example, with the double slit experiment, the classical assumption that the position/path of a particle is "unique" cannot explain observations. Feynman promoted paths from "fixed variables" into a probability distribution (path integral), and this solved the problem.

In second quantization, the "fixed variable" number-of-particles is promoted into a probability distribution ("grand-canonical-distribution"), and this solves the problem that when we scatter several particles at high energy (high enough to create new particles), it is impossible to have a path integral with a fixed number of particles that gives the experimental values.

The physical reason is that new particles are being created during the scattering, an the probability distribution of "paths" à la Feynman is not enough: one needs blobs that represent virtual particle creation. So, one needs a "probability distribution for the number of particles".

In formal terms, the quantization is usually defined as promoting variables into operators. And if one does this, everything works well. But intuitively, I like more the explanation of the promotion of variables into probability distributions (either paths or number of particles), since this explanation relates more to the physical reasons for the need of a new theory.

But what happens with gravity?

If we take the canonical approach, it seems obvious that what we have to do is to promote the metric into an operator, in analogy to the first and second quantization, which were successful that way. But we know that this canonical approach fails.

But what if we tried to find a new variable to be quantized, not as in the canonical approach (which says "the right variable to be quantized is the metric") but in the physical way:

What is the equivalent in gravity of paths or number of particles? In other words, what is fixed in QFT, but variable in gravity?

Of course, one answer to this question is "geometries", and this leads us to the canonical approach, which fails. But maybe there are more interesting answers?

In fact, in physics in general (in condensed matter in particular) the choice of variables to describe a problem is essential. Maybe the metric is not the right variable to be quantized, but something (slightly) different.

Are there proposals of other variables to be quantized for gravity, in the literature?

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