A quantum gravity replacing the time variable by an operator

Question

In this video (from 57:38 to 58:31) Aurélien Barrau talked about a simple obstruction for the existence of quantum gravity, stated in an unusual way (to me):

Original in French:

Ce qui m'intéresse ici c'est vraiment de regarder les aspects littéralement liés à la gravitation quantique. Alors pourquoi c'est si dur d'ailleurs? Bien sûr il y a des arguments techniques de non-renormalisabilité que vous connaissez sans doute, mais conceptuellement on pourrait aussi le dire de façon simple: Quand je fais la quantification canonique d'un système, je remplace x par un opérateur mais je ne remplace pas t par un opérateur. Vous voyez que la mécanique quantique distingue fondamentalement l'espace du temps, alors que en relativité générale l'espace et le temps c'est la même chose. Vous voyez que c'est trés compliqué d'avoir d'une part une théorie qui peut mélanger l'espace et le temps, et d'autre part une théorie qui ne traite pas l'espace et le temps sur un pied d'égalité. C'est une des raisons (parmi d'autres) conceptuelles qui rendent très difficile l'émergence d'une théorie de gravitation quantique; on peut dire aussi que le principe d'incertitude d'Heisenberg n'est pas géométrisable.

English translation:

What interests me here is really looking at the aspects literally related to quantum gravity. So why is it so hard? Of course there are technical arguments of non-renormalizability that you probably know, but conceptually we could also say it in a simple way: When I make the canonical quantization of a system, I replace x by an operator but I do not replace t by an operator. You see that quantum mechanics fundamentally distinguishes the space from the time, whereas in general relativity space and time are the same thing. You see that it is very complicated to have on the one hand a theory that can mix space and time, and on the other hand a theory that does not treat space and time on an equal footing . This is one (among others) of the conceptual reasons that make the emergence of a quantum gravity theory very difficult; we can also say that the Heisenberg uncertainty principle is not geometrizable.

Question: Is there an approach for quantum gravity where the variable t (for time) is replaced by an operator? 

Bonus question: Did you ever seen this obstruction stated like that (i.e. bolded sentence) before? Where? Is it relevant? Could it be the starting point of a new approach for quantum gravity?

Comments

It is a public lecture where things are simplified. To ramble, space and time are not always equivalent in RG. For qm and qft, if you consider the time prediction of an event, it is also a combination of operators, thus an operator. It's too broad to start a theory. A Barreau is a great lecturer.

---

You see that quantum mechanics fundamentally distinguishes the space from the time, whereas in general relativity space and time are the same thing.

Space and time are not "the same thing". Nowhere. They are distinguishable.

Canonical quantization deals with particle coordinates $\vec{x}(t)$, not with the whole space, so the wording is wrong. You see, time $t$ is the same for many particles whereas the particle coordinates $\vec{x}_n(t)$ are all different.

Underestimation of "non-renormalizability" and letting it stay are too serious errors in building a new "theory". A theory must be practical, not filled with "excuses".

---

I apologize, it is not a public lecture but an article on the state of the art in gravitation theories.The comparison is not to take litterally, it contributes to draw the theoretical and computational difficulties. Excellent review of the current research paths; made interesting for all publics.

---

I just posted the question on physics.stackexchange. It got a very interesting answer of Ben Crowel:
https://physics.stackexchange.com/a/519094/26397

---

I do not understand why one should quantize time if it is not quantized experimentally. There is nothing else to do?

Let's suppose there is a time operator $\hat{t}$ with eigenstates $|t_i\rangle$. If we are in an eigenstate $|t_i\rangle$, then time stays still: $t=t_i$, it does not change.

In physics there are macroscopic (inclusive) quantities that are not subject to quantization - by their classical nature.