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My understanding is that one of the driving forces behind the LQG approach is to maintain diffeomorphism invariance inherent in GR on the way to a quantum theory of gravity. Along the way it's often pointed out that this is incompatible with 'ordinary' QM in that it's formulated on a fixed background 'stage', along with the problem of time.

Sometimes the first part of this seems to imply that you cannot write a diffeomorphism invariant version of a quantum theory. However, would a QM equation (KG or Dirac) in curved spacetime not be invariant at least under spatial diffeomorphisms?

The way I think about that, which I admit may be incorrect, is that the 'physical law' should not depend on the geometry. So if we have an equation that can take any (reasonable) geometry, then is this not satisfied, at least in the spatial case? (I know the problem of time is more deeply rooted)

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In QFT in curved spacetime, the curved spacetime on which we are doing quantum mechanics/field theory is held fixed. In other words, the metric is a background field and is not dynamical. Diffeomorphisms acts on all the fields in your theory, including the metric. However, a symmetry is one which acts on all the fields on your theory such that

(1) The action/path integral is invariant (up to anomalies).

(2) The background fields are invariant.

In our case, the metric is a background field so general diffeomorphisms are not symmetries. However, those diffeomorphisms that leave the metric invariant are symmetries! These are precisely the isometries of the background.

As an example, standard QFT is defined on Minkowski spacetime whose isometries are the translations and Lorentz transformations. These are therefore the only actual symmetries of standard QFT.

A contrasting example is GR where the metric is a dynamical field so (all) diffeomorphisms are indeed a symmetry of the theory. More importantly, this is a gauge symmetry so only those fields that are diffeomorphism invariant are of import.

In a similar way, since the metric is dynamical in 'quantum gravity', diffeomorphisms should be a (gauge) symmetry of that theory. Therein lies the problem of quantizing gravity (among others of course). It is very hard to write down a diffeomorphism invariant quantum theory. For instance, such theories cannot have local operators since they are not diffeomorphism invariant.

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  • $\begingroup$ What about the idea that diffeomorphism is "trivial" and that any theory can be formulated in a coordinates independant way? $\endgroup$
    – Cham
    Commented Sep 27, 2019 at 1:24
  • $\begingroup$ @Cham - That statement is strictly classical. We know how to formulate basically any classical theory in a diffeomorphism invariant way. Its the quantization of such theories that has been difficult. $\endgroup$
    – Prahar
    Commented Sep 27, 2019 at 1:48
  • $\begingroup$ This is weird. It gives the impression that quantum theory is false in its most profound basement. Once an equation is true in some coordinates system, it should automatically be true in all other coordinates systems. $\endgroup$
    – Cham
    Commented Sep 27, 2019 at 1:53
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    $\begingroup$ I am confused by your comment (my fault in communication). All quantum theories can be described in a diffeomorphism independent way (though not always uniquely). It's the gauging of these diffeomorphisms that is hard. That last bit IS the problem of quantizing gravity. Quantum theory on fixed backgrounds (which is what we mean by QFT in curved spacetime) is perfectly well-defined. $\endgroup$
    – Prahar
    Commented Sep 27, 2019 at 1:56
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    $\begingroup$ @zylatis - ya that screws it up. To do QFT in curved spacetime, you need the spacetime to be globally hyperbolic which means that it contains one time-like Killing vector. This timelike Killing vector then provides a natural notion of time. I don't think it is known how to study theories in spacetimes that do not have this property. $\endgroup$
    – Prahar
    Commented Sep 27, 2019 at 5:41

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