Is there any interpretation where GR and QM can be compatible?

Question

In this article ^[1] , Dr. Sten Odenwald claimed that,

Quantum mechanics is incompatible with general relativity because in quantum field theory, forces act locally through the exchange of well-defined quanta.

Is there any hypothesis where both of them can be compatible?

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^{[1] - https://einstein.stanford.edu/content/relativity/a11758.html}

Answer 1

I don't agree with that quote. There is no problem with treating GR as a local effective quantum field theory, with gravitons being the quanta of the gravitational field (at least on a static background), below some scale.

The issue is not really about interpretation, it is more technical. We know how to quantize an effective field theory when the energy of processes being considered is less than the cutoff scale. In the case of gravity, the cutoff scale is the Planck energy, $E_p = \sqrt{\hbar c^3/G} \approx 2 \times 10^9\ {\rm J}$. For processes below this energy, there is no problem thinking of Einstein's equations as describing a local quantum field theory of the gravitational field. Above this energy, we don't know how to treat the theory quantum mechanically. There are various proposals, some examples and roughly speaking how they try to solve the problem:

• String theory replaces the spacetime metric and quantum fields describing elementary particles with new degrees of freedom. In some limits, these degrees of freedom are described by strings propagating in 10 spacetime dimensions. In other situations, this same theory describes other objects like branes or reduces to a theory of supergravity. The main issue string theory has is connecting to observations. That's true of all quantum gravity approaches, but since string theory introduces some new elements (extra dimensions, additional unobserved light fields, multiple vacuaa), one challenge is explaining what set of ingredients actually reproduces our world. Additionally, no one knows how to formulate string theory (or "M-theory") precisely in a non-perturbative way, outside of the special case of asymptotically Anti de Sitter spacetimes.
• Loop quantum gravity keeps Einstein's equations, but uses a different set of degrees of freedom besides the metric and changes the methods we use to quantize a theory. A major challenge for loop quantum gravity is to show that in an appropriate limit, it reduces to classical GR.
• Asymptotic safety proposes that Einstein's equations and the fundamental metric degrees of freedom can be quantized directly, and we "simply" have to learn how to quantize the theory when we can no longer use perturbation theory. More technically, the proposal is that the renormalization group for gravity has a non-trivial UV fixed point. A major issue here is showing that this UV fixed point actually exists (and it is not at all clear that it does).

It is worth noting that the holographic principle seems to imply that quantum gravity should have a degree of "non-localness" to it, which is not present in quantum theory. This is perhaps a reason to be skeptical of the asymptotic safety scenario.

Answer 2

Well, the problem is that there are no nontrivial diffeomorphism-invariant local observables for general relativity. C.G. Torre's "Gravitational Observables and Local Symmetries" (arXiv:gr-qc/9306030) proved this for the vacuum, and adding matter doesn't help things.

We can try to construct a class of observables by considering integrals over spacetime of scalar functions, but this family of observables do not posses a local interpretation. The scalar must commute with the generators of the diffeomorphism constraints. For a given scalar field $\phi(x)$, the diffeomorphism group acts as $\partial_{\mu}\phi(x)$ which vanishes if and only if $\phi(x)=\phi_{0}$ is a constant. Torre has shown general relativistic observables must include an infinite number of derivatives and hence are very nonlocal.

Quantum theory doesn't handle nonlocal obserables very well, and I believe this is what the OP's quote references: since there are Dirac observables for General Relativity, it doesn't play into quantum theory very well.

Loop Quantum Gravity uses holonomies to construct its observables, and holonomies are nonlocal. String theory has a similar implicit use of nonlocality.