# Are standard QFT and general relativity contradictory?

My professors say it's only a matter of finding the right mathematical formalism to unite GR and QFT, and that new physics can only possibly be found on extremely high energies and small scales. they consider GR to be a nice smooth approximation of QFT on macro scales.

I know QFT can be formulated on curved spacetime. But in GR spacetime is not only curved but curved dynamically, and with a dynamically changing background you lose certain conservation laws that are to my understanding essential for QFT. How is this not a contradiction?

I want to know how it's not a mathematical contradiction for example that one theory has the conservation of energy and other doesn't. There are more subtle and sophisticated apparent contradictions/paradoxes like no info conservation due to black holes etc. But seems to me that one would have to introduce drastic changes to one or both theories to avoid these obvious contradictions, at which point no sense is left talking about union of GTR and QFT.

• Can you be more explicit with which conservation laws are lost that you feel are essential to QFT? – J. Murray Apr 14 at 13:15
• Possible duplicates: physics.stackexchange.com/q/387/2451 , physics.stackexchange.com/q/55213/2451 and links therein. – Qmechanic Apr 14 at 13:45
• Thhese two posts talk about "inconviniences" of unification. But it seems to me there is an obvious contradiction here that cannot be a mere inconvenience.. – Matko Apr 14 at 14:04
• @j.murray I want to know how it's not a mathematical contradiction for example that one theory has the conservation of energy and other doesn't. There are more subtle and sophisticated apparent contradictions/paradoxes like no info conservation due to black holes etc. But seems to me that one would have to introduce drastic changes to one or both theories to avoid these obvious contradictions, at which point no sense is left talking about union of GTR and QFT. – Matko Apr 14 at 14:08
• QFT requires that observables in spacelike-separate regions commute with each other, but "spacelike separated" depends on the metric, which is dynamic in GR: it depends on what everything else in the spacetime is doing. How would we define a QFT if we don't know a priori which observables are supposed to commute with each other? That's not a "contradiction," but is it within the scope of the question? Also: some QFTs automatically include gravity, albeit in a higher-dimensional "emergent" spacetime. That's what the AdS/CFT correspondence is about. Is that within the scope of the question? – Chiral Anomaly Apr 15 at 20:33

The classical story

So the issue of conservation laws can already be understood on the level of classical field theory. For instance, consider a classical scalar test field $$\phi(x^\mu)$$ (i.e. a field that does not determine the geometry) moving in a space-time geometry dependent on time. It is an easy exercise to show that this field does not conserve its total energy on this background. Similarly, test fields evolving on backgrounds that break translation symmetries do not conserve their total linear momenta, and when rotational symmetries are broken, angular momenta are also not conserved.

A somewhat more complicated analysis can show you that similar statements hold when fields such as $$\phi(x^\mu)$$ do enter the Einstein equations as sources of gravity. As a simple demonstration of this fact, consider an isotropic homogeneous metric (the FLRW metric) coupled to a scalar field - you will come to the conclusion that total energy is not conserved in this universe.

So how do we ever come to conservation laws here on Earth, if they do not hold in the universe (which is modeled by a FLRW metric)? The point is that conservation laws hold locally on a curved background and you will never observe their violation as long as you are following processes over distances (and times) much smaller than the background curvature scale. Indeed, the statement that the covariant divergence of any stress energy tensor is zero, $$T^{\mu\nu}_{\;\;\;;\nu} = 0$$, means that for every space-time event with coordinates $$x^\mu_*$$ there is some set of coordinates $$x^{\tilde{\mu}}$$ such that:

1. the metric at the event and its linear neighborhood looks like the Minkowski metric, $$g^{\tilde{\mu}\tilde{\nu}}(x^{\tilde{\lambda}}(x^\kappa_*)) = \mathrm{diag}[-1,1,1,1], g^{\tilde{\mu}\tilde{\nu}}_{,\tilde{\gamma}}(x^{\tilde{\lambda}}(x^\kappa_*)) =0$$, and
2. the stress-energy tensor is locally conserved $$T^{\tilde{\mu}\tilde{\nu}}_{\;\;\;,\tilde{\nu}}(x^{\tilde{\lambda}}(x^\kappa_*)) = 0$$.

These sets of coordinates are known as Riemann normal coordinates and when one sets up a set of locally orthogonal coordinates, one approximately constructs precisely these coordinates. For comparison, the shortest curvature scales in the solar system are $$\sim 5 \cdot 10^8 \rm km$$; you have to study processes on comparable scales or longer to see curvature effects and the violation of conservation laws in the Solar system.

So one naturally takes any theory from flat space-time and extends it rather uniquely to curved space-time by requiring that the original theory holds locally in normal coordinates - this is where, in fact, we found and verified the theory in the first place. On the classical level this is more or less where the story ends, and one can understand most of QFT on curved background from this perspective.

The quantum story

However, on the quantum level one sees already in flat space-time that the choice of vacua matters. In particular, accelerating observers do not see a non-accelerating vacuum as empty, they see it as full of Unruh radiation. Similar issues with vacua arise in QFT on a curved background and give, for instance, rise to the prediction of Hawking radiation. It is true that the choice of the "correct" vacuum for QFT on a curved background can only be determined by global heuristic arguments. On the other hand, the observable consequences of the choices of vacua seem to mainly follow from their choice on the space-time boundary - and it is well known that boundary conditions are something that is traditionally provided "from above" in physics even in theories considered to be self-consistent.

So this is how one gets the behavior of QFT as a test field on a curved background and it is reasonably self-consistent. Another refinement is to consider semi-classical gravity, where the classical Einstein equations are sourced by the expectation value of the QFT stress-energy operator $$\langle \hat{T}^{\mu\nu}\rangle$$ and this, again, can provide you with concrete predictions.

Nevertheless, the most advanced conservative iteration of QFT+GR is to consider GR as a (non-renormalizable) effective field theory (EFT) and quantize it as such. The quantization of an effective theory comes with a regularization scheme where a part of the regularization parameters do not cancel out from final observables and can be set arbitrarily. However, one assumes that the values of these parameters are set by an underlying fundamental theory within certain bounds. In other words, the theory gives you all your predictions with a confidence interval.

On the other hand, by going to higher and higher loop orders in the computation, you can generate an infinite number of regularization parameters that enter your computation and these are all bounded by your assumptions. That is to say, the EFT quantization of GR comes with an infinite number of assumptions about certain new parameters of the theory. This is not necessarily an inconsistency, but certainly a drawback of the EFT-GR quantum theory. Then again, once you make peace with this, you can combine the Standard-model QFT with the EFT-GR QFT in a unified self-consistent framework that gives predictions within confidence intervals.

For certain cases the confidence interval can be very small, and there you are quite happy to use this effective theory; for others the prediction becomes essentially meaningless. This happens for for instance for processes with collision energies close to the Planck mass, and this is essentially what is meant by the statement that "GR breaks down at the Planck scale". It should also be noted that the size of the assumed confidence interval of the predictions is often mistakenly identified as "the size of the quantum-gravity corrections to GR", even though these may in principle be very different. (For example, a number in the interval $$[0,1]$$ is not of the size 1, it may also be exactly zero.)

This being said, the EFT approach provides a self-consistent theory that gives you amazingly accurate and specific predictions for any currently achievable experimental or observational setting. There are a few extreme experimental/observational settings, which we might not ever reach as a civilization, where this theory does not give specific predictions and that is the whole problem with quantizing gravity.

As far as I know the only consistent models,in that they can embed the standard model of particle physics, which has a QFT form, and also allow for quantization of General Relativity , are string theory models.

Maybe this power point presentation which shows the way Feynman diagrams are extended to string theories will help.

Unfortunately there is no definitive string theory model up to now, so quantization of gravity is phenomenologically used as your professor describes, assuming such parameters that the approximation can hold, as for example in cosmological models.

• How do you know they are consistent? I am not asking if GR is consistent with QFT. I am asking if there is a definitive contradiction when these two are taken together. – Matko Apr 14 at 14:00
• I strongly object to what is clearly Latex/Beamer being called a "power point presentation". :p – Federico Poloni Apr 14 at 14:20
• @FedericoPoloni sorry, It is because I saw repeated the same slides, and I am not familiar with all the possibilities, whatever latex/beamer is – anna v Apr 14 at 15:30
• @Matko I put commas to clarify what the consistensy is.( Also if things are consistent they cannot be contradictory in my vocabulary) – anna v Apr 14 at 18:49
• That's not what I'm saying. I was just highlighting that you can never know if a theory is consistent absolutely. – Matko Apr 14 at 20:07

Some people might say they are contradictory, and some people might say otherwise. That depends on what you mean by contradictory. In my opinion, there is nothing contradictory about GR and QFT. For instance, within GR it is possible to compute the radiative correction to the perihelion shift of Mercury, being a correction of order $$\frac{1}{10^{90}}$$, which we are not going to measure any time soon. Still, this is a genuine prediction of quantum gravity. The key point is that such calculation, as well as usual GR processes, is independent of a UV completion to the Einstein-Hilbert action. This means that one does not need to know the full theory in order to capture and reproduce the relevant phenomena. Therefore, at low energies GR works perfectly fine and captures all relevant phenomena. But at distances of the order of the Planck mass $$r\sim 10^{-33}cm$$ quantum corrections become important, and therefore there one needs to use the UV completion of GR. String theory is precisely a UV completion of GR.

In conclusion, GR is the only consistent theory of an interacting massless spin-2 particle, the graviton. It is a quantum theory, and it is non-renormalizable, and therefore non-perturbative for energies $$E\sim M_{Planck}$$ , but it is not inconsistent with quantum mechanics. Both theories are perfectly fine in their own range of applicability, which is how physics models are build. Would you say Newton's mechanics is inconsistent with Einstein's mechanics? I don't think so, I would prefer to say that Einstein's theory is a completion of Newton's theory, and Newton works remarkably well in its regime of applicability. In this sense, and this is my opinion, every theory of physics is an effective field theory, meaning that theories capture the relevant degrees of freedom and phenomena at a given scale, but out of such a scale the theory might not work well, and needs to be improved or left behind in favour of a new theory which works better at such a scale.

• No, quantized GR is not able to predict its radiative corrections quantitatively, these depend on regularization parameters. Of course, there is the asymptotic safety scenario, but that is currently on the level of conjecture and has never been used to give quantitative predictions. – Void Apr 14 at 16:13
• Thanks for your answer, I understand what you mean. I would like if you could elaborate more on why what I said is wrong. I am curious because I basically took these results from Schwartz's book "Quantum field theory and the standard model", chapter 22. Is the book wrong, or did I just misunderstand the explanation? Also, I would appreciate your opinion on a possible inconsistency between GR and QFT. What is your take on that? – Jordi Apr 15 at 14:57