# At which point is gravity inconsistent with quantum mechanics?

• I
• Gerenuk

#### Gerenuk

TL;DR Summary
Some beginners questions for the path to understand challenges of quantum gravity.
I'd like to understand how gravity does not combine with quantum mechanics. At least there is no accepted theory of quantum gravity, so I assume it is not solved? I'm only starting to learn QFT and eventually GR. Maybe, someone can already outline where those theories fail to combine and comment on the following thoughts (ideally in terms of concrete equations).

My understanding is that QFT has a Lagrangian describing all of the physics. And GR has the Einstein field equation. Shouldn't both describe some algebra to calculate "what happens" and the task is to find an algebra which encompasses both? Is that correct?

Or is there more than the QFT Lagrangian, because in canonical quantization there are these weird steps of making a coordinate dependent Hamiltonian and then replacing some terms by operators? Why are these operators not already in the Lagrangian anyway?

I've read answers that renormalization is a problem. So you can naturally combine QFT and GR in a Lagrangian? How concretely does it look then? When you calculate, then there will be infinities that you cannot remove? Is all this process written down somewhere? Is renormalization tied to a special way trying to solve the model (an additional assumption), or is the need for fixes already present in the original Lagrangian?

I also saw an answer that background dependency is an issue. How can I see this mathematically in the equations?

I've heard statements that string theory combines quantum mechanics and gravity successfully. So is it an algebra which does contain QFT and GR and can predict what happens? Or why is this not the accepted theory of quantum gravity?

• arivero
I could write down, and point to you some of the math stuff (like the lagrangians etc) but if you do not know either QFT or GR yet, what purpose would it serve you? Just trying to understand what kind of answer you expect to receive here. Do you know how and why renormalization in QFT works yet? Do you know the Einstein equation in GR? This is basically the Lagrangian you start with when you try to make a QFT of gravity https://en.wikipedia.org/wiki/Einstein–Hilbert_action

String theory is ONE quantum theory of gravity, but it not like Einsteins field equations pop out automatically as low energy limit.

• Demystifier, ohwilleke and dextercioby
I could write down, and point to you some of the math stuff (like the lagrangians etc) but if you do not know either QFT or GR yet, what purpose would it serve you? Just trying to understand what kind of answer you expect to receive here. Do you know how and why renormalization in QFT works yet? Do you know the Einstein equation in GR? This is basically the Lagrangian you start with when you try to make a QFT of gravity https://en.wikipedia.org/wiki/Einstein–Hilbert_action

String theory is ONE quantum theory of gravity, but it not like Einsteins field equations pop out automatically as low energy limit.
If I see the math stuff, I could look at it while learning QFT and GR and hopefully understand the challenges in a more focused way. Maybe I could understand which parts I can skip while learning. So any reference which pin-points the issue as directly as possible is appreciated, even though it will take me some time to understand it.

I don't know why renormalization works yet. The only question I have for now is if renormalization is unavoidable or if it's tied to way the Lagrangian is solved. I'm imagining that using perturbative approaches or writing out things in chains of operators is already an additional choice which in some theory could be avoided?

Thanks for the link. That's the kind of answer I'm looking for. And somehow combining that action with the QFT Lagrangian does not work?

Is there an answer possible why string theory is a quantum theory of gravity, but I cannot get the Einstein field equations from it? I naively thought you write down "math" and it should yield the field equations as a part of it. Otherwise, there is no reason to assume that it captures gravity. I don't understand how string theory is one theory of quantum gravity and yet not the accepted solution to quantum gravity.

• I can perhaps give you some reading ideas, I have these
Introduction to Quantum Effects in Gravity, by Mukhanov
Introduction to Quantum Field Theory with Applications to Quantum Gravity, by Buchbinder
Quantum Field Theory II, by Manoukian
Perhaps your university library have these?

Regarding why Gravity is not rernormalizable in QFT, then you need to know what it means and how you need to know what that concept means. But it steems from that sqrt term in the Einstein-Hilbert action, you need to write it as a power series and then introduce infinitely many counter terms. Nevertheless, you can study it as an effective field theory.

• • dextercioby, Gerenuk and PeroK
• LittleSchwinger and dextercioby
Try to learn QFT, GR and String Theory for their own sake. Sure its cool to have a goal in mind, but don't skip the basics.
Personally, I am trying to derive the Slavnov-Taylor identities for spontaneously broken non-abelian gauge theory, and learning about phase transitions in QFTs for cosmology applications, and grand unification of U(1), SU(2)L, SU(3) and things like composite higgs etc

• LittleSchwinger, PhDeezNutz, Klystron and 2 others
Try to learn QFT, GR and String Theory for their own sake. Sure its cool to have a goal in mind, but don't skip the basics.
Personally, I am trying to derive the Slavnov-Taylor identities for spontaneously broken non-abelian gauge theory, and learning about phase transitions in QFTs for cosmology applications, and grand unification of U(1), SU(2)L, SU(3) and things like composite higgs etc
Thanks a lot. I'll see if I can make it. It's rather a hobby and my actual job is different. It's tough to find enough time/motivation - not sure how anyone can know all the terms I read.

How can I imagine that you "try" to derive the identities? I imagine there are a set of algebraic rules and you either can transform equations into it or you cannot. Is it a longer process?

How can I imagine that you "try" to derive the identities? I imagine there are a set of algebraic rules and you either can transform equations into it or you cannot. Is it a longer process?
I mean trying on my own, filling in all details and calculations. Those identities are old and well known. You use those identies to prove certain things regarding renormalization of such theories. E.g. t hooft and veltmans nobel prize and I would like to that myself from scratch.

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Here is a survey of some answers to the OP, a topic upon which lots of both scientific journal article consideration and educated layman's level treatments have been devoted.

An Overview From Wikipedia

Wikipedia has a brief general treatment of the issue here. It notes that:
If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid-state physics, would be the other. However, how to reconcile quantum theory with general relativity is still an open question.

Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime. Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation leading to the possibility that they evaporate over time. As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.
The same article goes onto explain why it has been challenging to develop a quantum gravity theory despite numerous attempts using quite different approaches.
The demand for consistency between a quantum description of matter and a geometric description of spacetime, as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics. Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist. Projection of a Calabi–Yau manifold, one of the ways of compactifying the extra dimensions posited by string theory

Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. Some have argued that at low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity. At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative non-renormalizability"). Simple spin network of the type used in loop quantum gravity

One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects. The theory promises to be a unified description of all particles and interactions, including gravity; the price to pay is unusual features such as six extra dimensions of space in addition to the usual three. In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.

Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff. However, with the introduction of what are now known as Ashtekar variables, this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced, there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman Path Integral approach and Regge calculus, dynamical triangulations, causal sets, twistor models or the path integral based models of quantum cosmology.

All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.
Gravity Is Not A Gauge Theory

A Q&A about Gravity Probe B notes that:
Quantum mechanics is incompatible with general relativity because in quantum field theory, forces act locally through the exchange of well-defined quanta.
In other words, the forces in the Standard Model are governed by gauge theories, while Einstein's Field Equations are not. This is closely related to another problem with integrating GR and quantum mechanics discussed in the following 2018 scientific journal article which explains in its abstract that:
Some of the strategies which have been put forward in order to deal with the inconsistency between quantum mechanics and special relativity are examined. The EPR correlations are discussed as a simple example of quantum mechanical macroscopic effects with spacelike separation from their causes. It is shown that they can be used to convey information, whose reliability can be estimated by means of Bayes' theorem. Some of the current reasons advanced to deny that quantum mechanics contradicts special relativity are refuted, and an historical perspective is provided on the issue.
Marco Mamone-Capria, "On the Incompatibility of Special Relativity and Quantum Mechanics", 8(2) Journal for Foundations and Applications of Physics 163-89 (2018).

The position that quantum entanglement can be used to convey information superluminally, however, it should be noted, is a hotly disputed proposition and many scientists would disagree with that conclusion.

A Good Educated Layman's Introduction

The Guardian, a British newspaper, has a surprisingly good and balanced for a newspaper article take on the issue from November 4, 2015. A highlight in the longish, educated layman oriented article explains:
Basically you can think of the division between the relativity and quantum systems as “smooth” versus “chunky”. In general relativity, events are continuous and deterministic, meaning that every cause matches up to a specific, local effect. In quantum mechanics, events produced by the interaction of subatomic particles happen in jumps (yes, quantum leaps), with probabilistic rather than definite outcomes. Quantum rules allow connections forbidden by classical physics. This was demonstrated in a much-discussed recent experiment in which Dutch researchers defied the local effect. They showed that two particles – in this case, electrons – could influence each other instantly, even though they were a mile apart. When you try to interpret smooth relativistic laws in a chunky quantum style, or vice versa, things go dreadfully wrong.

Relativity gives nonsensical answers when you try to scale it down to quantum size, eventually descending to infinite values in its description of gravity.

Likewise, quantum mechanics runs into serious trouble when you blow it up to cosmic dimensions. Quantum fields carry a certain amount of energy, even in seemingly empty space, and the amount of energy gets bigger as the fields get bigger. According to Einstein, energy and mass are equivalent (that’s the message of E=mc2), so piling up energy is exactly like piling up mass. Go big enough, and the amount of energy in the quantum fields becomes so great that it creates a black hole that causes the universe to fold in on itself. Oops.
The first proposition in bold is primarily addressing quantum mechanical concept of entanglement, but there are also issues like tunneling and virtual particles that classical GR says should be possible.

One of the issues that is being flagged by the language in the second bolded section above is that the Standard Model assumes point-like particles, but a particle with finite mass and zero volume is problematic in classical General Relativity. A naive merge of the Standard Model and GR turns every fundamental particle into a singularity.

The third bolded section is addressing the disconnect between the large non-zero value of vacuum energy in quantum mechanics compared to the very tiny value of the cosmological constant (which nonetheless accounts for roughly three-quarters of the mass-energy of the universe in the LambdaCDM model).

Later in the Guardian article, another subtle issue is discussed. The Standard Model is basically a scientific theory describing sub-systems and components. General Relativity is basically a scientific theory describing the entire mass-energy content of a closed universe in a complete system as a whole.

For example, this poses the tricky issue of localizing or quantifying a gravitational field that is not present in ordinary QFTs.

The Nature of Time In the SM v. GR

A 2008 scientific journal article focuses in its abstract on stating that:
The aim of this work is to review the concepts of time in quantum field theory and general relativity to show their incompatibility. We prove that the absolute character of Newtonian time is present in quantum mechanics and also partially in quantum field theories which consider the Minkowski metric as the background spacetime. We discuss the problems which this non-dynamical time causes in general relativity, a theory characterized by a local dynamical spacetime.
Alfredo Macías, Abel Camacho, "On the incompatibility between quantum theory and general relativity" 663(1-2) Physics Letters B 99-102 (May 25, 2008).

Classical v. Quantum: An Issue But Not Inherently Incompatible

It bears mention, however, that GR is not incompatible with the Standard Model simply because one is classical and the other is a quantum mechanical theory. This plays into the analysis, but it is not inherently impossible for classical theories, in general, and quantum mechanical theories to play nice with each other. The problems of mixing GR and the Standard Model are particular to GR and not general to classical theories in general.

Indeed, for the most part, the theoretical incompatibility of GR and the Standard Model is not much of a problem, because in the domains of applicability of the theories where they are applied in practice, they don't overlap in troublesome ways.

There are even important ad hoc instances where elements of GR and the Standard Model are combined in a common sense sort of way under circumstances where the incompatibility of the theories isn't manifest, such as the theory of Hawking Radiation from black holes, in efforts to understand the equation of state (EOS) of neutron stars, and in the use of quantum electrodynamics and Standard Model neutrino physics to analyze of astronomy observations.

Renormalization

The renormalization question raised in the OP is discussed, for example, here:
We present a short and intuitive argument explaining why gravity is non-renormalizable. The argument is based on black-hole domination of the high energy spectrum of gravity and not on the standard perturbative irrelevance of the gravitational coupling. This is a pedagogical note, containing textbook material that is widely appreciated by experts and is by no means original.
Assaf Shomer, "A pedagogical explanation for the non-renormalizability of gravity" (December 3, 2007).

Note also that perturbatively non-renormalizable doesn't inherently mean internally inconsistent or internally flawed. For example, infrared quantum chromodynamics (the SM theory of the strong force) is also not capable of being addressed with perturbative renormalization, and instead used non-perturbative methods like lattice QCD. But, no one doubts that QCD is a mathematically rigorous and internally mathematically consistent theory. It's just that the mathematical methods perturbative methods that work so well for the electro-weak interactions in the SM doesn't work so well in low energy QCD.

String Theory and Quantum Gravity

I've heard statements that string theory combines quantum mechanics and gravity successfully. So is it an algebra which does contain QFT and GR and can predict what happens? Or why is this not the accepted theory of quantum gravity?

String theory addresses the renormalization problem and more generally avoids issues that QFT creates for GR by assuming point particles.

But, string theory is embedded in an overall mathematical structure whose unity and connectedness is part of its attractiveness. The generalization of string theory called M-Theory is conceived in a 10 or 11 dimensional space-time when we only observe 4 space-time dimensions,

Also string theory is really not just one theory but a whole legion of different possible "vacua" and no one has found one that corresponds to the Standard Model plus GR in its low energy approximation.

Indeed, there is basically no positive observational evidence that prefers string theory to the Standard Model.

It remains to be seen if the insights of string theory relevant to quantum gravity can be successfully extracted from the overall edifice of string theory as a "theory of everything" candidate.

Do We Have The Right Classical Target To Quantize?

Now, of course, we know that gravity actually works seamlessly and without glitches, so the problem of how to mathematically describe gravity has to be with our ability to use the right math to describe it and not with the fundamental unsoundness of the theory.

Also, the theoretical and phenomenological study of gravity includes many hypothetical very subtle tweaks to Einstein's Field Equations, such as, for example, "Conformal Gravity" or theories that consider torsion in a way not found in basic GR. Other examples are Whitehead's theory, Brans–Dicke theory, teleparallelism, f(R) gravity and Einstein–Cartan theory.

It could be that one of the problems with our efforts to develop a theory of quantum gravity is that the classical theory we are trying to find a quantum gravity counterpart for, is actually the wrong one in some seemingly inconsequential way that throws off the rigorous mathematical enterprise that is crafting a quantum theory of gravity. As an analogy, if you tried to create a quantum theory of Newtonian gravity, you could probably do it, but the end product would be wrong and might even have pathological issues in a rigorous mathematical analysis or a comparison to astronomy observables, because the classical analogy isn't quite right.

More generally, gravity seems to behave in a manner that is significantly more constrained than an unthoughtful quantization of Einstein Field Equations on their face would suggest.

Thus, for example, one fruitful path of inquiry into quantum gravity has been the QCD squared approach that finds that in many cases, setting up an analogous QCD problem to a gravity problem, and then squaring it, produces correct results.

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• gentzen and WernerQH
why not create a Gauge Theory that is gravity

why not create a Gauge Theory that is gravity
Lots of people would like to do so. But: as noted above "Despite major efforts, no complete and consistent theory of quantum gravity is currently known". The problem of figuring out how to do so is wickedly hard and has so far resisted all attempts create a gauge theory of gravity.

Lots of people would like to do so. But: as noted above "Despite major efforts, no complete and consistent theory of quantum gravity is currently known". The problem of figuring out how to do so is wickedly hard and has so far resisted all attempts create a gauge theory of gravity.
there is Ashtekar variables.

i also think of a Yang–Mills theory that reproduce gr

there is Ashtekar variables.

i also think of a Yang–Mills theory that reproduce gr
There have been efforts to come up with a quantum gravity theory with those tools. Nobody has managed to make it work operationally.

I also saw an answer that background dependency is an issue. How can I see this mathematically in the equations?
QM and QFT requires a background spacetime, from which measurements/experiments are made, in order to even be formulated, as this is where the "statistics" is also collected. The classical meaning of "background" is the coordinate frame that is naturally associated to an moving gedanken observer. But the "observer" is traditionally not given any other attributes beyond the associated frame of reference.

QFT extends QM to make sense for all intertial frames of SR. One can also consider what happens with curved backgrounds to some extent, but the bigger problem is that in GR the background metric is not fixed in a background lab, it's a dynamical object subject to evolution as per einsteins field equations.

This is both a conceptual and technical problem, differen research programs can be distinguished by their stances, like what do you need to twist to make it work, and wether one thinks we can solve QG independently from the full unification, or wether it goes together.

/Fredrik

• ohwilleke and kodama
QM and QFT requires a background spacetime, from which measurements/experiments are made, in order to even be formulated, as this is where the "statistics" is also collected. The classical meaning of "background" is the coordinate frame that is naturally associated to an moving gedanken observer. But the "observer" is traditionally not given any other attributes beyond the associated frame of reference.

QFT extends QM to make sense for all intertial frames of SR. One can also consider what happens with curved backgrounds to some extent, but the bigger problem is that in GR the background metric is not fixed in a background lab, it's a dynamical object subject to evolution as per einsteins field equations.

This is both a conceptual and technical problem, differen research programs can be distinguished by their stances, like what do you need to twist to make it work, and wether one thinks we can solve QG independently from the full unification, or wether it goes together.

/Fredrik
does string theory work with curved backgrounds

Superstringtheory works in 10 dimensions (11 in M-theory) so one issue is as we are used to only 4, howto hide the extra dimensions? This is the landscape problem. Another issue is that one initially chose background spacetime you put into the theory when formulating it, can be "excited", but as of yesterday the non-perturbative string theory and M-theory are not well understood so it's not at all clear to me if the very conceptual idea of how spacetime actually emerges dynamically in string theory will work out in the end, in a way that works out with standardmodel of particles physics. This would likely be related to a lot of the dualities and some interpretations of what they really mean. Perturbative string theory is I typically made around a flat background, but just like in QFT you can in principle try to perturb around a curved background as well.

So perturbative string theory is clearly background dependent, just like QFT and adds nothing to the conceptual QM foundations as far as I see. OTOH, it was not the ambition of string theory either.

String theory is certainly ambitious, which is good, but it is badly plauged by fine tuning due to it's large parameter space or solution space (depending if you talk about M-theory or not). My main issue with this, is that I think it is a problem that a paradigm generates more variety without a matching selection principle. My as far as I know it's yet missing.

plenty to read on this, one is here https://arxiv.org/abs/gr-qc/0410049
The well structured nLab pages also has a string FAQ https://ncatlab.org/nlab/show/string+theory+FAQ
Beyond this, resident string theorists on here that can give more faithful answers.

/Fredrik

• ohwilleke
does string theory work with curved backgrounds
Yes. You can build up a coherent states of gravitons by inserting the appropriate vertex operators on the world sheet.

• kodama and ohwilleke
Yes. You can build up a coherent states of gravitons by inserting the appropriate vertex operators on the world sheet.

Wouldn't this interact with the 6 extra dimensions that are curled up in a way that disagrees with observation?

Wouldn't this interact with the 6 extra dimensions that are curled up in a way that disagrees with observation?
If I understand your question correctly: that's a problem known as moduli stabilization. You already encounter it in 5D Kaluza Klein theory, where you need fluxes from the electromagnetic field to counter the tendency of the circular dimension to blow up or disappear.

More generally, gravity seems to behave in a manner that is significantly more constrained than an unthoughtful quantization of Einstein Field Equations on their face would suggest.

Thus, for example, one fruitful path of inquiry into quantum gravity has been the QCD squared approach that finds that in many cases, setting up an analogous QCD problem to a gravity problem, and then squaring it, produces correct results.
What is "QCD squared"? How does it mimic gravity? Do you have a reference? It sounds interesting.

This is good video that explains it all • • • pinball1970, Demystifier, DennisN and 1 other person
What is "QCD squared"? How does it mimic gravity? Do you have a reference? It sounds interesting.
See "Gravity is Yang-Mills Squared" by 4Gravitons, which notes:

for tree diagrams the message of this post is as literal as it can be. Using something called the Kawai-Lewellen-Tye relations, the result of a tree diagram calculation in gravity can be found just by taking a similar calculation in Yang-Mills and squaring it.

(Interestingly enough, these relations were originally discovered using string theory, but they don’t require string theory to work. It’s yet another example of how string theory functions as a laboratory to make discoveries about quantum field theory.)

Does this hold beyond tree diagrams? As it turns out, the answer is again yes!
The calculation involved is a little more complicated, but as discovered by Zvi Bern, John Joseph Carrasco, and Henrik Johansson, if you can get your calculation in Yang-Mills into the right format then all you need to do is square the right thing at the right step to get gravity, even for diagrams with loops!

• gentzen and WernerQH
I'm not sure if I'll be able to absorb this, but it is very interesting. Thank you very much!

• ohwilleke
• Demystifier
Is there an answer possible why string theory is a quantum theory of gravity, but I cannot get the Einstein field equations from it?
Sure you can get the Einstein field equations from string theory. One way they arise is as consistency equations to make the Weyl anomaly vanish.
I don't understand how string theory is one theory of quantum gravity and yet not the accepted solution to quantum gravity.
Note that string theory is not just a theory of quantum gravity, it is a TOE (Theory of Everything), meaning it does not only describe (quantum) gravity, but the other three fundamental interactions as well. However, string theory still suffers from a number of phenomenological and technical problems, that's why it is not yet the accepted solution.

• PeroK