What are the challenges with unifying gravity and quantum theory?

[Mr-T]

I am curious to know what the troubles are with unifying gravity and quantum theory.

 

[phinds]

I am curious to know what the troubles are with unifying gravity and quantum theory.

A trivial search on Google turns up this:

Quantum gravity is the effort in theoretical physics to create a theory that includes both general relativity and the standard model of particle physics. Currently, these two theories describe different scales of nature and attempts to explore the scale where they overlap yield results that don't quite make sense, like the force of gravity (or curvature of spacetime) becoming infinite. (After all, physicists never see real infinities in nature, nor do they want to!)

 

[Chronos]

Quantum gravity is a key element in creating a superunified theory [unify gravity with the other fundamental forces of nature]. The other forces submit to quantum treatments that produce well behaved calculations that match observations with great precision. Physicist desperately desire to do this with gravity. It has, however, thus far has resisted all efforts at renormalization, which is necessary to avoid infinities from emerging in the calculations. Physicists are allergic to infinities, as Phinds noted.

 

[atyy]

We do have a quantum theory of gravity that works at low energies, and matches observations. The problem is at high energies, near the Planck scale.

The gauge/gravity duality is conjectured to be a complete theory of quantum gravity at all energies. It does not however seem to describe our universe. It is hoped that studying it will help us generalize appropriately.

 

[Ilja]

Physicist desperately desire to do this with gravity. It has, however, thus far has resisted all efforts at renormalization, which is necessary to avoid infinities from emerging in the calculations.

Given the modern, effective field theory approach to renormalization, renormalizability is not necessary at all.

All you have to give up is the hope to obtain a theory which is valid for all distances. Instead, you have to accept that the theory is valid only for distances larger than a yet unknown critical distance.

So, all one has to give up is an utopian illusion. Because we are anyway unable to probe whatever theory for arbitrary small distances. So we cannot be sure, anyway, that there is no critical distance where the theory becomes invalid.

Moreover, given that general relativity contains infinities already in its classical version, it makes not much sense to hope that some quantum gravity based on GR will be free of infinities.

 

[tom.stoer]

physicist desperately desire to do this with gravity. It has, however, thus far has resisted all efforts at renormalization.

I don't agree. It is true that there is no known way towards a perturbative renormalization of gravity. But there are hints that gravity can be renormalized non-perturbatively. This would correspond to the existence of a non-Gaussian fixpoint and asymptotic safety (instead of a Gaussian fixpoint at G=0 and asymptotic freedom).

 

[A. Neumaier]

Difficulties in quantizing gravity

I am curious to know what the troubles are with unifying gravity and quantum theory.

Difficulties in quantizing gravity:

(i) (mathematical) No consistent interaction relativistic quantum
field theory is known in 4 dimensions.

(ii) (theoretical) The accepted ways to avoid divergences in
expressions for scattering amplitudes that work in simpler theories
all fail because of the lack of renormalizability. See, e.g.,
the references in Section 2.2 of
http://relativity.livingreviews.org/Articles/lrr-2002-5/

(iii) (theoretical) The theories for which a (perturbatively)
finite scattering theory is available have not been related
quantitatively to the established theories.
A convincing classical limit (to general relativity),
nonrelativistic limit (to a multiparticle Schroedinger equation with
Newtonian interaction), and low energy limit (at currently accessible
energies no new particles apart from the graviton) would be needed.

(iv) (conceptual) The three limits pose severe constraints on possible
quantum gravity theories, and it requires much imagination to come up
with a conceptual basis in which these limit make sense and are
tractable.

(v) (experimental) Quantum effects in gravity are so weak that no
experiments sensitive to quantum effects are in reach in the near
future, and the data from astronomy that may cast light on quantum
gravity are scarce. (Quantum gravity is not demanded by unexplained
data but only by the quest for consistency with particle physics.)

(Much more information can be found in the chapter on quantum gravity of my theoretical physics FAQ.)

 

[A. Neumaier]

It is true that there is no known way towards a perturbative renormalization of gravity.

This is not really true.
http://arxiv.org/pdf/1306.1058.pdf gives a fairly rigorous perturbative renormalization of gravity (as an effective theory).

 

[marcus]

This is not really true.
http://arxiv.org/pdf/1306.1058.pdf gives a fairly rigorous perturbative renormalization of gravity (as an effective theory).

Page 5 has an intriguing reference to "Convenient Calculus" as presented in AMS Survey 53, the book by Kriegl and Michor.
ABSTRACT. The aim of this book is to lay foundations of differential calculus in infinite dimensions and to discuss those applications in infinite dimensional differential geometry and global analysis which do not involve Sobolev completions and fixed point theory. The approach is very simple: A mapping is called smooth if it maps smooth curves to smooth curves. All other properties are proved results and not assumptions: Like chain rule, existence and linearity of derivatives, powerful smooth uniformly boundedness theorems are available. Up to Frechet spaces this notion of smoothness coincides with all known reasonable concepts. In the same spirit calculus of holo- morphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesies on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations.

This link has the abstract, the TOC, and references.
http://www.ams.org/books/surv/053/surv053-endmatter.pdf
Is there an on-line copy of the book itself?
I'd like to get out of making a special trip to the library just to look it over.

 

[marcus]

This is not really true.
http://arxiv.org/pdf/1306.1058.pdf gives a fairly rigorous perturbative renormalization of gravity (as an effective theory).

This article by Brunetti Fredenhagen Rejzner is interesting enough we should have the abstract to look at together:

http://arxiv.org/abs/1306.1058
Quantum gravity from the point of view of locally covariant quantum field theory
Romeo Brunetti, Klaus Fredenhagen, Katarzyna Rejzner
(Submitted on 5 Jun 2013)
We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin-Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.
51 pages, in this version: proof of the background independence corrected

This is new to me--I am just taking a first look. As a superficial first reaction, one thing like immediately about Brunetti Fredenhagen Rejzner is the frank down-to-earth style of their introduction section. It makes me think there is after all NOTHING wrong with gravity. (Statements about geometry are operationally meaningless below a certain length scale but we already heard about that from Loop gravity people and others--it is a familiar idea and accepted by many, if not all.)

 

[marcus]

The introduction of Brunetti Fredenhagen Rejzner gives a clear perspective on Quantum Gravity which I haven't heard much about lately. It's worth quoting in part.

==excerpts from introduction http://arxiv.org/pdf/1306.1058v3.pdf ==
The incorporation of gravity into quantum theory is one of the great challenges of physics. The last decades were dominated by attempts to reach this goal by rather radical new concepts, the best known being string theory and loop quantum gravity. A more conservative approach via quantum field theory was originally considered to be hopeless because of severe conceptual and technical problems. In the meantime it became clear that also the other attempts meet enormous problems, and it might be worthwhile to reconsider the quantum field theoretical approach. Actually, there are indications that the obstacles in this approach are less heavy than originally expected.

One of these obstacles is perturbative non-renormalisability [66, 73] which actually means that the counter terms arising in higher order of perturbation theory cannot be taken into account by readjusting the parameters in the Lagrangian. Nevertheless, theories with this property can be considered as effective theories with the property that only finitely many parameters have to be considered below a fixed energy scale [76]. Moreover, it may be that the theory is actually asymptotically safe in the sense that there is an ultraviolet fixed point of the renormalisation group flow with only finitely many relevant directions [75]. Results supporting this perspective have been obtained by Reuter et al. [64, 65].

Another obstacle is the incorporation of the principle of general covariance. Quantum field theory is traditionally based on the symmetry group of Minkowski space, the Poincaré group. In particular, the concept of particles with the associated notions of a vacuum (absence of particles) and scattering states heavily relies on Poincaré symmetry. Quantum field theory on curved spacetime which might be considered as an intermediate step towards quantum gravity already has no distinguished particle interpretation. In fact, one of the most spectacular results of quantum field theory on curved space times is Hawking’s prediction of black hole evaporation [43], a result which may be understood as a consequence of different particle interpretations in different regions of spacetime. (For a field theoretical derivation of the Hawking effect see [32].)

Quantum field theory on curved spacetime is nowadays well understood. This success is based on a consequent use of appropriate concepts. First of all, one has to base the theory on the principles of algebraic quantum field theory since there does not exist a distinguished Hilbert space of states. In particular, all structures are formulated in terms of local quantities. Global properties of spacetime do not enter the construction of the algebra of observables. They become relevant in the analysis of the space of states whose interpretation up to now is less well understood. It is at this point where the concept of particles becomes important if the spacetime under consideration has asymptotic regions similar to Minkowski space. Renormalization can be done without invoking any regularization by the methods of causal perturbation theory [28]. Originally these methods made use of properties of a Fock space representation, but could be generalized to a formalism based on algebraic structures on a space of functionals of classical field configurations where the problem of singularities can be treated by methods of microlocal analysis [13, 11, 45]. The lack of isometries in the generic case could be a problem for a comparison of renormalisation conditions at different points of spacetime. But this problem could be overcome by requiring local covariance, a principle, which relates theories at different spacetimes. The arising theory is already generally covariant and includes all typical quantum field theoretical models with the exception of supersymmetric theories (since supersymmetry implies the existence of a large group of isometries (Poincaré group or Anti de Sitter group)). See [14, 16] for more details.

It is the aim of this paper to extend this approach to gravity. But here there seems to be a conceptual obstacle. As discussed above, a successful treatment of quantum field theory on generic spacetimes requires the use of local observables, but unfortunately there are no diffeomorphism invariant localized functionals of the dynamical degrees of freedom (the metric in pure gravity). The way out is to replace the requirement of invariance by covariance which amounts to consider partial observables in the sense of [67, 22, 70].

Because of its huge group of symmetries the quantization of gravity is plagued by problems known from gauge theories, and a construction seems to require the introduction of redundant quantities which at the end have to be removed. In perturbation theory the Batalin-Vilkovisky (BV) approach [3, 4] has turned out to be the most systematic method, generalizing the BRST approach [5, 6, 72]. In the BV approach one constructs at the end the algebra of observables as a cohomology of a certain differential…
==endquote==

 

[Naty1]

Carlo Rovelli has some thoughts on unifying GR and QM:
He seems to feel there are major conceptual problems in our way...

Unfinished revolution

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

Introductive chapter of a book on Quantum Gravity, edited by Daniele Oriti,
to appear with Cambridge University Press

Carlo Rovelli
Centre de Physique Th´eorique de Luminy_, case 907, F-13288 Marseille, EU
February 3, 2008

A few excerpts:

In spite of their empirical success, GR and QM offer a schizophrenic and confused understanding of the physical world. The conceptual foundations of classical GR are contradicted by QM and the conceptual foundation of conventional QFT are contradicted by GR.

To make sense of the world at the Planck scale, and to find a consistent conceptual framework for GR and QM, we might have to give up the notion of time altogether

The key difficulty of quantum gravity may therefore be to find a way to understand the physical world in the absence of the familiar stage of space and time. ... quantum states must themselves determine and define a spacetime —in the manner in which the classical solutions of GR do.
Conceptually, the key question is whether or not it is logically possible to understand the world in the absence of fundamental notions of time and time evolution, and whether or not this is consistent with our experience of the world.

The difficulties of quantum gravity are indeed largely conceptual. Progress in quantum gravity cannot be just technical. The search for a quantum theory of gravity raises once more old questions such as: What is space? What is time? What is the meaning of “moving”? Is motion to be defined with respect to objects or with respect to space? And also: What is causality? What is the role of the observer in physics? Questions of this kind have played a central role in periods of major advances in physics.

 

[Naty1]

Quantum field theory on curved spacetime is nowadays well understood.

Can someone help me understand the distinction between this statement and the fact that the Standard Model with QFT is still [presumably] in flat [Minkowski] space?

Isn't the OP question ".. troubles ...with unifying gravity and quantum theory.."

equivalent to asking
"..trouble with unifying GR and the Standard Model..."

 

[tom.stoer]

"Quantum field theory on curved spacetime" is QFT on classical spacetime. "Unifying QFT / SM and gravity" means quantizing gravity.

I think there are rather general reasons why quantum field theory on classical spacetime is incomplete.

 

[marcus]

Naty, Tom answered your question. I will just add a few words expanding on what he said.

"Quantum field theory on curved spacetime" is QFT on classical spacetime. "Unifying QFT / SM and gravity" means quantizing gravity.

I think there are rather general reasons why quantum field theory on classical spacetime is incomplete.

QFT on curved spacetime (that Brunetti et al say is nowadays well-understood) is not QG because the matter does not react back on the geometry. You start with a FIXED curved spacetime geometry and you put matter fields on it and run the fields on this fixed curved environment. But the fields are not affecting the geometry the way that GR says they must!

That could be one of the "general reasons" that Tom has in mind for why QFT on curved spacetime isn't an adequate picture of reality. Plus there's the shortcoming that the geometry is described classically, the curved spacetime shape itself should be a quantum field interacting with the other quantum fields.

 

[tom.stoer]

The simplest reason is the following: suppose there is a spherically symmetric J=0 quantum state (e.g. a neutral pion) decaying into other particles. Suppose there is a spherically symmetric detector array. When preparing the experiment, spacetime is spherically symmetric, too. When the detector registeres the particles from the decay their quantum state "jumps" (*) to a state which is no longer spherically symmetric. Therefore the (induced) gravitational field must "jump" as well. But this "jump" is inconsistent with GR.

The conclusion is that both quantum fields and spacetime have to be described using a compatible formalism which is not "QFT + classical spacetime".

(*) "jump" means any process like collapse, decoherence, ...

 

[Naty1]

Marcus, Tom,

thanks for the last two posts...very helpful...

Checking back in my notes I found these apparently INCORRECT quotes from prior discussions in these forums... and that is what led me astray:

I did not record the sources but state them here in case others read them elsewhere in these forums.

" QFT is a synthesis of special relativity with quantum mechanics. Continuous QFT yields quantum particles and forces of the standard model of particle physics.

Quantum Field Theory is developed on flat Minkowski space with no gravity in the SM. It predicts an embarrassingly huge vacuum energy which we do not observe. "

Instead, I should be thinking about the Standard Model as QFT on a fixed classical curved spacetime geometry. So we need consistent quantized field descriptions...

Which raises the question why some in these forums insist spacetime is continuous rather than discrete, but we have been through that several times...

 

[A. Neumaier]

Page 5 has an intriguing reference to "Convenient Calculus" as presented in AMS Survey 53, the book by Kriegl and Michor.
Is there an on-line copy of the book itself?
I'd like to get out of making a special trip to the library just to look it over.

http://www.mat.univie.ac.at/~kriegl/Skripten/apbook.pdf

By the way, both Kriegl and Michor are like me from the University of Vienna, Austria. Just a few floors away from my office.

 

[martinbn]

The simplest reason is the following: suppose there is a spherically symmetric J=0 quantum state (e.g. a neutral pion) decaying into other particles. Suppose there is a spherically symmetric detector array. When preparing the experiment, spacetime is spherically symmetric, too. When the detector registeres the particles from the decay their quantum state "jumps" (*) to a state which is no longer spherically symmetric. Therefore the (induced) gravitational field must "jump" as well. But this "jump" is inconsistent with GR.

The conclusion is that both quantum fields and spacetime have to be described using a compatible formalism which is not "QFT + classical spacetime".

(*) "jump" means any process like collapse, decoherence, ...

This is not very convincing. Why should the quantum state (wave function) affect gravity? Imagine a Bohmian type formulation. The particle has a definite position and momentum, and any number of other properties, at any given time and they determine the geometry of spacetime, not the guiding wave. The wave only tells the particle how to move but does not appear in any way in the equations that relate matter an geometry (Einstein's equations or whatever modification is needed). Also you explicitly use time, not that there is anything wrong with that, but can you formulate the scenario only in terms of the spacetime manifold in a coordinate free way?

What Marcus said in post #15, is much more convincing, at least to me.

 

[tom.stoer]

I agree that you may be right in a dBB theory.

But except for that all other QM interpretations deal with states / wave functions only. In addition I have never seen any dBB-like theory to apply for general covariant QFT which requires fields / quantum fields. The Einstein equations formally G[geometry] = T[energy-momentum density] and I do not see how to formulate the r.h.s. in terms of particles instead of fields.

 

[martinbn]

I agree that you may be right in a dBB theory.

But except for that all other QM interpretations deal with states / wave functions only. In addition I have never seen any dBB-like theory to apply for general covariant QFT which requires fields / quantum fields. The Einstein equations formally G[geometry] = T[energy-momentum density] and I do not see how to formulate the r.h.s. in terms of particles instead of fields.

Yes, they deal with quantum states, but also with all kinds of other things, for example observables i.e. energy, momentum, spin...Why should the quantum state affect the geometry. My question is why should the right hand side of the equation involve in some way the wave function? It seems to me that your argument requires something like that. The right side may include energy, momentum and things of that sort but not the state vector. Of course that is still unclear given that they are operators.

 

[tom.stoer]

if you do classical field theory, e.g. GR with electrodynamics, the energy-momentum tensor T^ab is defined via the fields E, B and the current density j (and teh metric g)

combining RT with some sort of quantum (field) theory: how do you define the energy-momentum tensor T^ab w/o using the quantum state, its expectation value or something like that? what replaces the expression known from classical field theory?

what I am saying is that there are two options:
1) replace G=T by G=<T> where G is still classical, T is an operator expression and <T> is its expectation value; this leads to the above mentioned problem that <T> is subject to quantum jumps
2) replace G=T by an operator expression for both G and T, i.e. use QFT on the r.h.s. and QG on the l.h.s

what is your idea?

 

[martinbn]

I don't have any ideas, I am just trying to understand your argument. It seems to me that you insist that the right hand side T, what replaces it, must depend on the wave function. That may be obvious but I don't see it yet. Also you example suggests that the quantum jumps happen in time, at the moments of detection. So in some way the right hand side must be a function of t, which I don't know how to make sense of.

 

[tom.stoer]

In GR the tensor density T is a function of the 4-vector (t,x). In QG the operator T will still be a function of this 4-vector. In a canonical formulation T is defined over a spatial 3-foliation and will of course depend on time.

And of course the "quantum jumps" depend on time, regardless which interpretation you use (in a collaps interpretation the state collapses from a state |π°> to |2γ>; in the MWI it branches into several different "worlds", ...)

If T does not depend on the state, then please tell me on what else it could depend? How does a formal expression look like?

 

[ChrisVer]

What is the problem of GR and QFT?
I found these in Beckers and Scwarzs' book for String and M theory.
"In QFT one requires that two fields that are defined at space-time points with a space-like separation should commute (or anticommute if they are fermionic). In the gravitational context one doesn't know whether or not two space-time points have a space-like separation until the metric has been computed, which is part of the dynamical problem. Worse yet, the metric is subject to quantum fluctuations just like other quantum fields.
[...]
The most straightforward attempts to combine quantum mechanics and general relativity, in the framework of perturbative quantum field theory, run into problems due to uncontrollable infinities. UV divergences are a characteristic feature of radiative corrections to gravitational processes, and they become worse at each order in perturbation theory. Because Newton's constant is proportional to (Length)^2 in 4D, simple powercounting arguments (and detailed calculations) show that it's not possible to remove these infinities by the conventional renormalization methods of QFT."

The problem of dimensionful coupling constant is a problem for QFT and renormalization. The same problem also appeared in Fermi's 4point interaction. In that model, you had a neutron coming in and in a single point interacting giving the beta decay. The coupling constant wasn't dimensionless...We know today that by adding the degree of freedom of the interchanged boson, fixed that UV divergences and renormalized the theory...
In concept of the string theory, we are doing something in analogy... We add the degrees of freedom of the oscillation modes, and we get rid of point particle interactions, thus the UV divergencies of the Quantum Gravity disappear and the theory is somewhat renormalizable (in fact you get a lot of interaction terms, but you somehow can correlate them)

 

[tom.stoer]

This summarizes problems due to perturbative renormalization. It ignores the fact that non-perturbative renormalization (and asymptotic safety) may be a perfect solution.

 

[ChrisVer]

What do you mean with "non perturbative renormalization"?
The essence of renormalization is within the perturbative theory, and QFT should behave well under it, that means that you can somehow add up all possible interactions without inserting infinite parameters in your theory... The problem of QG is exactly that- you are unable to do that, so it's not a predictable theory as it is...

 

[tom.stoer]

The essence of renormalization is within the perturbative theory, ...

Renormalization is a much broader and deeper idea than just perturbation theory (which trivially fails for gravity, and which fails for other theories as well, e.g. for QCD in the low-energy regime). Renormalization says that given a theory at some energy scale E you can derive its properties at some other scale E' via renormalizing couplings g(E) and other parameters (no perturbation theory required).

Perturbation theory is used in the sense that at a given scale E you do perturbation theory and calculate the renormalized couplings at E'. So you have to assume that g(E) = 0 and g(E') = 0 are both valid starting points. In the case of QCD this is fine in the UV b/c g=0 g(E) → 0 for E → ∞ (this is asymptotic freedom).

But in the case of gravity there are indications that g(E) and other parameters do not run to 0 but to some other finite but non-zero value (this is asymptotic safety). Therefore g(E) = 0 is not valid as a starting point; and therefore perturbation theory must fail (in the UV).

Have a look at http://en.wikipedia.org/wiki/Asymptotic_safety_in_quantum_gravity and references therein

 

[martinbn]

I don't mean to be stubbornly stupid, but I still don't get the argument.

In GR the tensor density T is a function of the 4-vector (t,x). In QG the operator T will still be a function of this 4-vector.

This doesn't really make sense. Are you saying the spacetime manifold has to have a vector space structure?

In a canonical formulation T is defined over a spatial 3-foliation and will of course depend on time.

Yes, but this "time" is just a mathematical function, which need not have anything to do with what clocks measure.

And of course the "quantum jumps" depend on time, regardless which interpretation you use (in a collaps interpretation the state collapses from a state |π°> to |2γ>; in the MWI it branches into several different "worlds", ...)

Exactly, and time is not a natural concept in GR. So to me it is not clear whether this is a problem with the argument you present.

If T does not depend on the state, then please tell me on what else it could depend? How does a formal expression look like?

I don't know, but that is not enough to claim that it should depend on the state. In fact it is not at all clear what quantum state should be in this context. How do you assign a Hilbert space to a given quantum mechanical system? Does it depend on the underleing spacetime?

 

[tom.stoer]

Are you saying the spacetime manifold has to have a vector space structure?

...

Yes, but this "time" is just a mathematical function, which need not have anything to do with what clocks measure.

...

Exactly, and time is not a natural concept in GR. So to me it is not clear whether this is a problem with the argument you present.

I can present the same arguments using local observers (an observer field); so locally you can define a time as proper time of observers. And these observers can agree on the above mentioned experiment. So each observer will have to deal with a quantum jump and a smooth manifold. But this is contradictory. This is all I want to explain,

I don't know, but that is not enough to claim that it should depend on the state. In fact it is not at all clear what quantum state should be in this context. How do you assign a Hilbert space to a given quantum mechanical system? Does it depend on the underleing spacetime?

You have a Hilbert space describing a spherically symmetric quantum mechanical System. Then via a decay plus detection becomes non-asymmetric. You have to describe how this happens.

In QFT using SR it's simple b/c the quantum state (or its energy-momentum tensor defined as expectation value) does not affect spacetime. In QFT using GR the classical energy-momentum tensor affects spacetime. You have to explain how. If you calculate the expectation value <T> before and after the observation then you will find that it's no longer a smooth function in your local coordinate patch; there's a jump simply b/c before the observation you have an S-wave state, but after the observation you have two localized particles registered by the detector array, therefore <T> is no longer spherically symmetric.

The first conclusion is that due to quantum effects the classical Einstein equation G = T has to be modified; the r.h.s. cannot be an entirely classical quantity. The second conclusion is that due to the quantum jump observed by local observers the equation G = <T> cannot be correct. That's all what I am saying.

So the usual way out is to quantize both G and T, which we call quantum gravity, and for which we have a couple of research programs.

What else do you propose?