Are standard QFT and general relativity contradictory?

Question

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.

Answer 1

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.

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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.

Answer 2

The apparent collision between Quantum Theory and General Relativity - their apparent schism - centers on the issue how they treat time, and I emphasize the word "apparent" for reasons you're about to see.

The B-plot to this story - how to combine classical geometry with quantum theory, which is closer to your question - I'll also discuss, address (and answer?) further below.

Quantum Theory seems (again: "apparent") to want to treat time as the arena of dynamics, distinguishing it in an essential way from spatial dimensions, and to treat quantum systems as states that evolve in time. This approach stems from its earliest days where mechanics was what was first quantized, rather than field theory.

In mechanics - almost by definition - systems are described by a set of variables that evolve in time. The arena for this is a one-dimensional time space - there is only one independent variable, and the relevant equations are total differential equations that describe how a system unfolds in this arena: i.e. they are "equations of motion".

In field theory, systems are described similarly, but there are now four independent variables. You see this most clearly in the case of electromagnetic theory, where all the spatial coordinates as well as the time coordinate are treated as independent variables, and the properties of the system (the field, itself) are treated as functions of all of those variables. The arena is four-dimensional. Correspondingly, the equations describe the unfolding of the system in the arena are now partial differential equations.

Quantization was eventually done with field theory, by the 1920's. This development largely coincided with the passing from non-relativistic quantum theory to relativistic quantum theory. The primary reason you have the (mechanics :: field theory) = (non-relativistic :: relativistic) alignment is because you simply can't have just one independent variable in Relativity! If you try, you'll immediately be confronted with the question: what is its transform under a Lorentz boost? That breaks the narrative.

Nevertheless, it is possible to make field theory look like mechanics by layering up the four-dimensional arena into a temporal sequence of three-dimensional slices. The only complication that Special Relativity adds in is that this layering is no longer unique - but it can still be done. So, the usual textbook formulation even of quantum field theory is as a mechanics that just happens to have an infinite number of variables: the value of the fields at each position on any of the three-dimensional slices is counted as a separate variable, for the purposes of treating the system like a glorified form of mechanics. You can set it up as a system with a countably infinite number of variables by using a three-dimensional version of "time-frequency" analysis or "time-scale" analysis (i.e. wavelets); or otherwise you can set it up as a system with an uncountably infinite number of variables using singular functions and distributions.

In both cases, quantum theory seems to force a treatment of time that clearly sets it apart, while in general relativity, no such distinction is mandated, and no such distinction may even be possible if no layering exists on the underlying geometry (i.e. if the geometry is not "globally hyperbolic"). At present there is no consensus on how to do quantum field theory as a mechanistic "evolving in time" system, if the underlying geometry is not globally hyperbolic. It might not even be possible.

Now ... I said "apparent". The "schism" between General Relativity and Quantum Theory on the treatment of time is only apparent. That's not what the schism actually is, nor where it lies. It's been misidentified and mislocated.

In fact, Quantum Theory does not force this kind of treatment of time. Only the Schrödinger Picture does. The Heisenberg Picture does not! In the Heisenberg Picture, the equations that describe the unfolding of systems treats all the independent variables equally, when the arena is four-dimensional. Field theory in the Heisenberg Picture does not unfold as a system "evolving in time", and is not described by total differential equations with respect to just one variable. The Heisenberg Equation - when done for field theory - reduces to a set of partial differential equations that treat all the independent variables equally.

So, the schism is mis-located. It is not

Quantum Theory versus General Relativity

but

{ Schrödinger Picture } versus { Heisenberg Picture and General Relativity }.

The schism is a split within Quantum Theory, not between Quantum Theory and General Relativity. It's more like an instance of The 30 Years War in the West (with the East out of the loop) than it is like The Latin Massacre & Sack Of Constantinople of East versus West. It only looked like a split between Quantum Theory and General Relativity, because too many people keep (falsely) equating Quantum Theory = Schrödinger Picture and keep forgetting about the Heisenberg Picture!

The other issues which underlie the General Relativity - Quantum Theory split ... which I've totally neglected up to this point and relegated to being nothing more than a B-plot in the main story ... center mostly on the matter of how a classical geometry is to intermesh with quantum theory, when key attributes of the arena, itself, and its geometry (particularly: the connection), are now part of the dynamics of the overall system to be described.

The closest thing we had to a consensus is that geometry is to (somehow) give way. I would even surmise that the person most responsible for ingraining that point of view into the community was Einstein himself. The very last (published) thing he ever wrote was the following:

One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory. (Eeeeeee, plop!)

(The sound of him keeling over with pen in hand.)

I think it's just a rabbit hole, though. The idea certainly underlay the development of Loop Quantum Gravity; though not as much String Theory, which ostensibly worked on a fixed classical background. (It's hard to tell, because the "theory" had the amorphous consistency of a blob that conveniently changed every few years and was impossible to get a firm grip on; though I'm aware that there were some who tried to put some kind of system to background back-reaction into it.)

The ultimate issue is: back-reaction. How do you get a quantum system to geometry back-reaction installed? If the geometry is not classical - particularly if the light-cone field (or whatever passes for one, in the modified geometry) is not fixed, then you have light cone fluctuation and can't even distinguish between space-like and time-like. That's a problem, because that determines what commutes and what doesn't. And, you'll get causality violation. I'm not even sure that you'd be able to distinguish between globally hyperbolic versus non globally hyperbolic, if the light cone field is also included in the quantum dynamics.

So, if geometry is included in the dynamics, you could literally have a superposition of two states - one in which quantum theory can be formulated, and the other in which it cannot, because it's not even globally hyperbolic.

An instance of "what passes for a light cone field" would be the A. A. Robb style "before-after" partial order that's postulated in the so-called "causal sets" approach. That counts as a case of making the light cone field part the background, even if it is to be discretized.

The light cone field is determined (in four or more dimensions) by the Weyl tensor. It fluctuates if and only if the Weyl tensor is a q-number - which means the metric too.

Gravitational waves are quantized as gravitons, if the wave modes are q-number. The Weyl tensor modes are, themselves, the wave modes. So, to say "quantum gravity" in this sense is to say "q-number Weyl tensor" and "light cone fluctuation".

The light cone field is classical, if the metric is a c-number, or the frame field if you're using Riemann-Cartan geometry, instead of (pseudo-)Riemannian geometry. In the case of something like causal sets, amend that to "whatever passes for a metric or frame" in that formalism.

If the light cone field fluctuates, then you can tunnel under it, and now you have de facto causality violation. Maybe you want that to make something like Warp Drive more feasible, but it's still an issue.

If it is fixed, then you have a c-number metric. Now, you're faced with the problem of hybridizing a classical system (all the parts of the geometry that are described by c-numbers, but are included in the dynamics) with a quantum system (the matter and energy fields). The central issue, here, is: how do you get back-reaction from the quantum system to the classical system?

Now, if you step back and think about this a while, a realization will suddenly dawn on you. Any classical system, whatsoever, that is on the receiving end of a quantum to classical back-reaction qualifies as a measurement device for the quantum system. That's practically the definition of a measurement device!

The problem of hybridizing quantum and classical systems is exactly the same issue as central problem of measurement theory: getting c-numbers of some kind or another, from devices, out of quantum systems. Hybridization is measurement is hybridization.

Any solution that successfully combines a c-number geometry with a q-number system must also serve as a de facto solution to measurement theory. Conversely, a solution to the hybridization problem can be used to force-graft a c-number geometry and a q-number system complete with quantum-to-classical back-reaction.

There have been many attempts at solving the hybridization problem, Penrose and Diosi being the best-known figures in this field. There have also been many No Go results that crash all the "obvious" methods for hybridizing classical and quantum systems, and these two have not emerged from this unscathed. (I will try to line up and add in a bibliography on all of this in a later edit.)

But, I would be remiss if I didn't mention the latest figure to pop up on the scene, who (so far) seems to have dodged the minefield in the No Go zone:

A Post-Quantum Theory Of Classical Gravity? - ArXiv preprint from 2018-2023

that got peer-review-stamped a few weeks ago

A Post-Quantum Theory Of Classical Gravity? - Phys. Rev. X

and apparently, it got open-accessed (that costs author/institute-money), which I didn't know until ... like ... now:

A Postquantum Theory of Classical Gravity? - The Actual PDF

along with a companion:

The constraints of post-quantum classical gravity

that kind of serves as a "Must Go" riposte to the hybridization "No Go"'s. The riposte has experimental consequences that can be probed with "desktop" setups. A setup is "desktop" if it doesn't involve large consortia with billion dollar systems in outer space or on the ground taking up large chunks of real estate. I am not certain if this is the paper that discusses the experimental consequences of hybridization, though it lays out constraints that have to be in place to enable hybridization.

All of this is Oppenheim. So, 2023 had Oppenheim, and the movie Oppenheimer, and all we need now is Oppenheimest.

He's using non-unitary dynamics, so it's filled with Lindblad-this, Lindblad-that, which is alien to most anything you see in textbook treatments of classical or quantum theory. How necessary that is (as opposed, say, to doing something with coherent states), I don't know.

Initially, in the earliest versions of his 2018-2023 preprint, he put his stakes down on "no quantum gravity" and "no gravitons", even citing two potential "no graviton" results:

R. Lieu, Classical and Quantum Gravity 35, 19LT02 (2018) and R. A. Norte, M. Forsch, A. Wallucks, I. Marinković,

and

S. Gröblacher, Phys. Rev. Lett. 121, 030405 (2018).

but that seems to have fallen by the wayside and in his more recent lectures he backtracked, falling back to the position that maybe the classical-quantum force-grafted hybridization being a "classical limit" of a quantum gravity theory, after all. Nevertheless, as he pointed out in a lecture, a hybridized non-unitary dynamics does not, in general, embed inside a larger unitary dynamics, primarily because what counts as "positive definiteness" in the former is weaker than what counts as "positive definiteness" in the latter and may not carry over.

Since the apparatus for non-unitary dynamics is alien to most eyes - at least in the theoretical community, but also on the receiving end of those getting the text-book presentations of classical or quantum physics - then it might help to see how it comes about. These are two derivations I've found and am reading up on

Simple Derivation of the Lindblad Equation - Pearle

and

A simple derivation of the Lindblad equation - Brasil et al.

Oppenheim's treatment applied the classical-quantum grafting directly onto the ADM development of General Relativity - its Hamiltonian formulation (which assumes the geometry is globally hyperbolic). He didn't do it directly with the Lagrangian formulation either of General Relativity or the Standard Model with gravity included. That would have been more interesting to see, and it's on the To Do list of anyone who wants to experiment with this formulation of hybridized dynamics.

Answer 3

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.

EDIT dec 20203. the link to Witten's presentation is broken. If you search with "witten feynman diagrams in string theory pitp " it will come up and also a number of video presentations.

Answer 4

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.