Thursday, September 8, 2016

Taming Quantum Gravity

An Approach To Taming Quantum Gravity

Quantum gravity is a Holy Grail of fundamental physics that has eluded physicists for a century. This post considers a theoretical program that could make progress towards this problem by sacrificing theoretical purity in favor of results that produce useful actual calculations.

So far as I know, this strategy has not been heavily utilized by investigators in the fields of gravity and quantum mechanics, even though some interesting results have been generated in this fashion and even though similar approaches are used in QCD.  It is not even generally recognized as a distinct subfield in gravitational physics or quantum mechanics research.

I write this from a big picture perspective (one that professional researchers often lack the luxury to indulge in), fully aware that I am not competent to do much of the work suggested in this research program, and with no idea what this work would actually reveal if it was done. I'm not claiming to have solved the problem of quantum gravity that many greater minds that I have failed to master. Instead, I am suggesting a research program that is more humble in its ambitions than most of the research programs in quantum gravity currently producing publications, in the hope that investigators who are not once in a century geniuses might make useful progress with it by curbing their immediate ambitions and aspirations for their work

The Status Quo

For beginners, I provide some necessary background about the problem of quantum gravity in the context of fundamental physics and about the difficulties that have arisen in formulating a theory of quantum gravity below the fold.

The bottom line conclusion of that analysis is that the principle practical barrier to formulating a theory of quantum gravity based upon the quantum mechanics of a massless spin-2 graviton that has couplings to all fundamental particles proportional to their mass-energy is it cannot be renormalized, which means that you can't make any quantum mechanical calculations with a theory of this kind.

Is Modeling Quantum Gravity Really An Insurmountable Problem?

This seems crazy.

We know for a fact that Nature's true theory of quantum gravity doesn't actually create infinities (at least outside the singularities implied by general relativity and probably not even in all of those cases).

We know for a fact that outside some very highly specific circumstances, it is possible to use very simple calculations (typically using Newtonian gravity as a first order effect and a general relativistic correction, or by truncating classical general relativity in a manner that involves just one or two aspects of it with simplifying assumptions) to make extremely precise predications about all manner of gravitational phenomena.  People were making predictions with general relativity that have been confirmed by 2016 grade instrumentation back in the days when calculations were being done with slide rules and punch card computers.  We sent men to the moon and unmanned spacecraft to the limits to the solar system on exceedingly precise courses using gravitational calculations made with such meager computational power.

A Truncated Quantum Gravity Model Approach

How can this be overcome?

So, even if we are theoretically correct in assuming that Nature's gravity is governed by the quantum mechanics that flow automatically, naturally and elegantly from a massless spin-2 graviton that has couplings to all fundamental particles proportional to their mass-energy including their dynamical energy and preserving all vector components of that motion, from a practical perspective of developing a useable theory of quantum gravity, we are clearly doing it wrong.

There are a couple of instances, mostly in black hole physics, such as the physics of Hawking radiation from black holes, where we take some properties that we expect gravitons to have on an ad hoc basis to reach conclusions with an incomplete quantum gravity theory, but for the most part, we have not embarked on a comprehensive strategy of using incomplete or truncated aspects of a true quantum gravity to get something that is scientifically useful.

One plausible approach, which I can't recall seeing in any published papers or pre-prints is to follow the example of QCD and do calculations for practical purposes not using the complete version of the equations that we believe to describe Nature, but instead, by using an incomplete toy model version of the true equations and then developing a method to assess how much imprecision and inaccuracy is introduced into the result by doing so.

A Quantum Newtonian Gravity Toy Model

As a first iteration of this process, one could formulate a quantum gravity theory that has as its classical limit, Newtonian gravity, rather than general relativity. This can be done with a spin-0 graviton that creates a scalar field. It wouldn't be a perfect approximation, because Newtonian gravity propagates instantaneously while its quantum gravitational version would propagate only at the speed of light.  But, this "error" is actually a good thing, because in general relativity, which better reflects Nature than Newtonian gravity, gravity also propagates at the speed of light.

Using a Newtonian approximation of gravity is justified in a wide range of applications and is routinely done in circumstances where the relativistic modification of the Newtonian result is small because Newtonian gravity is much easier to calculate with than general relativity. Newtonian gravity, for example, is perfectly adequate for even quite sophisticated Earth based applications (e.g. locating underground water and oil reservoirs based upon slight differences in the pull of gravity above them because these liquids have less mass than the solid rock that is elsewhere).  Newtonian gravity also provides an excellent approximation of general relativity in many body calculations in astronomy from the dynamics of planets in the solar system to the dynamics of bodies in asteroids to the dynamics of large many body systems like galaxies.

The Newtonian approximation isn't perfect of course. Using general relativity slightly tweaks the perihelion of Mercury, it gives rise to frame dragging effects in the vicinity of rotating Earth which are just at the limits of state of the art instrumentation to measure, it gives rise to subtle gravito-magnetic effects caused by fast moving heavy objects, it gives rise to gravitational lensing, its cosmological constant gives rise to "dark energy" effects that accelerate the expansion of the universe, it gives rise to gravitational red shifts of photons, it is necessary to model the early universe around the time of the Big Bang, to model the behavior of black holes, and to evaluate pretty much any system with very strong gravitational fields like neutron stars.

Most importantly, there is no good reason to think that quantum gravity is really relevant in any circumstance where even general relativity can be ignored, so in almost every application where Newtonian gravity is an adequate approximation, classical Newtonian gravity is a more practical solution.

But, on the other hand, a quantum gravity version of Newtonian gravity is easy to formulate and to calculate with and can be integrated seamlessly with the rest of the Standard Model.  Also, by estimating theoretical error from the truncation of the infinite series in quantum Newtonian gravity and then adding in the imprecision arising from comparing classical Newtonian gravity to general relativity in any given situation, we can make a very credible estimate of how much precision we are sacrificing when we use quantum Newtonian gravity relative to Nature.

Truncated Quantum Gravity Models Beyond Quantum Newtonian Gravity

A second iteration of this process illustrates why we might want to bother with this approach.

You see, our toy model doesn't have to have as its goal mere replication of Newtonian classical gravity and indeed as we've already seen, even quantum Newtonian gravity automatically fixes one of its flaws by making it propagate at the speed of light rather than instantaneously.

It isn't hard to tweak our quantum Newtonian gravity model to incorporate more features of general relativity (although not all of them) in a manner that can still be renormalized.

For example, it is elementary (and indeed, more natural) to allow our spin-0 graviton to couple not only to particles based upon their rest mass, but also to the energy of photons and gluons in proportion to the energy of these particles and without regard to their direction of motion, which would restore most gravitational lensing (but not necessarily gravitational red shifting) to our model.

We can, and cosmology theorists routinely do, model the cosmological constant of general relativity as a separate quantum mechanical scalar field called "dark energy", rather than as a part of the equations of gravity itself in general relativity.

One of the aspects of the graviton, shared with the gluon, that makes it so much harder to model than photons, is the fact that gravitons couple to gravitons just as gluons couple to gluons, while photons do not couple to other photons since photons lack electromagnetic charge. But, it would be interesting to see what a toy model spin-0 graviton similar to the quantum Newtonian gravity graviton except that it could couple to photons, gluons, spin-0 gravitons and dark energy bosons, would behave like.

I strongly suspect that it would be possible, either by adding in additional kinds of gravitons, perhaps with spin-1, or by tweaking the properties of our spin-0 graviton, to add in further relativistic corrections to quantum Newtonian gravity.

For example, in addition to the dark energy bosons, one could image a model in which one kind of graviton captures most of the gravitational effects of the linear momentum components of the stress-energy tensor to general relativistic gravity, another kind of graviton captures most of the gravitational effects of the angular momentum components of the stress-energy tensor to general relativistic gravity, a third kind of graviton captures directional field flux components of the stress-energy tensor to general relativistic gravity, and our quantum Newtonian gravity spin-0 boson captures the contributions of the rest mass component of the stress-energy tensor to general relativistic gravity.

Perhaps, to simplify the mathematics, the gravitons making vector contributions (i.e. linear momentum, angular momentum and flux) would be modeled as not having interactions with other gravitons, while the spin-0 graviton could be modeled as either interacting with itself, or even interacting with itself and with the other three gravitons making vector contributions to general relativistic gravity.

Admittedly, this approach would be inelegant and would still not capture the cross-interactions of components of the stress-energy tensor with each other and would probably even sacrifice some of the "sacred" foundational axioms of general relativity, for example, by somehow perhaps subtly introducing some kind of frame dependence.

But, it seems to me possible that one could capture not only the first order gravitational effects of general relativity that are reflected in a quantum Newtonian gravity toy model, but also a significant share of the post-Newtonian general relativistic effects of general relativity in such a model without sacrificing the practical necessity of a renormalizable field with which you can do calculations.

And, after all, even in classical general relativity, physicists almost never do analytical calculations using the full complexity of the theory.  Instead, the most widely used calculations truncate full general relativity by using symmetry and other non-physical assumptions and choices of inertial frames in which to do calculations that make the math of one particular aspect of general relativity tractable.

One could also develop means to accurately estimate the magnitude of aspects of general relativity that are ignored by this toy model.  This could be done by direct comparison with classical general relativity in either the toy model calculations actually being done, by direct comparison of the toy model with classical general relativity in the case of a simplified example likely to have comparable general relativistic effects, or most ambitiously by creating analytical expressions of the components of general relativity that are omitted from the quantum gravity toy model (which could be evaluating numerically even if they weren't possible to solve analytically).

What Could Be Achieved With Truncated Quantum Gravity Models?

Suppose then, that we do develop truncated quantum gravity models that deliberately omit the aspects of general relativity that make it impossible to renormalize, while estimating the magnitude of the effects of the omissions in some credible way.

The hope would be that these models could be used to create a workable substitute for Nature's quantum gravity, that is amenable to numerical calculation, whose phenomenology could be used to understand how true quantum gravity would behave in a significant class of circumstances in which (1) it is plausible that there are measurable quantum gravity effects, and (2) we can determine that the magnitude of the general relativistic effects omitted in the truncation are much smaller than the magnitude of the measurable quantum gravity effects in the truncation.

For example, I suspect that such a model could work quite well to evaluate the phenomenology that a true quantum gravity theory would have in weak gravitational fields relative to general relativity, which would have potentially widespread practical relevance in particle physics and astronomy. For example, very heavy exotic matter created in particle accelerators might create weak gravitational fields that are not so weak that they can be entirely ignored in precision calculations. And, these models could be used to create middle ground between using pure Newtonian gravity in many body galaxy and cosmology models and using full fledged classical general relativity which is often mathematically intractable.

Indeed, it might be possible, by using a variety of different truncations, to credibly measure the magnitude and nature of multiple quantum gravity effects (even in the same hypothetical physical system), even though it would be impossible to renormalize a quantum gravity model that included all of these general relativistic effects at the same time.

Most idealistically of all, one might hope that by making piecemeal progress in understanding what quantum gravity does and does not do in particular applied circumstances, that the scientific community could develop well founded expectations about quantum gravity phenomenology, and insights into the nature of the quantum mechanical calculations which could shed light on terms in the fully quantum gravity of Nature that are and are not important, that might lead to insights that make it possible to do full fledged quantum gravity of Nature calculations, for example, by developing good intuitions about how to cause immaterial terms in the full quantum gravity calculations to cancel out or to be segregated from important terms, in order to make full quantum gravity mathematically tractable.


There are several absolutely central unsolved problems in fundamental physics.

One is to fully characterize the oscillation properties, absolute rest masses, and nature of the masses of neutrinos. Another is to understand dark matter and dark energy. Another is to figure out how the quarks, charged leptons and neutrinos in the universe came to be in a manner that allows almost all of the quarks and charged leptons in the universe to be matter rather than antimatter when our laws of physics seem to insist that new quarks and leptons can be created only when antimatter counterparts are created as the same time. Another is to understand "inflation", a process hypothesized to explain why the universe is more homogeneous than a naive version of the Big Bang theory would suggest. Another is to explain the fundamental forces of the universe as arising from a single unitary mathematical structure.  Another is to explain from first principles far more of the constants of the Standard Model that are now determined experimentally.

And then, there is the subject of this post, coming up with a theory of quantum gravity that puts the force of gravity on a footing that is compatible with the three forces of the Standard Model.

Developing a theory of quantum gravity is something that has taunted scientists ever since Einstein made foundational discoveries leading to quantum mechanics (which is what he actually got his Nobel prize for) and came up with general relativity, a theory of gravity and space-time formulated classically (i.e. on a non-quantum mechanical basis) that remains the state of the art in the field a century later. Einstein spent a large share of his later years trying to unify general relativity with the rest of fundamental physics and failed. So have countless scientific geniuses who have followed in his footsteps.

One of the biggest breakthroughs was made when I was a little kid, shortly after Standard Model of particle physics reached its current form (except for the lack of neutrino masses), as part of the very early stages of research into the idea of string theory.

This was the discovery that it appeared that general relativity appeared to be the classical limit of the quantum mechanical properties of a massless spin-2 boson with a coupling strength proportionate to the mass-energy of the particles (including itself) with which it interacted, which was dubbed the "graviton."

The Trouble With The Graviton

What went wrong with the program of developing quantum gravity by generalizing the Standard Model to include a spin-2 graviton?

There are several big ones and as a practical matter, the third one is the one that is of a concern to us.

1. In general relativity, gravitational energy is conserved globally but is undefined locally and does not contribute to the stress-energy tensor of general relativity on an equal footing with all other sources of mass-energy in the universe. But, the whole idea of a "graviton" is that gravitational energy is localized and that the gravitational energy in gravitons gravitates in the same manner as any other particle.

2. General relativity is fundamentally "background independent" and works in the total absence of any coordinate system for space-time.  General relativity operationally describes gravity as the consequence of deformations of space-time caused by having mass-energy in it, rather than as a force which arises from interactions between particles via a force carrying boson the way that the three Standard Model forces do.

An alternative way to attempt to formulate a theory of quantum gravity (which also avoids the issues involving localization of gravitational energy to some extent) is to define quanta of space-time rather than realizing quantum gravity with a carrier boson like a graviton (although gravitons could be an emergent phenomenon in these theories, which are usually described as "loop quantum gravity" theories although there are a number of kindred approaches with different names like causal dynamical triangles). It isn't obvious that the graviton approach is truly background independent in the manner necessary to accurately reproduce general relativity.

3. A massless spin-2 graviton is not "renormalizable".

The Standard Model is build on a foundation of particles moving and interacting in a quantum mechanical manner.  In quantum mechanics, particles move in a manner computed via a "path integral".  A path integral calculates the probability that a particle will end up at each possible new location by considering every possible way that the particle could travel from its starting point to its destination (something that is in principle infinite) in a manner that weighs and sums up the relative probabilities of each possible path. In another interesting seeming contradiction with general relativity, the path integral of quantum mechanics for the photon must include paths at which a photon travels at more than, or less than, the speed of light, even though general relativity dictates that photons always travel at exactly the speed of light, although this detail of the calculation method doesn't show up in the final answer from the calculation which provides the probability that the particle will end up at the destination in question.

But, if you do this, the equations in the path integral blow up with infinities that can't be calculated.

A mathematical trick called "renormalization" provides a way to do those calculations that cancels out those infinities and produces answers which are consistent and match what we observe to extreme precision, even though there are actually myriad variants of the way that you do the "renormalization" whose specific details don't appear in the final result, all of which are equivalent to each other.

One fundamental consequence of renormalization which is reflected in the way that Nature behaves is that the strengths of the fundamental forces and the masses of the fundamental particles of the Standard Model change based upon the momentum transfer involved in the interaction being studies, which as usually described as "running" based upon the energy scale of the interaction.

Another non-fundamental practical reality of all path integral calculations (which also apply to interactions of particles as well as their movements in a bit more complex way) is that the ultimate formula used to determine the probability of something happening that arises from an evaluation of the proper path integral at the final stage before the answer is totaled up takes the form of an infinite series of terms whose terms correspond to the number of "loops" considered in the calculation. The one loop terms consider only the simplest and most direct paths.  The second loop terms consider paths with an additional level of complexity.  And so on.

The more loops you consider, the more accurate your result will be.

It turns out that in the case of electromagnetism (i.e. the quantum mechanics of photons being exchanged by electromagnetically charged particles), you can get extremely accurate answers by considering only a small number of loops because the terms in the infinite series get smaller very rapidly. In the case of the weak force (i.e. the quantum mechanics of W and Z bosons being exchanged by left parity particles and right parity antiparticles), the accuracy is still very good with a modest number of loops, but not quite as good as electromagnetism.  The calculations related to the Higgs boson's interactions likewise rapidly converge in a manner comparable to weak force interactions, which make sense because weak force interactions are in some respects special kinds of Higgs field interactions because W and Z bosons arise and acquire mass through the Higgs mechanism.

In the case of quantum chromodynamics (i.e. the quantum mechanics of quarks and gluons interacting with each other by exchanging gluons), the situation is much different. You can renormalize these quantum mechanical calculations, but the infinite series that results from this calculation converges much more slowly, so you need to consider far more terms to get a result with comparable accuracy. A next to next to next to leading order calculation (the current state of the art in QCD that considers terms through the first four loops) provides answers with a mere 1% to 0.1% or so precision, while comparable electroweak calculations would provide answers with precisions in the parts per trillion or more with that many calculations.  And, these calculations take months to do even when done by the most clever scientists added by state of the art computer resources.

Fortunately, most QCD calculations are one off affairs. Once you calculate how the strong force will behave in a particular situation, your done, and can reuse the result of your calculations regarding, for example, the mass of the proton calculated from first principles, or the probability that a D meson or sigma baryon will decay in a particular manner, that conclusion will hold for all time and never has to be repeated. The phenomena that QCD calculate are typically localized and repeat themselves over and over again in exactly the same way.

Indeed, the kind of path integral calculations used in electroweak theory only work for QCD at high energies and are called "perturbative QCD" calculations.

At low energies, the perturbative QCD approach, which analytically determines the exact result up to the number of loops considered is so cumbersome and inaccurate that instead of doing these continuous space-time calculations, a completely different math called "lattice QCD" is used. In lattice QCD, discrete particles are situated on a grid called at lattice with a hypothetical spacing between points and a discrete approximation of the Standard Model QCD equations is used to calculate the evolution of the particles in the gird over time.

Often, QCD calculations, especially in lattice QCD are done using incomplete versions of QCD that omit interactions with one or more kinds of heavy quarks that have only slight impact on low energy QCD interactions, assume for sake of calculation simplicity that up and down quarks have identical masses (which again only slightly impacts the results), do a series of calculations with different counterfactual mass scales for the pion (to extrapolate the trend to the physical pion mass) and/or assume that there are more QCD color charges than the three found in Nature (along with their three corresponding anti-color charges) and again extrapolate the trend as the number of color charges are reduced to the physical number of color charges.


Ryan said...

"Newtonian gravity also provides an excellent approximation of general relativity in many body calculations in astronomy from the dynamics of planets in the solar system to the dynamics of bodies in asteroids to the dynamics of large many body systems like galaxies."

It seems to break down entirely for large body systems like galaxies actually, does it not?

andrew said...

Newtonian gravity and GR predict essentially the same dynamics for galaxies. The observed dynamics differ either because both are wrong or because dark matter influences those dynamics.