Why can't Einstein and Quantum Mechanics get along?

Despite our successes at describing the inner workings of the universe (Higgs, anyone?), there are still some gaping holes in our knowledge. Where's our Grand Unified Theory or our Theory of Everything? And why is Einstein's General Relativity still at odds with Quantum Mechanics? Why should we want to unify them anyway?

In this week's "Ask a Physicist," we'll find out.

Top image: NASA.

Virtually everything we know about the laws of physics falls into one of two piles. In one, there's quantum mechanics, from which we've developed the "Standard Model," including all of the fundamental particles we've yet detected, and three of the four interactions: electromagnetism, and the weak and strong nuclear forces.

In the other pile, there's Einstein's theory of General Relativity, which describes the fourth force, gravity, and gives us black holes, the expansion of the universe, and the potential for time travel.

Can they coexist peacefully? Reader Tony Lund puts it rather succinctly when he asks:

We've all been told that Einstein's Theory of GR doesn't work with Quantum Mechanics. Why? Where exactly does the theory break down?

I should warn you in advance that we don't ultimately know how Quantum Mechanics and General Relativity will be combined into a theory of "quantum gravity." And although there are some good ideas that I might be persuaded to write about in a future column, for today, I'm going to focus on why we need a theory of Quantum Gravity in the first place.

The Two Domains

Quantum Mechanics and Relativity typically operate on vastly different scales. Quantum mechanics, for instance, was unknown to science for so long because it normally becomes important only on the scales of atoms. If you're clever, you can imagine scenarios where quantum mechanics governs the destiny of a cat, but that tends to be a stretch.

Relativity, on the other hand, tends to be important in strong gravitational fields. Time, for instance, gets slowed near the surface of the earth compared to far away; light gets bent around clusters of galaxies. These effects can be largely ignored unless you're talking about the surfaces of neutron stars and the like. In other words, General Relativity typically kicks on on large-ish scales, from stars all the way on up to the entire universe.

But there are some very interesting corners of spacetime where General Relativity and Quantum Mechanics collide.

Black holes tend to be pretty good astrophysical laboratories, in large part because they are both small, and have extremely strong gravitational fields. Indeed, the first attempts to successfully combine both gravitational and quantum effects occur on the edges of black holes, the famous Hawking Radiation, which will ultimately (in quadrillions of years) evaporate even the biggest black holes and lead inevitably to the heat death of the universe.

Outside, we do okay. As we move further and further in to the centers of black holes, however, we have less and less of an idea how physics really works.

Singularities

Once you drop something below the event horizon of a black hole, not only can it never escape, but it will be drawn inexorably inward. The upshot of that is that in a world where gravity is the only (or at the most important) game in town, everything you throw into a black hole will ultimately end up confined to a literal point – the so-called "singularity." The instant of the big bang has the same sort of problem: incredibly high density (so strong gravity) confined to a very small space – in the first instant, presumably infinitesimally small.

We've never seen a so-called "naked singularity" directly (and there's good reason to suppose that we never will), which is unfortunate from the perspective of understanding them, but rather fortunate from the perspective of not being ripped apart by the gravitational tidal forces.

The picture from general relativity is that the cores of black holes have literally zero radius, but quantum mechanics says something entirely different. In quantum mechanics, there's an "Uncertainty Principle" which says, among much else, that you can't ever determine the exact position of anything. In practice this means that even things that we call "particles" can't be arbitrarily small. According to quantum mechanics, no matter how hard you try, a mass as large as our sun can't ever be confined to a region smaller than about 10^-73 m.

Insanely small, but not zero.

If this were the only collision between quantum mechanics and gravity (and I suspect it's one a lot of io9 readers were already aware of), I could forgive you for being underwhelmed by the magnitude of the problem.

But the real conflicts between quantum mechanics and relativity run even deeper than a space of 10^-73 m.

Classical and Quantum Theories

General relativity is what's known as a classical field theory, which describes the universe as a continuous distribution of numbers – exact numbers, if you had the tools precise enough to measure them – that can tell you all about the curvature of spacetime everywhere and everywhen. The curvature, in turn, is described completely and exactly by the distribution and motion of mass and energy. As John Wheeler famously put it:

Mass tells space-time how to curve, and space-time tells mass how to move.

But quantum theories are totally different. In quantum theories, particles interact by sending particles between them. Electricity, for instance, sends photons between charged particles, the strong force uses gluons, and the weak force uses the W and Z bosons.

We don’t even need to dive into a black hole to see the conflict between classical and quantum theories. Consider the famous “double-slit experiment.” This involves shooting a beam of electrons (or photons, or any other particles) through a screen with two small slits etched out. Because of quantum uncertainty, there is no way to figure out which slit a particular electron travels through: An electron literally travels through both slits at once. This, in and of itself, is kind of nuts, but in the context of gravity, it gets even stranger. If the electron goes through one slit it presumably creates a very slightly different gravitational field than if it goes through the other.

How does it know?

It gets even stranger when you realize that according to Wheeler's delayed choice experiment it's possible to set up the experiment so that after you've already run the experiment, you can retroactively observe the system and force the electron to travel through one slit or another (though you can't choose which). Crazy, no?

Put another way, the world of gravity is supposed to be entirely deterministic, but quantum mechanics is anything but.

Gravity is Special

There's an even deeper issue: unlike with, say, electricity which only affects charged particles, gravity seems to affect everything. All forms of mass and energy respond to gravity and create gravitational fields, and unlike with electricity, there aren't negative masses to cancel out the positive ones.

We can imagine a quantum theory of gravity, at least in principle. Like with the other forces, there would be a mediator particle, proactively called the graviton, which would carry the signal.

We could even imagine probing smaller and smaller scales, and seeing more and more virtual gravitons being sent between particles. The problem is that on smaller scales, there are higher and higher energies. The nucleus of an atom requires much more of a punch to break apart than peeling an electron off the outside, for instance.

On the smallest scales, the swarm of insanely high energy virtual gravitons would produce an incredible energy density, and that's where we really run into problems. Gravity is supposed to see all forms of energy, but here we are generating an infinite amount of highly energetic particles which in turn generates a huge gravitational field. Maybe you see the difficulty. At the end of the day, every calculation involves a whole bunch of infinities flying around.

In electromagnetism and the other quantum interactions, calculations get severely confusing at a very small scale known as the "Planck Length," around 10^-35 m – far smaller than an atom. I am required by long tradition to point out that we have no freakin' clue how physics is supposed to work on scales smaller than the Planck Length. On those scales, quantum mechanics says that miniscule black holes can pop into and out of existence through sheer randomness, suggesting spacetime itself gets pockmarked if you look at it too closely.

We try to avoid these collisions of theories through a process known as "Renormalization" (thrown in as fan service for the experts). Renormalization is simply a fancy way of saying that we only do the calculation down to a certain scale and then stop. It gets rid of the infinities in most theories, and allows us to carry on with our lives. Since most forces only involve taking differences between two energies, it doesn't really matter if you add or subtract a constant to all of your numbers (even, ostensibly, if the constant you're adding is infinity). The differences work out fine.

Not everyone is so sanguine with this. The great Richard Feynman noted:

The shell game that we play . . . is technically called ‘renormalization’. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

Those objections aside, things get even worse when we talk about gravity. The thing is, because (unlike with electromagnetism) gravity affects all particles, those infinite energies mean different curvature. Renormalization doesn't even seem to be an option for gravity. We can't make the infinities go away.

What we do know

So we don't have a theory of quantum gravity, but we have some idea of what a successful theory must be like. For instance, there needs to be a graviton, and because gravity seems to be able to extend over all space, the graviton (like the photon) needs to be massless. Massive mediators (like the W and Z bosons) can only operate over a very short range.

But there's more (although it's a little more technical so the squeamish may want to turn away). It turns out that there is a unique relationship between classical and quantum theories. For instance, electromagnetism is generated by electric charges and currents. The sources are described mathematically by a vector, and it turns out that vectors produce spin-1 mediator particles. It turns out that mediators with odd spin produce forces in which like particles repel. And indeed, two electrons will repel one another.

General relativity, on the other hand, is known as a "tensor theory" because there are all sorts of sources related to the pressure and flow and density of an energy distribution. The quantum versions of tensor theories have spin-2 mediator particles. So whatever else, the graviton will be spin-2. And, you guessed it, even spin mediators attract like particles. And lo and behold! Like particles do attract gravitationally!

So yay! We know a little something about what gravitons must be like. But as for all of those infinities, damned if we really have any idea.

Dave Goldberg is a Physics Professor at Drexel University. His newest book, The Universe in the Rearview Mirror is here! You should definitely become a fan on facebook, follow him on twitter, or better yet, send a question about the universe.