The Knotty Problem of Quantum Gravity: Part 1

You may have been in this situation:

You have two friends, your two best friends, both witty, fun, and thoughtful. An hour or so spent with either one leaves your brain buzzing with new ideas and insights. “I gotta get these two together,” you think.

You arrange a dinner, but the evening is a disaster. Your friends are incompatible in ways so deep, so fundamental, they can barely stand to be in the same room.

Now imagine your friends are physics theories; and not just any theories, but the two most revolutionary theories of the twentieth century, explaining entire libraries of data, predicting new, mind-blowing phenomena. They are quantum mechanics and general relativity.

And they do not play nice with each other.

Over the course of three posts, I will tackle the problem of quantum gravity. In this, the first post, I will investigate why quantum mechanics and general relativity are so at odds with each other. In the second, I will outline one of the most popular approaches to quantum gravity, string theory–a theory that even its most ardent advocates will admit is as yet incomplete. And in the final post I will discuss a couple of recent proposals that provide alternatives to string theory and have in the past year set the physics world abuzz.

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First, let me introduce our two amazing friends-who-cannot-befriend-each-other.

General relativity is the physics of the cosmos, a grand theory of gravity that describes black holes, the expansion of the universe, gravity waves, and so on. In general relativity, gravity is not really a force at all, but an illusion caused by the warping of space and time: a planet orbiting the Sun actually thinks it is going in a straight line (the fancy-shmancy term is geodesic) and it’s just outside observers who disagree, because their personal, local geodescis disagree. It’s general because it deals with accelerating bodies, in contrast with its more narrow-minded elder brother special relativity, limited to constant velocities.

The basic principles of special relativity were stumbled upon by Poincare, Lorentz, Fitzgerald, and others, but a patent clerk (3rd class) named Albert Einstein made the derivation and the consequences of special relativity dramatically clear. For an encore he spent the next eleven years dreaming up general relativity, with maths so difficult he had to seek help from a professional.

In the other corner is quantum mechanics, the physics of things so small you can’t see them. Unlike general relativity, quantum mechanics had many fathers–Bohr, Planck, Schrodinger, Heisenberg, Born, de Broglie, and good old Einstein himself–and the theory itself shows this complicated heritage in its jumble of strange, arbitrary rules. Eventually, you get used to them. But quantum mechanics, like general relativity, has been tested to inhuman precision.

Quantum mechanics, or more properly its descendent quantum field theory, sees forces very differently from general relativity. Instead of a rubber-sheet deformation of spacetime guiding planets and projectiles in their courses, in quantum mechanics forces are the result of tossing back and forth particles with energy and momentum. For electromagnetic forces this particle is the photon; for the weak nuclear force, in charge of beta decay and neutrinos, it is the W and Z vector bosons; for the strong nuclear force between quarks, it is a gluon.

Both theories are breathtaking in their beauty as well as in the difficulty of their maths; both theories are so well established by experiment they have the sheen of Truth upon them. We know two Beauties do not always mix, but we want to believe that two Truths always ought to be compatible. But general relativity and quantum mechanics clash like a bitterly divorced couple.

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Physicists are smart but lazy. We don’t want to memorize 206 bones of the human body, or dozens and dozens of taxa and genera. Trying to recall the difference between ethyls and aldehydes and carboxyls bores us silly.

Instead we prefer a compact view of our discipline, ideas like origami cranes that look small and elegant but can be unfolded, and unfolded, and unfolded,  yielding more and more at each step. The basic equations of quantum mechanics and general relativity are H? =E? and G ? ? =8 ?T ? ? . Of course it takes months of study just to decipher these equations, and years to learn how to use them.

The ideal would be a single equation that one could unfold into Schrodinger’s and Einstein’s equations, an equation that unifies quantum mechanics and general relativity. Combining many theories into one is a vaunted goal of physics. It started with Newton’s laws unified the laws of motion for the heavens and the Earth, and continued with James Clerk Maxwell unifying electric and magnetic forces, and predicting radio in the process.

In the early days the goal of unification looked promising. Paul Dirac showed how to unify Schrodinger’s equation with special relativity, giving rise to his eponymous equation and, not incidentally, predicting antimatter. In fact, it led to the idea of pairs of particles and antiparticles popping in and out of existence, making even a vacuum a rather crowded and lively thing.

Keeping track of an ever-changing population of particles and antiparticles required new mathematical tricks, and led to the new discipline of quantum field theory. Quantum mechanics treats particles via wavefunctions, but the fields and forces between them were still classical. In practice this means solving differential equations, in one variable x or sometimes three x,y,z; not trivial, but once you get the hang of it, you can solve one-variable differential equations to pass the time on an airplane, or a sleepless night, or a bad date. I know I do.

Treating the electromagnetic and others fields quantum mechanically, however, really ups the game. Instead of solving for the wavefunction of an electron, one has to also include the wavefunctions for an indefinite and possibly infinite number of photons, as well as the wavefunctions for an infinite number of electron-positron pairs.

I’m good at differential equations, but not that good. No one is. Instead, physicists have to resort to various approximations. The famed Feynman diagrams, little squiggly sketches of photons, electrons, and so on bouncing around, are actually graphical shorthand for the mathematical instantiation of the approximation.

Easy, no?

A simple representation of hard maths

The problem is, the answer is often infinity. I’ll go into more details why in my next post, but it’s no shocker, given that one has suddenly gone from one variable in plain vanilla quantum mechanics to an infinite number of variables in quantum field theory.

In the 1940s and ’50s, Schwinger, Feynman, Tomonaga, and others found a way to get around the problem of getting infinity as an answer–in essence, they invoked a miracle. But the miracle worked; not only did they get physically meaningful answers, the answers agreed with experiment. This was a great triumph.

Moreover, Dutch theorist Gerard ‘t Hooft showed that for a significant class of quantum field theories, called gauge field theories, one only had to invoke a single miracle to make the entire theory consistent. This is much like science fiction, where one is allowed to ‘break a law’ of physics, but then work out the consequences consistently.

Gauge field theories include quantum electrodynamics, the quantum theory of electromagnetism; quantum chromodynamics, the theory of the force between quarks; and the weak nuclear force that governs nuclear beta decay and neutrinos.

But gravity? Alas, no. It turns out that for gravity, one must invoke an infinite number of miracles…and that takes us from science fiction to pure fantasy.

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Next time I’ll take up the problem of quantum gravity and one possible solution, string theory. But before I go: why worry about a quantum theory of gravity at all?

For scientists, the impetus is intellectual curiosity. We do believe that, ultimately, two Truths must not conflict. Furthermore, as we try to understand the birth of the Universe in the Big Bang, the intense densities require us to understand what happens to gravity at very small scales, and may even give us a clue as to how the Universe came about in the first place.

And one never knows where science may actually have applications. To use GPS to streetlevel accuracy, one needs to take into account small but important relativistic corrections.


(Source: http://xkcd.com/808/)

And this is where we shade over into SF–for SF authors naturally hope that a quantum theory of gravity might provide a natural way to hop from star to star.  Hard SF rock star Stephen Baxter has invoked GUT (Grand Unified Theories–the mother of all gauge theories, literally) drives and SUSY (supersymmetric string theory) drives in his stories.

What I hope to do with this series, though, is not simply provide you with Cool Names for Star Drives, but give you a little bit more insight into the science that goes into your fiction.

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