Quantum Gravity, Part 2: A Thread ( or String) Leading Out of the Maze?

A good public relations campaign can do wonders.

Science is empirical. If there is no experiment, no observation, then an idea is truly relegated to “it’s just a theory.” [1]

Yet, consider string theory, a mathematical exercise so intricate Einstein’s general relativity is easy in comparison, and with no experimental evidence backing it whatsoever.

In the popular imagination, however, string theory dominates modern physics. Popularizations of string theory have topped bestseller charts. Friends and neighbors ask me about string theory. Students tell me they want to be string theorists, even though they, along with most of the public, are unsure what string theory even is.

In this essay I’ll attempt to untangle string theory for you, explain what it’s good for, why there are such devoted proponents, and what the skeptics say.

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As I explained in a previous post, quantum mechanics and general relativity (Einstein’s theory of gravity), are the two most revolutionary and experimentally successful theories of the twentieth century. Yet combining the two into a quantum theory of gravity has proved challenging.

A somewhat metaphorical explanation of the mathematical problems goes like this:

In ordinary quantum mechanics, one solves for the wavefunction of, say, an electron orbiting an atom, which depends on the x,y,z coordinates of the electron. Though it sounds hard, one can mathematically calculate the wavefunction by solving Schrodinger’s differential equation. In some cases it can even be solved with a pencil, a lot of paper, and a big eraser. But a simplifying assumption has been made. The electric field in which the electron moves is treated classically, that is, there is no wavefunction for the electric field.

The next step up is quantum field theory, where one treats the electric field as well as the electron with quantum mechanics. This is much harder. Quantum mechanically, the electric field is created by a sea of photons flickering in and out of existence. But while there is only one electron, there can be between zero and an infinite number of photons. Furthermore, these photons can, briefly, cause an electron and anti-electron to come into existence, only to swallow each other up and make another photon.
If this sounds complicated, it is: instead of only three coordinates (x,y,z) of

the electron, we know have an infinite number of coordinates.
And when one tries to solve the problem, the answer itself tends to be infinity.

Physicists are nothing if not stubborn and clever. Early on they tried subtracting infinity from the math of quantum field theory. While this sounds rather suspicious–after all, infinity minus infinity could equal anything– they found they could get a finite, reasonable result. More important, they only had to subtract infinity once and all the answers thereafter would be consistently finite. (This process is called renormalization.) And the answers agreed with experiment, often to many decimal places, and even predicted new phenomena such as the Casimir effect.

Quantum field theory has been successfully applied to electromagnetism and to nuclear forces, with theory and experimental verifications winning Nobel prizes left and right.

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Alas, when one tries to apply quantum field theory to gravity, it blows up in your face. You can’t get away with subtracting infinity just once. You have to subtract an infinite number of infinities. Even for theoretical physicists this is too much.

Why the explosion of infinities? Remember when we went from just an electron to an electron plus electric fields (which is really adding photons and pairs of electrons and positrons), we added an infinite number of coordinates? Well, in Einstein’s general relativity, space and time are not simply grid marks, but are themselves dynamic, and what we perceive of as gravity is the warping of spacetime. But for quantum gravity this means that, in addition to particles and fields, the very coordinates of those particles and fields must be quantized, adding another whole layer of infinities.

Ow.

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I’ve been coy in describing exactly how these infinities arise, mostly because of years of encountering “Ahh! Math! It makes my brain bleed! Stop it! STOP IT!” when trying to explain how to balance a check book. Well, lay on the ibuprofen and bandages, we’re diving in.

The math gets complicated even for professionals, and Richard Feynman’s eternal gift to physics was to devise a pictorial way to keep track of the math.

What such Feynman diagrams represent are complicated integrals. Each line and intersection has a specific interpretation.

In particular, whenever there is an interior closed loop as above, the intermediate particles, photons, electrons, what have you, can take on arbitrary energies and momenta. This is allowed under the Heisenberg uncertainty principle: the shorter an interval of time, the greater the fluctuation of energy allowed, or the smaller the region of space, the greater the fluctuation in momentum.

It is the integrals over infinite ranges of energy and momentum that give rise to infinities.

In renormalization of ‘ordinary’ quantum field theory, one literally stops or cuts off the integral before it becomes infinite. The key to making renormalization sensible is a mathematical shell game that makes the answer independent of where the cut-off in the integral occurs. (If that isn’t obvious, don’t worry; that’s why they awarded Nobel prizes to Feynman, Schwinger, and Tomonaga for figuring it out.)

Unfortunately this kind of shell game doesn’t work as well when it comes to quantum gravity; the infinities becomes too overwhelming for the usual cut-off trick to work. (To explain why I really would have to write down the integral, and I don’t want to hurt your brains that much.)

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There is another solution. As I mentioned before, Heisenberg’s uncertainty principle allows, as the interval of space gets smaller, for the fluctuations in momentum to get larger and larger.

But what if fundamental particles are not infinitesimal points, but little wriggling strings with a fixed length and thus an guaranteed upper cut-off of momentum? This, my friends, is the motivation for string theory.

The string length is tiny. It is likely to be a hundred billion billion times smaller than the size of the atomic nucleus, far out of reach of any current experimental apparatus to measure. This is a problem, because I like a little experimental evidence with my theory.

If you think infinite integrals and renormalization and using strings to cut off infinite momenta is headache-inducing, that’s nothing. For better or for worse, invoking strings is not enough to cure the infinities in quantum gravity. One also needs to invoke “supersymmetry.”

All fundamental particles are grouped into two classes, fermions such as electrons, neutrinos, and quarks, and bosons such as photons and gluons. Supersymmetry postulates that for every fundamental fermion there is a “superpartner” boson, with weird names such as scalar electrons or selectrons, sneutrinos, squarks, etc, and similarly for bosons there are superpartner fermions, photinos, gluinos, etc.

The existence of these invisible dance partners creates mathematical cancellations that eliminate the last of the wild infinities in quantum gravity.

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There’s always a fascination with achievements beyond those of mere mortals: running marathons and ultramarathons; families with fifteen children and couples celebrating seventieth wedding anniversaries; understanding string theory.

The obsession with string theory goes beyond its brain-numbing difficulty. It is touted as the theory, the final theory, the Theory That Would Explain It All If Only You Hadn’t Flunked Calculus.

Is it?

Part of the problem is that there is, as yet, no experimental evidence either in favor or against string theory. Part of the problem is that string theory is so difficult it’s not easy to make it connect to “simpler” (yeah right) theories like quantum chromodynamics and electroweak theory, so it is hard to make concrete predictions where string theory diverges from these.

There are some important, nontrivial predictions. The most important of these is the proliferation of superpartners, and accelerators at FermiLab and at CERN are constantly looking for evidence for these particles. Particles predicted by supersymmetry are among the leading candidates for nonbaryonic dark matter, which makes up 25% of the composition of our universe. (Baryons, meaning you and I, make up only 5%.)

The most obvious area of application of a quantum theory of gravity would be in the intense densities of the Big Bang; but recently string theorists have identified 10500 possible string-theory models of the universe, and so some have literally given up on predictions as a hallmark of a physical theory and instead embraced the anthropic principle, which is a fancy way of saying, “I’m here because if I wasn’t I’d be over there.”

This has led to criticism of string theory, such as Peter Woit’s book and blog, Not Even Wrong, which in turn has garnered some savage attacks and mockery on the part of string theory advocates. String theorists state they are the ‘only game in town’ with regards to a quantum theory of gravity.

This isn’t true. After thirty years, string theory is the best developed approach to quantum gravity, but this is partly because it most closely follows the up-to-now successful model of quantum field theories. Given that three decades have passed with no real consensus on the final form of the theory–or, worse, 10500 versions to choose from–some of the shine has faded from string theory.

And there are alternate approaches to quantum gravity, some of which have stirred enormous excitement and controversy in the physics world. In a month, I’ll discuss those.

Footnotes:
[1] Evolution, the Big Bang, relativity, quantum mechanics, and many other “theories” in fact have enormous piles of data supporting them.

[2]In recent years some string theorist have proclaimed in the media experimental tests of string theory. But these are simply applying some of the interesting mathematics of string theory to totally unrelated problems.

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