## 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 10^{500} 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, 10^{500} 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 t*otally unrelated problems. *

So… let me just see if I’ve got this straight…

The motivation is finding out where a particle is.

You can look at just the particle, which is (comparatively) simple.

You can look at the particle,

andthe field it’s in. You need to map the field to work out where the particle is. The magic of maths allows you to define the field to some degree, and thus still find the particle.Or, you can look at the particle, and the field it’s in,

andthe space-time they both occupy. Then, you need to map the field, and the space-time the field and particle exist in. The space-time creates another layer of infinities which you can’t derive away.However, if you treat the fundamental particles as minute strings, then that limits their momentum. Because the momentum of the particles of the field and space-time is finite, then you can make the maths work.

Although string theory explains a lot of what we’ve already proven, it’s not made any of its own experimentally testable predictions yet. The closest it comes is necessitating the existence of the supersymmetry partners, which people are still looking for. The supersymmetry partners are needed to balance the string theory equations.

Is that right?

I don’t think, overall, it’s any worse than my hand-waving explanation. Keep in mind we are using rather vague metaphors to the math, and that a metaphor is an

approximateway to understand something.Just a couple of small things, which I didn’t explain well due to compressing a semester long graduate course into 1500 words . Heisenberg’s uncertainty principle, which comes out of basic quantum mechanics, states that if you narrow the window or uncertainty in space, the window or uncertainty in momentum must get much larger. Putting in a string rather than a point particle changes this somewhat. It’s not that choosing strings limits momentum per se… And this is where not using math really hobbles me. For point particles, you must have integrals over all momenta, but with strings, the finite size introduces a finite momentum cutoff. But that probably doesn’t help at all.

Second, string theory has a great deal of trouble even explaining what we already know. It does lead to general relativity. It readily accepts a group structure that easily accomodates the particles we already know such as electrons, quarks, neutrinos, photons, gluons, and so on.

On the other hand, it is difficult to show that all the results of, say, electromagnetism arise naturally from any one given string theory. This is unlike many other theories–general relativity can be readily shown to be approximated by Newtonian gravity, and quantum mechanics reduces to Newtonian mechanics in the appropriate limit. Doing this for string theory is not so easy.

And this is one reason why it is difficult to make solid predictions for string theory. In general relativity, for example, you can show that for relatively small masses you get Newtonian gravity, so as the masses increase you can plot the increasing difference between the two theories. But since we aren’t always sure how, or even if, string theory reduces to quantum electrodynamics, finding the difference that we could test for is very hard.

Supersymmetric partners, on the other hand, are a very clear and distinct prediction.

Does that help?

“For point particles, you must have integrals over all momenta, but with strings, the finite size introduces a finite momentum cutoff.”

I tried to understand this, but I kept running into walls of maths. It wouldn’t be so bad, but I didn’t even understand the symbols they were using… I’ll just take your word for it. Wikipedia started talking about strings having one dimension and more dimensions than currently exist in our universe. I know where my limits lie

But the thing about string theory not breaking down into quantum theory the way quantum theory breaks down into Newtonian physics makes sense.

So, if string theory has so many holes, then why is it so popular? And will the lack of supersymmetric partners disprove it?

You know, I thought this was funny when I first saw it. But now I’m wondering if it was public service announcement instead of a joke…

http://xkcd.com/435/

So, if string theory has so many holes, then why is it so popular?Good question.

The main reason is familiarity. String theory uses the same tools as “ordinary” quantum field theory, and inasmuch as quantum field theory has been very successful, it seems reasonable to keep using the same plan. Other approaches are significantly more unorthodox–and to be fair, they also suffer similar problems. In a month I’ll have a final post where I talk about a couple of currently popular alternatives.

It’s hard to prove something

doesn’texist—hence millions of children who believe in Santa Claus and [insert your favorite political joke here]. But if searches for superpartners consistently fail to turn up something, the explanations for not finding them will eventually wear thin.[...] recent post by Calvin Johnson on the Science in My Fiction blog about quantum and sub-quantum mechanics has got my head back in the stars again (galaxies and quanta are pretty much the same thing in my [...]

I think the two major claims of string theory proponents are that it gives 1) a possible entry point to a grand unified theory that includes gravity and 2) the ability to predict from principle (rather than arbitrarily) the various properties of elementary particles.

If the theory (theories) cannot do so without additional fudgets and widgets, much of the attraction is lost even at the theoretical level — to say nothing of the lack of testable predictions.

Speaking of lack of testable predictions… Mini black holes, pocket dimensions and exotic particles all absent from the LHC tests which were looking for them:

http://io9.com/5714210/string-theory-fails-first-major-experimental-test

As the article says, it doesn’t disprove anything. Not quite the knock-out blow I’m sure some where hoping for, though.

(I know you’ve probably all seen it already, but I thought it was worth sharing on the off-chance someone hasn’t…)

The first line of defense will be “We need higher energy,” the second “variants of the theory predict micro-black holes with different properties than those testable by the LHC.” Either/both may be true, but eventually there will be a concrete contradiction or confirmation, rather than absence of evidence. As long as there’s no experimental evidence, this is in an “uncollapsed” state of hazy probability. More than zero, but very hard to pinpoint! *laughs*

Oh, that made me laugh!

My partner asked me what I was laughing at, and all I could say was that it wasn’t quite worth a lecture in quantum mechanics to get to the punchline…

What, a golden unforced opportunity to teach QM basics and you threw it away?? *shakes fist*

I’ve tried to explain them in the past, believe me! But I’m not very good at explaining it and she doesn’t really like particle physics, so I’ll get about ten minutes of blank looks before, ‘you know, I stopped listening about half-an-hour ago’. I mean, she’s smarter than me, she just far prefers science she can touch and feel and be a part of. It’s kind of hard to be part of quantum mechanics without your own particle accelerator.

I think the two major claims of string theory proponents are that it gives 1) a possible entry point to a grand unified theory that includes gravity and 2) the ability to predict from principle (rather than arbitrarily) the various properties of elementary particles.Actually, what many string theorists claim is that string theory is the

onlyentry point to a quantum gravity theory of everything.(Terms of art: most particle physicists use “grand unified theory” (GUT) to refer to a theory that encompasses everything

butgravity, while “theory of everything” (TOE) also throws gravity into the mix.)String theory can accomodate particle properties. As to predicting them from first principles, that doesn’t seem likely. Right now one has to choose, by hand, the appropriate group structure. We already have to do that in the Standard Model of particle physics. The Standard Model, however, does make predictions that are not included in the choice of group, for example the existence and mass of the neutral vector boson. Right now string theory can’t even do that. Indeed, this is part of the reason behind the recourse to invoking the anthropic principle.

Einstein said he wanted to know if God had any choice in the universe, assuming implicitly that mathematics has a Platonic reality that even God couldn’t change. String theory doesn’t resolve this issue, and when the anthropic principle is invoked, it takes the opposite tack: not only does God roll dice, She does so over and over again until She gets finally gets the outcome She wants.

(…and this just gave me an Idea for a story…)

When someone writes an piece of writing he/she retains the image of a user

in his/her mind that how a user can know it. So that’s why this post is outstdanding.

Thanks!

When I initially commented I clicked the “Notify me when new comments are added”

checkbox and now each time a comment is added I get several e-mails with the same comment.

Is there any way you can remove people from that service? Cheers!

Hello, constantly i used to check weblog posts here early in the daylight, because i love to learn more and more.