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Miracles prove the divine power of string theory

No, I didn't mean Jesus Christ.

Pete Wilton, an Oxford science writer, wrote a piece introducing the new website WhyStringTheory.com:

Pulling the strings (Oxford science blog)
Three creative folks behind the website say various things. Edward Hughes, a Cambridge UK student, says that string theory is fundamental, beautiful, and he wanted to communicate the excitement.

Charlotte Mason, an Oxford student, says that she has mixed feelings about string theory. The picturesque ideas are beautiful but she describes the maths exactly in the way you would expect from a somewhat randomly chosen girl. No, Charlotte, the true beauty of string theory may only be revealed and understood when all the relevant maths is added. It should be done peacefully and beautifully – and not necessarily in the Greg-Moore-like heavy formalism way – but the maths is still critical for the beauty.

Finally, Joseph Conlon is the only "senior" person behind the project – he's at Oxford faculty. And he says some interesting – although not quite new – things that I want to spend some time with. It has something to do with the miracles.




Conlon says that strings are too small to be seen directly, so lots of good luck or theoretical or experimental breakthroughs may be needed to see them. He says the usual things about string theory's ability to cure the short-distance problems of quantum field theory by replacing points with strings; unify gravity with other forces and matter; about the correct counting of black hole entropy string theory provides us with, and a few other achievements in which string theory remains (and, most likely, will always remain) unmatched.

However, two of the paragraphs have a kind of a cool religious spin:
Joseph Conlon of Oxford University, another member of the team, explains that part of the theory's appeal lies in 'string miracles', these are 'calculations that look like they are going to fail and show that the theory is inconsistent, but then something comes in and suddenly saves the day. Once you see this happening several times you realise that the theory has a very deep structure and your understanding of it only scratches the surface.'

...

Yet string theory has a habit of turning up surprises, as Joseph says: 'Working on it is also good for humility, you are perennially aware that the theory is smarter than you.'
You might say that these sentence describe string theory by similar words that other people use for God. And in some sense, you would be right. The only difference is that the power of God is supported by personally and verbally communicated superstitions – sorry, believers – while the power of string theory boils down to objectively functional calculations that everyone may verify and that reveal a striking degree of internal coherence and compatibility with all known qualitative concepts and phenomena observed in Nature.

String theory is also able to turn water into wine – well, its nuclear physics approximation is enough for that because you just need to produce some carbon nuclei from the hydrogen and oxygen nuclei, aside from a few trace elements.

More seriously, the mathematical miracles behind string theory are numerous and initially unbelievably surprising. Later, they may be understood as consequences of a smaller number of technical properties of string theory's maths. While those explanations reduce the seemingly "supernatural" character of the miracles, the ultimate "conceptual explanation" always boils down to the existence and consistency of string theory.

So the ultimate miracle – the very existence of this rich mathematical structure that contains all good ideas and physics but remains fully consistent despite its incredibly richness – remains a miracle even today. We may spend years by philosophical musings on the reasons why the Universe exists at all; however, it's much more likely that we may discover a shocking and valuable insight if we ask why string theory exists at all.

The partial, individual miracles were parts of string theory's CV from its very birth. In fact, the first characteristically stringy formula – the Veneziano amplitude (for the tree-level scattering of four open-string tachyon, using the modern terminology) was found by Veneziano by demanding the first miracle, the "world sheet duality" (that's the modern terminology; they would call it just "duality" in the late 1960s, a word that is used somewhat differently today, and that's why string theory was initially known as "dual models"). What was this zeroth miracle of string theory?

Imagine that you collide 2 particles elastically. So there are 4 external legs in the Feynman diagram. You may write down diagrams in which a particle is exchanged in the \(s\)-channel. That will make the amplitude depend on \(s\). However, this intermediate particle may also have derivative interactions with the 4 external particles and in this way, the amplitude may acquire some \(t\)-dependence, too. You may get a rather general function of \(s\) and \(t\).

Then there are also diagrams with the \(t\)-channel exchange in which the role of \(s\) and \(t\) is interchanged. Veneziano boldly demanded that the sum of all \(t\)-channel diagrams is actually the same as the sum of all the \(s\)-channel diagrams. It turned out it was possible even though the "channel" variable enters through denominators and the "derivative interaction" variable enters through numerators. The Euler Beta function that Veneziano finally found in a library had the required property. In fact, you should only count the \(s\)-channel diagrams or only \(t\)-channel diagrams, otherwise you're double-counting the amplitude.

It was a miracle, something that couldn't appear in a quantum field theory with finitely many particle species (or fields). The Euler Beta function was soon derived from the assumption that the particles were actually open strings – and a closed string counterpart of the Veneziano amplitude, the 4-closed-string-tachyon Virasoro-Shapiro amplitude, was soon found as well. Once you know that the particles are open strings, the Veneziano miracle has a simple geometric interpretation: Both the \(s\)-channel and \(t\)-channel diagrams may be viewed as the particle limits of a string diagram, a disk with four strips coming out of it. The topology is the same for \(s\)-channel and \(t\)-channel diagrams so it's not too surprising that a naturally calculated "full tree amplitude" already contains both \(s\)-channel and \(t\)-channel terms.

You probably don't understand the previous paragraph – unless you have understood it for years. In that case, I want to say that there is a geometric explanation why certain sums of quantum-field-theory-like diagrams end up being \(s,t\)-symmetric even though such a symmetry looks insane from the quantum-field-theoretical viewpoint.

But much more typical examples of miracles in string theory have something to do with fully canceled anomalies, divergences, or harmful discontinuities. The first superstring revolution started with one of these miracles: Green and Schwarz found the precise cancellation of all the gravitational, mixed, and gauge anomalies in type I string theory in \(d=10\) assuming that the gauge group is \(SO(32)\). You're only allowed to choose one number, \(n=32\) "half-colors" of the quarks carried by the open string endpoints, and the theory manages to cancel five coefficients in front of these anomaly terms even though there are lots of contributions and the numerical constants that contribute are as complicated as \((n-496)/725760\) – and indeed, the dimension of the \(SO(32)\) group is \(496\). Thank God, or thank string theory, more precisely.

A sixth-order invariant in \(SO(32)\) from a gaugino loop seemed impossible to cancel in general. However, Green and Schwarz found a sub-miracle, the Green-Schwarz mechanism. They noticed that the sixth-order invariant factorizes to the product of a fourth-order and second-order invariants exactly when the group was \(SO(32)\) – and to cancel the remaining product of these fourth- and second-order terms, they discovered a previously neglected Feynman diagram resulting from unusual gauge transformation laws for fields – laws you wouldn't expect in low-brow perturbative quantum field theories.

So the Green-Schwarz miracle was five-fold and it had sub-miracles, too. You could think that no sensible scientist would assume this many miracles but you would actually be wrong. As Schwarz has repeatedly recalled, Green and Schwarz have done the tedious calculation because they already lived in the string theory "belief system". String theory was beautiful, they reasoned, so it couldn't possibly have any anomalies. And indeed, a long calculation with miraculous cancellations has shown that the belief was true even though they weren't able to write a crisp mathematical proof of the belief at that time.

Of course, within two years, this miracle was largely demystified, too. For the \(SO(32)\) gauge group, one may show that the "disk" cancels the "projective sphere" – some simple world sheet diagrams contributing world sheet anomalies of a kind. So the theory is free of world sheet anomalies and it's enough to prove that the resulting spacetime amplitudes will have all the desired physical properties including the spacetime anomaly cancellation, too. The spacetime calculation looks complicated because, in some sense, it's not the easiest or most fundamental one here: the world sheet objects and anomalies are simpler and more profound in those calculations.

In the following year, in 1985, the heterotic string was discovered. Exactly when you combine left-movers from bosonic string theory and right-movers from the only other known credible string theory, superstring theory, you obtain a hybrid string whose extra left-moving bosons have exactly enough freedom to produce the weight lattice of \(Spin(32)/\ZZ_2\), the right way to write \(SO(32)\) in this context, and exactly this weight lattice miraculously turns out to be even and self-dual. That's a new way to construct a string theory with an \(SO(32)\) symmetry, one that was shown to be equivalent (S-dual) to type I string theory ten years later – and this S-duality boils down to lots of miracles because all the objects and couplings that must match do match. Also, the heterotic string theory of 1985 had another version, based on the other self-dual even lattice, and it gives an \(E_8\times E_8\) gauge group in the spacetime which (now already less miraculously) cancels all the spacetime anomalies as well and that miraculously produces realistic spectra when compactified on Calabi-Yau manifolds.

I could tell you hundreds of examples of Conlon's general theme that many calculations may proceed for a long time and until the very end, they may keep an infidel physicist who does the calculation worrying that the final answer will be meaningless, ambiguous, inconsistent, full of divergent, ambiguous, regulator-dependent, and ill-defined factors, bullocks. But right before you're finished, all the sources of problems get canceled against others, usually many others (and new objects that you may have stupidly overlooked because they are purely stringy although they always admit a rather comprehensible explanation) and string theory's perfect coherence is saved.

For example, string theory implies that the topology of spacetime – of the extra dimensions, to be concrete – may continuously change. However, such a change would lead to a discontinuous jump of masses of some string modes, something that isn't allowed, because the masses depend on intersection numbers of various "cycles" and those change if the topology gets transformed (through a singular Calabi-Yau space whose topology is ambiguous). However, there's a contribution from world sheet instantons that string theory clearly tells you to include as well (it's a part of the Feynman path integral) and with these world sheet instantons, the continuity of the masses is restored. Also, on a similar manifold, one may be afraid of a singular formula for the Hamiltonian before she realizes that it's exactly what one gets from integrating out massless D3-branes wrapped on a "shrunk 3-cycle", thus proving that the full theory before the D3-branes are integrated out is completely smooth near that point in the configuration space. The new "stringy building block" – either the world sheet instanton or the D3-brane – has exactly the right properties that are needed to cure a disease that would be certainly lethal in quantum field theory or any other generic "related" theory which isn't quite the full string theory. But these new stringy building blocks aren't ever added in an ad hoc way. They're always the same blocks, satisfying the same unambiguous laws in all the environments. There's nothing to adjust about these building blocks and the laws they obey; nevertheless, their properties are always exactly right to cure all potential diseases.

The possibility to describe string/M-theoretical vacua via Matrix theory, matrix string theory, AdS/CFT correspondence, and the correct black hole behavior, including the thermodynamic properties, much like the very fact that the path integral summing over stringy histories fully reproduces general relativity including the non-linear corrections aren't just slogans. They are totally well-defined calculations and each of them presents numerous examples of Conlon's miracle theme. Tons of things could go wrong and pretty much "any" generic theory containing fields, particles, or other pieces of string theory – but mixing them in a slightly different way so that it is not "quite" string theory – would almost inevitably end up with an anomalous, ambiguous, inconsistent, non-unitary, or otherwise pathological result. But string theory always passes without any flaw. You may always take it to the limits. Any limits. It gets connected with some new degrees of freedom, new possible processes, possibly new branches of maths, and the answer always makes sense at the very end.

String theory isn't the first example of a physical theory that has this property. Quantum field theory itself was showing a similar internal strength – and it wasn't the first framework, either. However, in string theory, these miracles are much more diverse, multi-dimensional (in the figurative or imaginative sense), and they make the theory resilient under a much wider spectrum of inequivalent tests than any previous framework in physics. It's hard to imagine that this perfectly functional mathematical structure doesn't actually fulfill any functions in the inner workings of our Universe.

As I have said many times, many of the "miracles" have been demystified. They have been reduced to some maths whose statements we can prove and understand even without any "religious" assumptions on string theory. But it's a somewhat analogous situation to mathematicians' ability to prove Fermat's Last Theorem but only for certain exponents. What is really shocking – and unproved so far, assuming you can't use "religious arguments" in the proof – is that whatever you do with string theory, whatever new vacua or objects or processes you find by taking the previous ones to the limits or finding new solutions to the exact equations that constrained the old well-known objects, you will always cancel all the candidate inconsistencies. So far it's been the case.

I said that the "universal miracle" – the omnipresent cancellation of all anomalies and pathologies in all sectors of string theory – is somewhat analogous to Fermat's Last Theorem (which had to be proved for all exponents simultaneously). But it's probably much deeper and more important than that; Fermat's Last Theorem is a piece of recreational mathematics in comparison (despite the abstract concepts that Wiles' proof had to introduce). And the proof may be much harder or non-existent – or it may also be much easier, although no one can imagine what such a simple proof of string theory's "universal miracle" could look like. If it exists, it's bound to be conceptually profound.

The "partial proofs of stringy miracles" that we can actually write down have been compared to exponents in Fermat's Last Theorem; you could also compare them to theorems proved for individual patches of a manifold. However, we don't have any proof that the "miracles" apply to the whole manifold. After all, this manifold isn't just a simple manifold – it's a much richer structure (I am talking about string/M-theory) whose "global definition" remains elusive although we have always been able to extend the manifold from the known "patches" to the adjacent ones.

There are lots of things to write here but this text has already gotten pretty long. I would love to understand "why" string theory really exists and remains well-behaved regardless of the directions (not only on its configuration space) in which we take it. Because string theory's spectrum and parameter spaces and lists of objects etc. (and even lists of relevant branches of mathematics) change as you move on its moduli space (or as you switch in between dual descriptions), the proof – if there's any – must say something about the continuity and closedness of a highly flexible, chameleon-like mathematical structure. I don't know what the proof is and whether it exists at all. But despite the absence of a complete proof – i.e. despite the fact that we have just hundreds of "anecdotal pieces of evidence" – I am a clear believer. The coherence of string theory is perfect, is here with us to stay, and implies that curious, sufficiently mathematically talented people will always be intrigued by it and will continue to do research of it.

Of course, I am less certain about the continuing existence of curious and sufficiently mathematically talented (and motivated) people. ;-)

None of those miracles rigorously proves that string theory is the right physical theory describing this Universe. But I find it implausible that a theory which shows this degree of internal coherence, precision, and co-operation between its pieces as well as this degree of qualitative agreement with the features of Nature as we know them – including the general relativistic and quantum field-theoretical approximations, and beyond – exists by an accident. It would just sound utterly bizarre. It would be like if you found an alien rocket that seems to contain a perfectly streamlined and sophisticated engine that is capable of going through most of the difficult stages of an interstellar flight (while for some of them, we can't verify whether it can do what it needs to do) – but learning that this alien rocket has never been outside the Earth. Well, the analogy isn't perfect. Someone may produce a fake alien spaceship and impress people who know less than the constructor. But string theory clearly has no anthropomorphic constructor. All the new pieces of its engine are "objectively out there". So a better analogy would be that the alien spaceship seems capable of doing all the tasks even to the best constructors on Earth. If we found such a spaceship, it would have to mean it's an alien spaceship, wouldn't it?

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reader Dilaton said...

The Oxford Science blog article is nice and I thank god that it is not possible to comment there ... ;-)

The rest of this nice TRF article I'll enjoy later to happily finish my day after watching 3 TBBT episodes on PRO 7 tonight :-D

Cheers


reader James Gallagher said...

I really wish you well that the genius sophistication of this theory is the one required to describe Nature.


But fear that it will end up as mundane, not so well fitting, combinatorial and differential geometry arguments that will be called the New Standard Model, shortly before some alien race arrives to wipe us out for our fucking incompetence


reader Peter F. said...

Maybe the string/M-theoretical framework deserves to be described as: What the face or the profile (not to sound overly fanciful) of eternity formally looks like close-up through a least-possible distorting marriage between a mathematical macro and a ditto micro lens.
? %-}


reader Peter F. said...

This was one of those most impressive and fascinating (as usual Mottled for the best and most illuminating effect possible) blog-posts!!
It sure exemplifies/proves that even topics as demanding (in respect of the relevant fine details and the the massive amount of them) as this one can be made accessible and enjoyable to not so stupendously mathematically talented, trained and motivated people. :-)


reader Shannon said...

It has to be alien...


reader Mephisto said...

I found an interesting site with links to introductory string theoretical papers. I wish I had the time to read at least some of those. But maybe others might find something of interest there
http://www.nuclecu.unam.mx/~alberto/physics/stringrev.html


reader Eugene S said...

Oh Mephisto, what happened to your avatar? Listen, a few months ago you posted a very nice link to an article on the Higgs that started with a historical lead-in (Landau's work) and discussed the "Mexican hat" at length. Not too primitive (dumbed down), not too difficult: just right for me. Would you happen to have that link still?


reader Mephisto said...

Hi, my avatar is well and alive, only while writing from work, I did not remember my password. The links
http://plus.maths.org/content/secret-symmetry-and-higgs-boson-part-i
http://plus.maths.org/content/secret-symmetry-and-higgs-boson-part-ii
They have also some nice introductory articles on Schödinger equation
http://plus.maths.org/content/schrodinger-1
http://plus.maths.org/content/schrodingers-equation-action
http://plus.maths.org/content/schrodingers-equation-what-does-it-mean


reader Shannon said...

Cool articles Mephisto.


reader Eugene S said...

Thank you, much obliged!