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Midsize miracles of perturbative string theory

A big part of the reason why all competent high-energy BSM+QG theoretical physicists are basically certain that string theory is the correct unifying theory of this Universe is that

* it makes the right qualitative predictions (without assuming something equivalent) of the qualitative traits of the approximate laws of physics that were determined empirically
* it solves problems with consistency and finiteness in a way that looks "infinitely unlikely" or "miraculous" in a generic similar yet non-stringy framework (and the alternative theories are the greatest examples)
* it interpolates between various kinds of effective theories in ways that would also look miraculous to someone not familiar with the "stringy exception"

Concerning the first point, we must realize that the progress in physics has been incremental and for quite some time, physicists were switching from a working hypothesis to a "theory that is one step deeper" than the previous one. String theory really does nothing else than that, too. You may ask whether the step is greater or smaller than previous steps but this question is operationally meaningless because there exists no way to compare the steps between completely different pairs of situations.



Click the screenshot to 3D navigate the 6D Calabi-Yau with your mouse

Was the electroweak theory a bigger step than Maxwell's theory of electromagnetism? One cannot say, the conclusion is purely subjective or convention-dependent because the two advances are not commensurable.



OK, string theory improves upon the previous framework which is a combination of

* a renormalizable quantum field theory, the Standard Model, loosely paired with
* the classical general theory of relativity due to Albert Einstein

In practice, both may be encoded in the action, \(S = \int d^4x \,{\mathcal L}\), where the Lagrangian density \({\mathcal L}\) is a sum of non-gravitational and gravitational pieces. Both may be interpreted as quantum theories and they should be because the world is quantum mechanical. In the case of the non-gravitational forces and matter, it's really essential (atoms exist etc.; fermionic fields are classically zero); in the case of the gravitational fields, the quantization leads to problems because Einstein's theory is not renormalizable.



The gravitational action is something like the Einstein-Hilbert action\[ S_{EH} = \int d^4 x\,\frac{1}{16\pi G} R \sqrt{-g} \] which is the integral of the Ricci curvature scalar (multiplied by the invariant 4-volume measure), the only natural curvature invariant. On the other hand, the non-gravitational part, the Standard Model, is a sum of the kinetic terms for spin-1 gauge fields corresponding to the group \(SU(3) \times SU(2) \times U(1)\), fermions transforming as three copies (generations) of some reducible representation of that group, and scalar fields (the Higgses), along with the interactions (electromagnetic implied by the gauge invariance [for fermions and the Higgs] plus the similar Yukawa interaction; did I miss some? You bet, the Higgs quartic self-interaction).

A deeper theory should derive these traits without assuming them (or without assuming all of them). How do the proposed candidates succeeed in doing so? Take a theory by Lionel Volk, the magnetic raspberry (it's very important that it is not a strawberry). Does it predict the features of the Lagrangian above? Do we get the kinetic terms for the Yang-Mills fields out of the magnetic raspberries? No, we don't, of course (imagine that a one-hour-long monologue is inserted here, making fun of the fact that the raspberries don't even imply the right Dirac equation or anything else... nothing works). So Lionel's theory fails but it gives you magnetic raspberries. Too bad, if you do experiments with raspberries, you will find out that they are not really magnetic.

Out of the long list of Lionel's competitors, let's pick Stefan Tungsten. He has a Rule 30 theory. Define a binary coloring of the checkerboard based on some logical rule and you get something that Stefan finds surprising: the resulting coloring is neither "almost completely white", nor "almost completely black", nor "periodic", or otherwise simple. It is unsolvable. On top of that, after 3 bottles of wine, it looks like the skin of a tiger. Is that a theory of everything?

Well, no. He still didn't get the Yang-Mills action correctly. Or any physics, for that matter. The fact that a rule led to an unsolvable, "complex", system isn't because Rule 30 is interesting or worth attention. Instead, it is because Stefan's expectation was absolutely wrong and kind of shockingly dumb, too. Almost all equations or rules lead to "unsolvable" outcomes, solutions that cannot be expressed analytically or constructively (think about the three body problem in Newton's theory of gravity). So he got nothing surprising at all. And again, he got absolutely nothing out of the Maxwell-Dirac-Higgs-Yukawa-... action. Not even an epsilon about it.

There is a more elementary problem with the very basic "program" of all these armchair physicists. They seem impressed – and they want you to be impressed – with some solutions' appearances that resemble some objects in the real world like raspberries or the tiger's skin. Will a theory deeper than the Standard Model or Einstein's theory directly imply tigers and raspberries? Well, if that were the case, the Standard Model and Einstein's theory would be totally circumvented. These two (and hundreds of similar) geniuses' theory would construct the macroscopic object in the real world directly. It would mean that the theory is capable of making "several steps" at the same moment so that all the 20th century progress in physics could be ignored and replaced by something totally different. It is an extraordinary claim that requires extraordinary evidence. If you claim to have something deeper than the Standard Model or general relativity, you need evidence that is deeper, broader, and/or more precise than the evidence supporting those 20th century approximate theories. Of course, the geniuses don't have anything of the sort. They can't even get 1% of the correct predictions that the 20th century theories made correctly. They are self-evidently on the wrong track. Their whole thinking is an absolute non-starter.

One way to describe the failure is to say that they just don't understand reductionism. Before string theory, we didn't know the final theory but we still knew a lot about the structure of tigers and raspberries and the mechanisms that create them (evolved them and replicate them) and that keep them alive. And these mechanisms just don't follow from elementary magnetic raspberries; or elementary bits on a binary checkerboard. We know that these are wrong models of these objects, even as approximations. Natural processes may produce the structures on the tiger's skin that have similarities with Rule 30 of Stefan Tungsten. But we know that the mathematical ability of such hypothetical processes, even if they agreed (which they almost certainly don't), isn't a feature of the fundamental laws of physics. The processes that generate raspberries and the tiger's skin are self-evidently emergent i.e. not fundamental. You can't determine the fundamental laws of physics out of them directly!

There are lots of other armchair physicists who have their objects – playing the same role as LEGO pieces – and they are imagining that God constructed the world out of them. Again, the problem is that the only "correct" prediction of their paradigm is that "something is made of something" but this prediction was already incorporated as an assumption so it is not really a prediction at all! They only got what they assumed. And everything else is wrong. They don't get any electromagnetic fields, waves, Yang-Mills fields, Higgs fields, fermionic generations, electromagnetic, Yukawa, and Higgs quartic interactions.

There is a whole category of armchair physicists whose plan is similarly hopeless. Even though many of them are much older than 10, they still mentally play with LEGO and they imagine that everything is made of it. They don't ever get a single encouraging sign and they fail to notice. They haven't understood physics at all – or where it stands in the 21st century.

How does string theory succeed? String theory is a unique theory and since the 1990s, we have known tons of things about its nonperturbative behavior (that puts all the perturbative flavors into one theory that is even more connected and sexy than previously thought). But even if you focus your attention on perturbative string theory (where the goal is to calculate particles' scattering amplitudes as a Taylor expansion in the string coupling constant \(g_s\)), the success in reproducing the previous layer of the approximate theories is just amazing.

First, it's important to realize that perturbative string theory doesn't assume the existence of any "spacetime fields", especially not "fields with a nonzero spin". Nevertheless, their quanta are obtained – as strings in some energy eigenstates of the vibration – and the properties of these quanta exactly match the qualitative Ansatz we started with, namely the Higgs, Dirac-Weyl, Maxwell or Yang-Mills, and graviton fields with the desired interactions (and nothing else at low energies).

The fields' quanta are formally point-like, parameterized by their location (or momentum). A string becomes effective point-like in a long-distance approximation in which the dimension of the string seems negligible. It has some squared mass \(m^2\) that counts the amount of internal vibrations (which is quantized due to the usual magic of quantum mechanics, basically quantum harmonic oscillators). The string's (or particle's) spin is similarly nonzero (but a multiple of \(\hbar/2\)) and it is realized as some "orbital angular momentum of the string bits' relative motion".

These qualitative properties could be obtained for any "extended" object that replaces the point-like particles. But once you study the interactions between these particles (originated from strings), you find something way deeper, more specific, and more aligned with the world as we know it (the Lagrangian we started with). Unlike little green men, the strings directly imply what the interactions between all the particles are (at all energies, without adjustable parameters) and the resulting interactions precisely match the electromagnetic/gauge, Yukawa, and Higgs quartic interactions we started with. It's really a miracle of a sort because if a particle were a little green man, it could have a similarly quantized mass but the allowed interactions for the little green men would display (almost) no similarity with the Lagrangian of the Standard Model or Einstein's theory.

Fine. First, you get the qualitatively right spectrum of particles (which are fields quanta in quantum field theory; but they are derived "more directly" from strings, without fields, in string theory). In heterotic string theory using the RNS formalism, there is ground state tachyon \(\ket 0\) of a string that is ultimately filtered out; but the surviving excitations include \[ \alpha_{-1/2}^\mu \tilde \alpha_{-1/2}^\nu \ket 0 \] which has two indices and is capable of producing a spin-two graviton, too. By choosing \(\mu\) or \(\nu\) along some compactified dimensions, you may lower the spin to one or zero, too. And there are fermions in the sector with the periodic fermions, too. Let me not go into details. You may derive the correct spins between 0 and 2 at the massless level; all higher-spin particles are massive (which means very massive, comparable to the string scale which may be close to the Planck energy scale).

Fine, the single string's spectrum is a problem with infinitely many quantum harmonic oscillators and it's fun to play with it. But the true success and nontrivial checks validating string theory only begin once you start to consider the strings interactions. The strings (or particles that are made of strings) start to interact as soon as you allow the topology of the two-dimensional world sheet to be variable. Note that locally on the world sheet, the theory is already determined, so when you say "let the strings be free to have any topology of the world sheet", you don't really add any information about the interactions. The interactions are directly encoded in the rules defining the spectrum. It is not quite the case because one needs to determine the allowed boundary conditions and sometimes there are several options but it is still true that you are insanely far more constrained than if you were just defining a theory with "some objects" where you would have to invent "some interactions". The interactions of the strings are mostly determined automatically.

We have observed that spin-two massless particles are found among the excitations of the string. What are the interactions of these spin-two particles? Once you seriously learn the math from the first chapters of a string theory textbook, you will know that these interactions of spin-two particles are exactly what you would deduce for the interactions of gravitons, the quanta of the metric tensor field, that follow from the interaction terms involving the metric tensor and other fields. And those interactions of fields are determined by the equivalence principle. You get the same form of the interaction for the graviton-strings; you may derive the equivalence principle just from strings! This is looks like a miracle but once you know how to calculate with the stringy mathematics, it is a straightforward and rigorous proof.

Similarly, you may prove that even though the strings (e.g. propagating on a flat space time a torus) don't seem to have any non-Abelian isometries, some of the string states will behave as Yang-Mills fields and their interactions will be... exactly what you expect from the gauge symmetry! The proof is really completely analogous to the equivalence principle case in the previous paragraphs. It's not quite a new relationship between general relativity and Yang-Mills theory: already in the 1920s, the Kaluza-Klein theory knew how to get Yang-Mills theory from general relativity with extra dimensions. When some indices of the tensors go along the extra dimensions, you reduce the (four-dimensional) spin of the particles but there are still interesting patterns left. Gravitons may be reduced to gauge bosons; the equivalence principle is reduced to the gauge symmetry.

OK, you may classify all possible massless particles coming from string theory and their interactions. You get the usual spin 0 up to 2 particles (yes, including the spin-3/2 gravitino that is waiting to be discovered but we don't know whether it is light enough to make it in this century) with the right renormalizable, long-distance interactions: Yang-Mills self-interactions, gauge interactions with fermions, gauge interactions with scalars, Yukawa interactions, Higgs self-interactions. And some gravitino interactions that are almost completely determined by supersymmetry.

The gauge symmetry or the equivalence principle was assumed when we were building the Standard Model or general relativity. In string theory, these things may be derived from a completely different set of assumptions (namely that we have strings whose world sheet topology may be nontrivial). And it's equally important that we have a smaller number of assumptions to determine the theory. We only assume that there are the relativistic strings; and we may deduce the qualitative features of the Lagrangian of our (seemingly unrelated) approximate theory or theories.

Once you get to this point, you realize that the theory is really telling you much more than the approximate theories do. It's almost true that in the approximate theories, the spectrum and interactions were determined by direct measurements. They are not quite arbitrary and the equivalence principle and gauge symmetry are sort of needed for the consistency; but they may only apply once you start with fields with a spin and this assumption may look like a contrived one by itself. And all the pieces (fields of different spins) are independent building blocks that are coupled by interactions which are mostly independent, too. In string theory, all elementary particle species and their interactions come from the same string dynamics.

The stringy calculations are capable of calculating the interactions at arbitrarily high energies and the short-distance (ultraviolet, UV) divergences are completely avoided. It's another midsize miracle that you get. Again, you could imagine that particles are little green men and the extended nature of the little green men is enough to make the theory UV-finite. But it isn't really the case. The generic interactions will be UV-divergent just like the point-like particles' interactions. And even if you only allow some interactions that reflect the softness of the little green men (to get a soft behavior at high energies), you will hit virtually unavoidable problems. Such as those with causality.

If you integrate out all the excited modes of the little green men, you will unavoidably obtain a nonlocal theory for the particles as functions of their center-of-mass. That is true for string theory, too. But such a nonlocal theory will be almost always acausal. A particle at point P of the spacetime will interact with a particle at point Q which may be spacelike-separated as well as timelike-separated. That is a complete nightmare because a generic acausal and nonlocal interaction like that basically depends on arbitrarily high derivatives of the fields. So you need to specify a field and its infinitely many time derivatives. The initial data will depend on the reference frame etc.

You may see that string theory totally circumvents these deadly problems. And it's basically due to the local character of the string interactions – they are local on the world sheet. Locally, the world sheet always looks the same, as we said, so you can't really say where two strings join in the (Euclideanized) spacetime: there is no privileged point on the pants diagram. You may analyze possible "elementary objects" of higher dimensions than the 1D strings. You will find out that nothing works as a foundation for a perturbative string-like theory, basically because the worldvolume gets too high-dimensional and therefore suffering from the limited consistency of non-stringy yet gravitational quantum field theories in \(D\gt 2\).

There is a nice way to see why the UV divergences are absent in (closed) stringy one-loop diagrams. Topologically, the diagram is a torus (toroidal world sheet) and the shape of the torus is parameterized by \(\tau\). The imaginary part of \(\tau\) looks like a Schwinger parameter in one-loop quantum field theory diagrams. But such a Schwinger parameter \(u\) is integrated from \(0\) to \(\infty\). The UV-divergence arises from the \(u\to 0\) region. Nicely enough, the integral over the \(\tau\) plane only goes through the fundamental region\[ |\tau| \gt 1,\quad {\rm Re}(\tau) \leq \frac 12 \] which includes all the non-equivalent conformal shapes of the torus. The region with a small real part of \(\tau\) is completely avoided. It is due to the \(SL(2,\ZZ)\) symmetry of the torus, the "modular invariance". The modular invariance is responsible for removing the UV-divergences in a consistent way. The little green men (or any other objects you invent to be inside particles) fail to have this properties because the little green men don't have any modular invariance. If you imagine the world volume of the little green men in a periodic Euclideanized time, it's still a "circle times a little green men" and there is no other way to rewrite the Cartesian product because the two factors are so different. Because the string geometrically looks just like the circular periodic time coordinate, there is an extra symmetry and that symmetry is sufficient (and almost necessary) for the UV-finiteness. With some extra assumptions, you may say that this is a proof that you need strings in a similar UV-finite theory.

String theory does much more than to produce the qualitatively correct spectrum, the qualitatively correct interactions of particles, and UV-finite expressions (including loops of gravitons) at all energies. It really unifies all allowed "field theory Lagrangians" of this kind. In quantum field theory, a new Lagrangian means a new theory. The objects living in two QFTs must be said to be totally disconnected. People don't respecting the most fundamental laws of Nature are even more alienated aliens than aliens who believe in a different religion or irreligion. You won't ever meet them.

In string theory, you will find out that there is still a "landscape" of possible derived effective QFTs that comes from the non-uniqueness of the shape of extra dimensions (which is very constrained, however), from the extra branes and fluxes which may be added (which are also constrianed), and from some discrete-torsion-like discrete parameters that determine some allowed sets of boundary conditions and/or the phases they contribute to the Feynman path integral. This landscape seems rich enough to contain the Universe around us, including the more detailed properties of particles than the qualitative ones that were obtained correctly in the "previous sketch".

And what's amazing – another collection of midsize miracles – is that all these possibilities are ultimately connected into one theory. The connectedness of the possible "string theories" into something that is really just many vacuum states of one string theory becomes particularly striking once you appreciate the nonperturbative phenomena, topological transitions, and dualities. But even at the level of perturbative string theory, there exist many transformations of "one vacuum into another" that would look stunning or impossible in point-like particle-based quantum field theories.

Two of the five ten-dimensional flat-space supersymmetric string theories are the heterotic string theories. In the \(D=10\) spacetime (which is compactified in the environments like the Universe around us), there is a supergraviton multiplet and a super-Yang-Mills multiplet. The latter has a gauge group that must be either \(E_8\times E_8\) or \(SO(32)\). These are two possible gauge groups, two possible heterotic string theories. They cancel some anomalies which may already be calculated in quantum field theory; the anomaly cancellation is much more automatic and "derived from something more fundamental" in string theory.

Both of these gauge groups have the rank equal to 16; and, more interestingly, the dimension equal to 496. Note that\[ 248+248 = \frac{32 \times 31}{2 \times 1}. \] They actually share several other, harder to define, invariants. Now, at the level of quantum field theory, a quantum field theory with these two gauge groups would be "analogous to one another" but otherwise as disconnected as you can get. When you assume the gauge group to be \(E_8\times E_8\), it just can't be changed to \(SO(32)\) or vice versa because the gauge group is a part of the most unchangeable DNA of a field theory, model builders in QFT normally think.

But within string theory, you may see that this assumption is wrong. String theory is miraculously able to change one of the gauge groups to the other – and the intermediate steps are as consistent as the limiting ones. Just compactify one of the 9 spatial dimensions on a circle. Break the gauge groups to \(U(1)^{16}\) by the Wilson lines. The spectrum of possible Wilson lines and related \(B\)-fields may be adjusted to restore the symmetry either to \(E_8\times E_8\) or \(SO(32)\). So a \(U(1)^{16}\) theory at a point in between may be derived as a spontaneously broken theory with \(E_8\times E_8\) or from \(SO(32)\), via the Higgs mechanism, but the broken theory is exactly the same! At all energies!

String theory allows you to interpolate between the different options. Topology change is possible – flop transitions, conifolds. Even perturbatively, you may see T-duality that says that a toroidal compactification with the radius of a circle \(R\) is equivalent to another with a circle of radius \(\alpha' / R\), and so on. There is a large number of such amazing features of string theory that wouldn't hold if strings were replaced with the little green men (or raspberries or tigers or octopi or anything else, even membranes which are slightly more promising than the previous three). With some extra assumptions, you may really see that strings are the only fundamental objects "to be hidden inside elementary particles" that may generate these great properties. None of these proofs of the inevitability of string theory is completely rigorous and complete; but the number of highly suggestive patterns or nearly complete proofs is high and they seem very independent of each other.

The picture gets stronger within non-perturbative string/M-theory. Even the boundary conditions on the strings and other things we used to define "a perturbative string theory" become derived from a more elementary or more fundamental starting point. And the broader theory is more tightly unified and constrained than anything before. A person who looks for a deeper understanding of Nature within the wisdom outlined by string theory is really centuries if not millenniums ahead of a would-be competitor who wants to ignore string theory and hope that his tigers, octopi, or raspberries will miraculously do a better job than string theory. Be sure that they won't and they can't.

And that's the memo [Sorry for typos, I don't plan any proofreading].

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