Yo Dawg, I Heard You Liked Quantum Field Theory so I Made You a String Theory

String theory may sound strange and exotic, with its extra dimensions and loops of string. Deep down, though, string theory is by far the most conservative attempt at a theory of quantum gravity. It just takes the tools of quantum field theory, and applies them all over again.

Picture a loop of string, traveling through space. From one moment to the next, the loop occupies a loop-shaped region. Now imagine joining all those regions together, forming a tunnel: the space swept out by the string over its entire existence. As the string joins other strings, merging and dividing, the result is a complicated surface. In string theory, we call this surface the worldsheet.

Yes, it looks like Yog-Sothoth. It always looks like Yog-Sothoth.

Imagine what it’s like to live on this two-dimensional surface. You don’t know where the string is in the space around it, because you can’t see off the surface. You can learn something about it, though, because making the worldsheet bend takes energy. I’ve talked about this kind of situation before, and the result is that your world contains a scalar field.

Living on the two-dimensional surface, then, you can describe your world with two-dimensional quantum field theory. Your two-dimensional theory, reinterpreted, then tells you the position of the string in higher-dimensional space. If we were just doing normal particle physics, we’d use quantum field theory to describe the particles. Now that we’ve replaced particles with strings, our quantum field theory describes things that are the result of another quantum field theory.

Xzibit would be proud.

If you understand this aspect of string theory, everything else makes a lot more sense. If you’re just imagining lengths of string, it’s hard to understand how strings can have supersymmetry. In these terms, though, it’s simple: instead of just scalar fields, supersymmetric strings also have fermions (fields with spin 1/2) as part of their two-dimensional quantum field theory.

It’s also deeply related to all those weird extra dimensions. As it turns out, two-dimensional quantum field theories are much more restricted than their cousins in our four (three space plus one time)-dimensional world. In order for a theory with only scalars (like the worldsheet of a moving loop of string) to make sense, there have to be twenty-six scalars. Each scalar is a direction in which the worldsheet can bend, so if you just have scalars you’re looking at a 26-dimensional world. Supersymmetry changes this calculation by adding fermions: with fermions and scalars, you need ten scalars to make your theory mathematically consistent, which is why superstring theory lives in ten dimensions.

This also gives you yet another way to think about branes. Strings come from two-dimensional quantum field theories, while branes come from quantum field theories in other dimensions.

Sticking a quantum field theory inside a quantum field theory is the most straightforward way to move forward. Fundamentally, it’s just using tools we already know work. That doesn’t mean it’s the right solution, or that it describes reality: that’s for the future to establish. But I hope I’ve made it a bit clearer why it’s by far the most popular option.

7 thoughts on “Yo Dawg, I Heard You Liked Quantum Field Theory so I Made You a String Theory

  1. Thorsten


    thanks for the article!
    I am an experimentalist, but I guess within the string community this 2d -> 10d behavior is generally seen as an “emergence” of the 10 dimensional theory from a 2-d QFT on the world sheet. At least, this is what Witten has said in some talks, e.g. in a recent conference in stony brook here http://media.scgp.stonybrook.edu/video/video.php?f=20150507_2_qtp.mp4, where he emphasizes that 2d dimensions is the “minimum” worldsheet dimensions you need to cure a lot of problems .. and he also mentions that this is the best example of emergent spacetime he thinks exists.

    Nima mentioned in one of his recent talks , https://www.youtube.com/watch?v=C9LwPkijT10,
    that even the worldsheet picture (thickened feynman diagrams) might also emerge from an amplituhedron analogue, and he gets asked by nathan Seiberg in the end (@ min 19) if this is not similar to the worldsheet->10d emergence.

    He answers that the Amplituhedron picture is much more removed from this usual way of spacetime emergence, and I guess this is related to the fact that one does not start with a space time at all. I mean, in the standard 2d->10d emergence, the 2d theory can also be viewed as its own (1,1) spacetime (with 10 scalers+fermions), but contains the usual lagrangian language.

    I am especially confused abut the relation, since it was said in another talk (https://www.youtube.com/watch?v=n0FMgNQfwEY, the very end of the talk, Nima gives a comment) that the amplituhedron picture might be connected to the scattering equations .. but on the other hand the scattering equations follow from a worldsheet picture as skinner has shown https://www.youtube.com/watch?v=KnynWh6VuCw.

    Forgive me if I am mixing up things here, since I am not working in the field myself 😉

    What do you think about the relationship between these two different types of “emergent spacetimes”?



    1. 4gravitonsandagradstudent Post author

      Good question!

      There are, indeed, a couple of different things at work here. It’s important to keep in mind that the amplituhedron is currently quite limited, in that it only works for planar N=4 super Yang-Mills. When Nima talks about an amplituhedron-like picture of the worldsheet, he’s thinking about the possibility of applying the philosophy behind the amplituhedron to string theory, but it’s hard to know what this would look like beyond very broad strokes, namely the core idea of rephrasing things in terms of geometrical objects.

      The scattering equations do indeed follow from a worldsheet picture. Nima’s comment makes sense on one level, in that the structure of the scattering equations probably should be visible from the amplituhedron perspective, but that’s not the same as saying they’re going to reduce to the same thing in the end: currently, the scattering equations only work for purely bosonic theories at tree level, while the amplituhedron covers N=4 sYM at loop level, so even if the two are connected it’s not going to be true that the amplituhedron is “just” a worldsheet theory.


      1. Thorsten

        Hmm yeah that makes sense.
        So if an amplituhedron analogue for N=0 YM would be found, it should in princple be possible to find the connection to the scattering equations in the tree amplituhedron?

        Do you think in the next few years this direct link between the scattering equations and the Amplituhedron will be discovered?

        Another thing thing that is not clear to me is how these perturbative developments connect to all-loop results, e.g. by Pedro Vieira et al (e.g.

        I mean, calculating the amplituhedron volume for each loop level, adding them up should in principle come close and closer to the all-loop calculation using integrability methods, or not?
        Both pictures are studied for N=4 SYM.

        Sorry for all these questions, ever since I have heard of these developments end of 2013 (through natalie wolchovers amplituhdron article and nimas SUSY talk) I cannot stop following these things and think about them.. to me it seems these connections might tighten the bridge between string theory and standard QFT, such that even more and more experimentalists can appreciate string theory .. especially since these developments were motivated by string theory ( at the moment certainly most experimentalists I know do not).


        1. 4gravitonsandagradstudent Post author

          If an N=0 amplituhedron were found, yeah, it would probably have a more direct connection to the scattering equations. (Alternatively, if super-scattering equations were found, they could be linked to the amplituhedron.) It’s probably going to be quite some time until an N=0 amplituhedron is found, though, probably not in the next few years. They’re currently working on getting a nonplanar amplituhedron, and while they’ve made some progress it looks like it’s slow going.

          The connection between the amplituhedron and integrability is pretty indirect. Remember, the amplituhedron is just computing an integrand, which still has to be integrated to get the amplitudes, which then would have to be added up loop level by loop level to get the integrability results. Basically, they have to connect up to my stuff before they can hope to reach integrability. I’ve been talking with Jaroslav Trnka about this, and we’ve got a few tentative ideas, but the topic is still very young.

          If you want experimentalists to appreciate string theory’s contributions to QFT, you may want to focus on things that have already worked. 😉
          In particular, generalized unitarity was first developed in N=4 sYM, but now is the state of the art for one-loop LHC physics via the BlackHat collaboration. There are currently a bunch of people working on extending this to two loops, and their work couldn’t have happened if people hadn’t explored N=4 at two loops first.


          1. Thorsten

            Thanks for the reply!

            Hmm yeah the BlackHat publications was actually what I meant. I had heard somewhere that the BCFW recursion relations (maybe they are also part of BlackHat) are used for some calculations for the LHC. .. which would then show a link to the amplituhedron as one particular triangulation in the case of N=4 SYM. The application to LHC results convinced me so I am convinced that other experimentalists can be convinced as well via this route yes ;).


  2. Wyrd Smythe

    Interesting, but my limitations as an interested amateur came into play here big time, so there wasn’t much illumination for me. But string theory is something I only dimly grasp.

    One thing I do totally get: It can be a really sheety world sometimes. 😮

    Liked by 1 person

  3. ohwilleke

    “But I hope I’ve made it a bit clearer why it’s by far the most popular option.”

    ‘fraid not. It makes it a bit clearer what string theory involves, but does not make it at all clear why it is popular.



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