One, Two, Infinity

Physicists and mathematicians count one, two, infinity.

We start with the simplest case, as a proof of principle. We take a stripped down toy model or simple calculation and show that our idea works. We count “one”, and we publish.

Next, we let things get a bit more complicated. In the next toy model, or the next calculation, new interactions can arise. We figure out how to deal with those new interactions, our count goes from “one” to “two”, and once again we publish.

By this point, hopefully, we understand the pattern. We know what happens in the simplest case, and we know what happens when the different pieces start to interact. If all goes well, that’s enough: we can extrapolate our knowledge to understand not just case “three”, but any case: any model, any calculation. We publish the general case, the general method. We’ve counted one, two, infinity.

200px-infinite-svg

Once we’ve counted “infinity”, we don’t have to do any more cases. And so “infinity” becomes the new “zero”, and the next type of calculation you don’t know how to do becomes “one”. It’s like going from addition to multiplication, from multiplication to exponentiation, from exponentials up into the wilds of up-arrow notation. Each time, once you understand the general rules you can jump ahead to an entirely new world with new capabilities…and repeat the same process again, on a new scale. You don’t need to count one, two, three, four, on and on and on.

Of course, research doesn’t always work out this way. My last few papers counted three, four, five, with six on the way. (One and two were already known.) Unlike the ideal cases that go one, two, infinity, here “two” doesn’t give all the pieces you need to keep going. You need to go a few numbers more to get novel insights. That said, we are thinking about “infinity” now, so look forward to a future post that says something about that.

A lot of frustration in physics comes from situations when “infinity” remains stubbornly out of reach. When people complain about all the models for supersymmetry, or inflation, in some sense they’re complaining about fields that haven’t taken that “infinity” step. One or two models of inflation are nice, but by the time the count reaches ten you start hoping that someone will describe all possible models of inflation in one paper, and see if they can make any predictions from that.

(In particle physics, there’s an extent to which people can actually do this. There are methods to describe all possible modifications of the Standard Model in terms of what sort of effects they can have on observations of known particles. There’s a group at NBI who work on this sort of thing.)

The gold standard, though, is one, two, infinity. Our ability to step back, stop working case-by-case, and move on to the next level is not just a cute trick: it’s a foundation for exponential progress. If we can count one, two, infinity, then there’s nowhere we can’t reach.

5 thoughts on “One, Two, Infinity

  1. Wyrd Smythe

    Heh! Some underlying math philosophy there. From an abstract point of view, there is only zero and one (nothing and something; yin and yang). And then infinity. 🙂

    Finite numbers are just special cases of one; specific instances of something.

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    1. Adel Sadeq

      Another way to look at it is to say that nothing(zero) has be differentiated to have a meaning, hence something(one) comes about. but now the zero and one has created two things and hence two is born. but now you have three entities. Well, you get the idea.
      As a matter of fact, existence is nothing but a consequence of this truth. The assorted differences between these “quantities” generate the mathematical structure that creates reality.
      here is an example
      http://fqxi.org/data/essay-contest-files/Sadeq_realityfinal2_1.pdf
      http://fqxi.org/community/forum/topic/2884

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      1. Wyrd Smythe

        I don’t know about linking math with existence, but with regard to math, definitely it’s zero, one, infinity. 🙂

        One a related topic, I recently ran into an interesting objection to the idea of the Platonic realm versus our concrete world. The objection is that, on a directed graph illustrating the situation, with one box labeled “Platonic” and the other labeled “us”, an immediate question is why two, and only two, boxes? What’s so special about two?

        This suggests the possibility of a multi-verse of realms, the more “Platonic” ones informing those more concrete ones below. Or that the Platonic duality (perhaps any duality) is false.

        Interesting take, I thought, but I do think Yin/Yang is a fundamental aspect of reality, so I’m comfortable with two-ness and duality. 🙂

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  2. Ann O'Neamus

    For nonexperts like me, this article is too cryptic or metaphoric to extract information (“infinity” becomes the new “zero”? :-)). I guess the numbers you talk about count loops in an expansion? And you calculate certain amplitudes order by order until you notice a pattern that allows you to deduce the answer for any number of loops with much less computational effort than conventional methods that don’t use the pattern? And the same pattern might occur in a large class of similar problems, so you can then go to a new or a more general class of theories that challenges your pattern recognition abilities in new ways?

    Assuming that is roughly what you mean, I wonder what one should expect from amplitude computations in a not too special theory (without supersymmetry, like the Standard Model): Should we expect that there is always a pattern one can see at some finite number of loops and that allows us to compute the answers at higher orders much more easily than by conventional methods? And in a case where such a pattern exists, at which loop order should we expect to notice it, if the theory is not too special? 2? 17? 2^2^1000? (Probably the number grows with the number of interactions in the theory. Can you estimate how?)

    Putting as much detail into black boxes as possible, can you give an example of such a pattern that lets you “extrapolate from two to infinity”? (To understand the basic idea, I’d find one explicit example more useful than a metaphor, even a very good metaphor.)

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    1. 4gravitonsandagradstudent Post author

      Part of the reason the post is ambiguous is that I think this is a general pattern, not just one in amplitudes.

      In amplitudes, sometimes the story is how you describe: we see a pattern at one loop and two loops, and that lets us say something about what the all-loop result should look like. That’s sort of what I’m hoping for in my current work, but we’ve needed more loops than two to do it.

      More often, it’s in the form of a method. People use generalized unitarity at one loop and two loops, and at the time the method was still quite rough. But after understanding two loops, people began to understand how the technique could be applied more generally, and showed that unitarity cuts always contain enough information to fix the full amplitude. Note that actually applying this technique at higher loops can still be cumbersome, the “infinity” here doesn’t mean we now know the answer to any order, just that we know how to use the technique to any order.

      But I do think this structure, “one two infinity”, applies more generally. When Basso, Sever, and Vieira, were figuring out the Pentagon OPE they figured out the behavior of one excitation, then two excitations. Once they understood that, they could propose the rules for an arbitrary number of excitations.

      In some sense, “one two infinity” is just the structure of a proof by induction. You prove the base case, you figure out what you need for case n->case n+1, and that gives you what you need for all n.

      In each of the cases I’ve talked about, once the general case is known you don’t need to go up order by order any more, so you look for a generalization in another direction. With generalized unitarity, that meant going to more general theories, to more general dimensions, filling in gaps in the method and making it more efficient. With the Pentagon OPE, it meant studying correlation functions with the Hexagon OPE. And so on.

      I get the impression this kind of thing happens in pure math too, with the caveat that sometimes “two” and “infinity” really require completely different methods. But I don’t have clean examples in mind for that.

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