Running this blog, I regularly check referrers – that is, the sites who’s links to GM/BM actually bring readers to the blog. It’s interesting to see who’s linking, and it helps give me an idea which posts people find most interesting. Also, every once in a while, it gives me something interesting to blog about. Like today. I got linked to by the Flat Earth Society!
The FES is the grand-daddy of hopeless crackpot organizations. They are, in all seriousness, a group of people who believe that the earth is flat. They’ll tell you that the space program is a total fraud. Every picture you’ve seen from space is faked. No one ever went to the moon. Depending on which flat-earther you talk to, satellites are either a fraud, or they’re just great big balloons floating above the earth’s surface. It’s all an eloborate conspiracy for some nefarious reason.
The FES set up a group of forums. And in one of them, someone posted something about that old -1/12 nonsense. It’s the worst misunderstanding/misrepresentation of the idea behind that that I’ve seen so far, and let me tell you, that’s really saying something.
I’ll keep this short so that it might get read.
Analytic continuation allows us to approximate an infinite series as a function. The more terms you add to the series, the more accurately you describe the function. So at the limit, when you have all the terms, that series can be considered equal to that function.
This allows us to assign meaningful numerical values to infinite series, even divergent ones.
For example, the infinite series “1 + 2 + 3 + 4 + 5… ” can be evaluated. However, the result is disturbingly counter-intuitive. If you stop adding terms at any finite point, you’ll have a larger and larger number as the result. However, if you “evaluate this after an infinite number of terms, you’ll find the sum to actually be -1/12.
There are a variety of proofs for this, and this number is demonstrably a meaningful . This result is actually seen in physics. Furthermore this result is a foundation in string theory for the number of required dimensions.
Analytic computation isn’t about converging infinite series. It’s not about adding more and more terms. What he’s describing is, simply, the idea of an infinite series that converges to a value. For example, consider the following:
That series taken to infinity is equal to 2. You can keep drawing it out. At any point, the sum of the series is less than 2. 1 1/2, 1 3/4, 1 511/512, 1 65535/65536, on and on, forever. But if you keep following it, using the mathematical concept of limits, it’s easy to show that the sum of the full series is exactly 2.
You don’t need to bring analytic continuations into the picture to show that – just limits. In fact, analytical continuations don’t help with solving that problem – they’ve got nothing to do with it.
Analytical continuation is both much simpler (in concept) and much harder (in practice) than convergence of infinite series.
Analytical continuations come into play when we have an partial function for something, which we’re using as an partial solution for the value of something else. There are lots of problems where we don’t know a perfect, total closed-form equation for some function that we’re interested in. But we’ve managed to find a closed-form equation that matches the function we’re interested in a lot of the time. Then, using some pretty hairy complex (in the sense of complex numbers, although it’s also pretty complicated) analysis tricks, we can figure out what the value of the target function should be at places where our partial solution is undefined. So even though the partial solution doesn’t work some of the time, we can use that partial solution to derive the actual solution. That process of using analysis of the partial solution to get at least some of the undefined points where it the partial solution doesn’t work is analytical continuation.
In the case of the infamous -1/12 argument, we’re trying to probe at a really important thing called Riemman’s zeta function. The Zeta function describes fundamental properties of prime numbers. It’s an important deep thing which ends up having a lot of applications when you’re dealing with number theory, topology, and differential equations, among other things. Because of how it describes some fundamental properties, any concrete application of those fundamental properties ends up involving Zeta as well – including things like string theory, which propose a topological structure for the univese.
We don’t know, in general, how to write a simple equation for Zeta. We do know how to write an equation that is a partial subset of the zeta function – a equation that describes a function that works much of the time, but which is not zeta, and which is not defined in some places where zeta is.
Using analytical continuation, we can compute the value of the zeta function in places where the equation that we use for a partial approximation does not work, where that equation cannot and will not ever produce a result.
Using analytical continuation, . At , the equation that we know for computing zeta in some places does not work. At all. But analytical continuation shows us what the value of is at that point: it does not tell us the value of computing that equation at -1: that’s doesn’t work.
(In terms of physics, the application of this is interesting. The zeta function is used in a lot of places in string theory, and it’s normally “defined” in physics text as being precisely the series that gives us the partial approximation of zeta. So in string theory physics, when you find that series, it’s not actually that series; it’s zeta, which was expanded out into that (technically incorrect) series. Since it’s zeta, you can replace it with zeta. Since zeta(-1)=-1/12, that means that in those physics equations, you can effectively pretend that 1+2+3+4+5+…=-1/12. It isn’t, but a combination of a notational convenience and a shortcut to avoid a long-winded explanation of analytical continuation means that too many physicists are taught to believe that it is.)
So, shocking as it might be: flat earthers – they’re not just wrong about geography!