I’m reading Lawrence Krauss’ Hiding in the Mirror, about the allure of extra dimensions in theoretical physics. On page 132 there’s an odd statement:
No details on what those “mathematical tools” are or how this might work. Can somebody explain what he’s talking about?
No appropriate method will show that the sum of the natural numbers is -1/12. Ther is a proof that, in the 2-adic numbers, 1 + 2 + 4 + 8 + … = -1; perhaps Krauss is confused with that?
Now how can this be?
Huh? Can you elaborate a little? How is can 1 + 2 + 4…= -1?
It’s a reference to zeta function regularization.
For s > 1 (technically Re s > 1), the series
1 + 1/1[sup]s[/sup] + 1/2[sup]s[/sup] + …
converges to a function of s called the Riemann zeta function. That function can then be uniquely defined for s = -1 by a process called analytical continuation. The answer turns out to be -1/12.
That’s not to say that this is the sum of the series. But it can be the case that you know you’re looking for a function with certain properties, one of which is that it has this series expansion when s > 1. In such circumstances, it’s therefore a reasonable method to define the function in this region using the series expansion. Analytical continuation is then a way of taking the function in the region you know it in and extending it to find its value in a region where you don’t. The function of interest has a sensible value at s = -1, even though it doesn’t have a meaningful series expansion there.
This particular case is famous because it cropped up in early string theory. There are other methods of deriving the same ultimate result without going through this particular piece of gymnastics.
Make that: There are other methods of deriving the same ultimate result there without going through this particular piece of gymnastics.
Ah, I see. Interesting.
This is exactly the context in which it came up. I guess Krauss was simplifying the description, but it’s a pretty weird thing to say without elaboration. He should’ve at least mentioned the Riemann zeta function.
Actually, zeta(-1) = -1/12
Doesn’t zeta have poles at the negative integers?
As for the 2-adic integers, you can view them as a sequence of remainders mod 2[sub]n[/sup]. So 1 = (0, 1, 1, 1, … ), 2 = (0, 0, 2, 2, …), 4 = (0, 0, 0, 4, …). Add those together and you get (0, 1, 2, 4, …) which is equivalent mod 2 to (0, -1, -1, -1, …), and that’s equal to -1.
Nope, it can be analytically continued everywhere but a pole at 1. It has zeroes at the negative even integers. I think you may be confusing it with the Gamma function, which was designed to have poles at the negative integers, but is more well-known for growing like the factorial function in the positive direction.
Incidentally, you might think string theorists are being weird to use an analytic continuation to get this result, and weirder still for taking it seriously. However, quantum field theorists, and even some normally-sober analysts analytically continue the dimension they’re working in – a natural number if ever I’ve heard of one.