String theory has no experiments, so has no touchstone to justify dodgy maths. String theorists routinely depend on the infinite sum 1 + 2 + 3 + ... = -1/12, without apology. It's anybody's guess how well that will turn out in the end, but it hasn't interfered with grants, publication, or tenure, so it's OK so far.
If memory serves, 1+ 2 + 3 + ... = –1/12 also figures in the estimation of the Casimir force between two electrically uncharged plates (which has been experimentally verified).
It's a rigorous result if you define it as an analytic continuation of the Riemann zeta function defined in the normal way. It doesn't take much to prove, but the way it's used in physics it's not always immediately clear that interpreting infinite and divergent sums as analytic continuations of finite sums is justified.
This is still a lie, but the truth is that F(s) does make sense for a large domain of s, and there is a unique and canonical way of extending F(s) to a larger domain which contains 1. And then that new function evaluated at 1 is equal to -1/12.
The trick is that normal everyday summation is only defined for convergent series, so a slightly different definition of summation (called zeta function regularization) is used to assign a value to this divergent series.
Physicists seem to be suggesting that our universe behaves in accordance with this second definition of summation.
Mainly that in physics, these sums crop up naturally, using normal summations over real numbers, often sums of energies over different modes.
For instance, if you look at an idealized quantum violin string, each mode of vibration has a minimum energy (zero point energy) proportional to the frequency of vibration - classically they can all be 0, but not quantum mechanically. When you try to ask, then, what the minimum energy of the string is, you end up with a term that is literally 1+2+3+... and no particular reason to intepret those as complex or anything. But in a lot of ways, if you just barrel through and treat it as if you can do the sum, you get real results - the Casimir effect is an example where a real force can be predicted and measured based on calculations that are zeta regularized.
It's also worth noting that the dimensionalities in various string theories tend to hinge on the exact values of these infinite sums. Bosonic string theory being 26 dimensional comes out of, IIRC, a consistency equation that ends up including -1/12 because of a 1+2+3... sum. If memory serves, that result can be rigorously established in other ways, as well.
Thanks for the insight. I'll have to look at this more closely sometime... not sure I can grok it casually.
I've been watching the PBS spacetime videos on youtube a lot lately. The recent ones have been diving into zero-point energy (and related stuff), which has been fascinating.
Some people say that they are using "=" to mean something else, and relying on context to distinguish which "=" they mean. But it's less fun to say that the sum of the natural numbers "is associated with" -1/12, especially if you have to say how, exactly.
It’s the Ramanujan Summation of the Euler Zeta Function (not to be confused with the Riemann Zeta Function, which deals with complex values, which is a generalisation thereof) when s = 1.
