Gadianton wrote:
So what I think you're telling me is there are these procedures ("normalizations"?), "zeta" being one of them, that pace techniques like a limit, or Taylor series, that work when n-> infinity and converge. But unlike the college freshman techniques, these "procedures" give finite answers to divergent series -- somehow. Is this accidental? Did someone get tricky here and intuitively work their way out of the muck?
Mathematically it is all quite rigorous. Philosophically nothing spooky is happening.
Now, it might be occasionally mysterious why one way works for a physical problem, but other times it is pretty clear.
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That's a great question. I don't know. I know some procedures come with formal proofs, so if there is a formal proof for the procedure then yes.
Mostly, we have proofs and precise definitions for the mathematical parts. On the other hand, physics seems to always be on the edge so they might be using some stuff that lacks complete proofs.
Witten's approach to Chern-Simon theory is like that.
Let's just see how you feel after we get to the point of understanding Zeta function regularization.
First, I am going to ask all interested parties to look up the meaning of complex numbers and complex sequences, complex series. In particular, complex power series.
The first main point will be to consider the function of a real variable given by
1/(1-x^2) (Here x^2 means x squared)
Now we have that 1+x^2+x^4+x^6+......converges to 1/(1-x^2) but only for x between -1 and 1. Now, it is not surprising that something goes wrong as x attempts to move outside that interval since obviously 1/(1-x^2) is undefined at +1 and -1 (we say it blows up).
But we also have that 1-x^2+x^4-x^6+......converges to 1/(1+x^2) but again only for x between -1 and 1. But this time the expression 1/(1+x^2) gives a function that is perfectly fine at -1 and 1 and in fact everywhere else on the whole real number line. There is no immediate hint that the interval of convergence of 1-x^2+x^4-x^6+...... should be trapped to lie in the interval -1 to 1.
But, if we look at things using a complex variable z=x+iy then we see something a little bit enlightening. You see, every complex power series has it's own radius of convergence, say R.
We can consider powers of (z-a) for a fixed complex number a which will happen to be zero below. If z is within the circle of radius R, centered at a then the series will converge and will diverge if strictly outside.
Now consider 1+z^2+z^4+z^6+......converges to 1/(1-z^2) for z within some circle centered at 0=0+0i. Can you guess the radius? How big can it be without bumping into a point (complex number) that 1/(1-z^2) cannot handle?
OK, now replace z by iz to get 1-z^2+z^4-z^6+...... which converges to 1/(1+z^2). Now ask the same question. How big can the disk centered at 0 get before 1/(1+z^2) has trouble?
Hint: i and –i.
You can already see something in the offing: If we put z=x=2 into 1+z^2+z^4+z^6+......we get 1+2+4+16+64+…..which doesn’t converge. But if we put z=2 into 1/(1-z^2) we get -1/3.
Hmmm. To make this more interesting we need a stronger connection between the left and right hand sides. (The series and the function). We need the notion of analytic (a.k.a. holomorphic) function and the notion of analytic continuation.
I think I messed up earlier, the infinite sum thingy that kinda gives -1/3 is as in this post.