Question for Tarski

[_Gadianton]

It turns out that 1+2+3+4+5....to infinity =

:drumroll:

-1/12.

I find this disturbing, please do some apologetics for the institution of mathematics.

(fyi for others, there is a 1+2+3... link on wiki and a couple websites come up that throw around basic concepts but nothing i found on page 1 with a good explanation)

[_Tarski]

LOL

I guess this is zeta function regularization.
In order to appreciate this, folks would have to realize that the series 1+2+3+4+......
is divergent by ordinary standards (besides, why would it sum to a negative number??).

How far back do I have to go?
Complex numbers?
Holomorphic functions?
Analytic continuation?

I can give a quick expo of this stuff but I need it all to make sense of your example.

What about the easier example of summing up the squares of all positive integers and getting -1/3?
Edit: oops, I meant the sum of all even powers of 2. See below

This stuff is actually used in physics. It seems God avoids unwanted infinities by regularization too!

[_Gadianton]

Yeah, I raise the question because Krauss brings this up in his discussion of dual strings and the 26 dimension model. sigh. this is a lot of work just to figure out what I think of Krauss's argument against God.

It's been over a decade since I've done math but I'd like to learn this one.

Complex numbers? no (but do so for lurkers who might say yes)
Holomorphic functions? yes
Analytic continuation? yes


What about the easier example of summing up the squares of all positive integers and getting -1/3?

sure, if that's a better place to start. However, I would eventually like to see the -1/12 enchilada, given it's importance.

[_Tarski]

Yeah, I raise the question because Krauss brings this up in his discussion of dual strings and the 26 dimension model. sigh. this is a lot of work just to figure out what I think of Krauss's argument against God.

It's been over a decade since I've done math but I'd like to learn this one.

Complex numbers? no (but do so for lurkers who might say yes)
Holomorphic functions? yes
Analytic continuation? yes


What about the easier example of summing up the squares of all positive integers and getting -1/3?

sure, if that's a better place to start. However, I would eventually like to see the -1/12 enchilada, given it's importance.

OK, but being a tiny but busy, I am going to drag it out into a series of tidbits and won't really start until tomorrow.

However, to prime you to the sense of things, consider the following situation:

Suppose that you have a mathematical procedure that you can apply to a list of numbers a1 a2 a3 .....(infinite list) that has the following properties:

1) The procedure is elegant, natural, and connects to already known important mathematics.
2) Whenever the infinite sum a1+a2+a3+.... converges in the usual sense (the sequence of partial sums converges to some number in the usual delta epsilon sense) then your procedure gives that very answer as well.
3) Your procedure gives finite answers for a large class of cases where the infinite sum diverges (so no answer in the usual sense).

Would it make sense to adopt a stipulative definition to the effect that in a case that is normally divergent we nevertheless say that a1+a2+a3+.... =A where A is the number that your procedure gives?

What if there were more than one such natural procedure (a normalization) but that disagree outside the circle of traditional convergence?

[_Gadianton]

Tarski,

work is also killing me, so i'm low on time, and when it gets like this it's also intellectually draining for me, the little hampster running around in my head gets tired. I've thought about your post though, and I hope you keep going.

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?

Would it make sense to adopt a stipulative definition to the effect that in a case that is normally divergent we nevertheless say that a1+a2+a3+.... =A where A is the number that your procedure gives?

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. Otherwise, there would have to be some other kind of intuition at work to convince me that the method can be trusted. That's why I asked above how diliberate the discovery of these normalizations were. For instance, if someone was looking for a new method to solve a convergence series, accidently applied it to a series that diverges and wow, it comes up with a finite answer, but there isn't a proof for the procedure or good reason why, and the answer is totally unintuitive, like -1/12, then my gut instinct would not be to bet the farm on it.

What if there were more than one such natural procedure (a normalization) but that disagree outside the circle of traditional convergence?

So multiple "normonalization" procedures that all agree on what we already know, but disagree on the hard questions? Shouldn't we pray in this case and ask which one is true? If you're implying there are in reality multiple methods -- methods that are not just restatements of each other in different guise -- that agree with each other for convergence and divergence, then that's another interesting question. Assuming no formal proofs of the methods, can we accept math propositions based on inductive reasoning like we do in science? (i mean statistical, not the math induction)

[_Tarski]

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.


[

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.

[_Gadianton]

Ok Tarski,

I had to watch a few youtube videos and do some review but I think I'm with you.

One question: Is it a "radius" because on the complex plane it traces out a circle rather than just having the "interval" along x axis?

So where you're at by the end:

you've produced a function that represents a series. you've plugged in 2, and the series diverges, but converges for the function. But I'm a little confused. it's unsurprising that i and -i would be the radius for complex numbers, given the distance from the origin is 1 for x before you plugged z in, so this makes a circle. But my understanding is that everying within the circle converges, everything on the edge could go either way depending on the series, but then what's clearly on the outside of the radius of 1, which 2 is, diverges. Plug in 1.1, that diverges but the function shows it converges.

If what you're going to do is prove the function is right, I'm going to have a hard time accepting it. Look at how crazy this is: say we plug in 3, and its -1/8.
1+9+81+729 diverges worse than when plugging in 2, but it comes out a smaller number. if you plug in infinity, it's -1/infinity^2, or approaches 0?

This holomorophic stuff better be magical...

[_Tarski]

But I'm a little confused. it's unsurprising that i and -i would be the radius for complex numbers, given the distance from the origin is 1 for x before you plugged z in, so this makes a circle.

The radius is one. The distance to i (and -i) is one.
Power series converge inside circles but what they converge to often agrees with -or is equal to--a holomorphic function. Indeed, as long as we are looking at a so call simply connected domain, any such extension is unique.

Now suppose we have a series that converges to an analytic function on some disk or radius say R. Now go out to some point z_0 on the disk near the edge of the disk. There will be a new power series centered there ( expanding in powers of (z-z_0)) that also converged to that same function but not the radius of convergence may be big enough that we have extended the function outside the original disk (or maybe we already know what this function is as in our example).

If what you're going to do is prove the function is right, I'm going to have a hard time accepting it. Look at how crazy this is: say we plug in 3, and its -1/8.
1+9+81+729 diverges worse than when plugging in 2, but it comes out a smaller number. if you plug in infinity, it's -1/infinity^2, or approaches 0?

This holomorophic stuff better be magical...

It is more like the lesson is that the holomorphic function itself is the important reality and the series is just trying to get at it but can only do so a disk at a time.

Holomorphic stuff is magical. Read complex magic here:
http://staff.washington.edu/freitz/penrose.pdf

The stuff about analytic continuation and the functions I mentioned is all in there.

[_Tarski]

Let me just make it clear that when people say that 1+2+3+4+5....to infinity =

-1/12, they are being a bit cutsy. I am sure Krauss presents this in a provocative way because it sounds deep make it a better read. Physicists seem to love to make their subject spooky.

To oversimplify, one has a physical theory that involves equations whose solution can only be obtained by an approximation scheme such as a perturbation expansion.
The series expansions that arise give infinity when the relevant numbers are inserted. This is bad.
Now, the divergent series wasn't just handed to us, it arose in a specific way. The question is, can one obtain an answer that

1) respects, or maintains an interesting connection with the way in which the divergent series arose and
2) Gives a finite answer that agrees with experiment (or is at least reasonable and consistent with known facts).

If the numerical series was obtained by plugging a specific number into a function series (like a power series or a zeta function type series) and if the series converges to a holomorphic function on some disk that excludes the point of interest, and if that holomorphic function itself is known to actually make sense for the complex number we plugged that resulted in a divergent numerical series, then we just plug into the holomorphic function and use that answer.
This either gives a physically reasonable answer or not.
We have traded in the divergent series for something directly connected to it but that is well behaved for a larger set of complex numbers. To be as deflationary about this a possible, one could insist that 1+2+3+4+5....to infinity =-1/12 only in a sort of metaphorical sense. Or just say it is so by definition and let all that stuff about holomorphic functions serve as simply motivation for the definition.

[_Gadianton]

To be as deflationary about this a possible, one could insist that 1+2+3+4+5....to infinity =-1/12 only in a sort of metaphorical sense

hmm. wondering if this is good. You know some apologist out there must be considering that if you add up all the numbers from alpha to omega, it sums up to Jesus presiding over the twelve apostles.

I see you've linked to the entire text of a book I've mentioned a couple of times is above my head as a reference. ha.

well, it looks like series stuff is discussed fairly early in the book so maybe I'll give it another try for at least that portion.