Can Zeta Regularization Explain Divergent Sums in Physics?

[tahayassen]

Can anyone listen to two or three of his videos and tell me if he made any mistakes in his videos?

 

[morphism]

He is being intentionally fast and loose with his math, probably to stir up controversy and gain more views. The fact of the matter is that whenever you deal with infinite sums in such a manner, you must pay attention to issues of "convergence". You simply cannot manipulate divergent (and even some convergent) infinite sums as you would finite sums and expect everything to be fine.

If you tell us what your mathematical background is we might be able to suggest some reading on what's going on.

 

[tahayassen]

I'm taking Grade 12 Advanced Functions. I'm going to start Calculus next semester. We're almost done this semester.

 

[Jamma]

I looked at his videos, and they seem fun and mostly correct (although he skips a lot of details for you to learn all that much).

Note that you shouldn't say that this limit converges to -1. When he says that this sum IS -1, he's just being controversial, but in some sense we can view this sum as being -1

http://en.wikipedia.org/wiki/1_−_2_+_3_−_4_+_·_·_·

I don't know all that much about the uses of this, but apparently there are uses. But just remember, this sum is equal to -1 in only a very abstract way and he should really have said this since the sum doesn't converge.

 

[Containment]

Well there is one that talked about the arrow of time and it gave an idea to ponder so I guess I like them :)

 

[Dickfore]

He is using an analytic continuation of the geometric series:
$$f(z) = \sum_{k = 0}^{\infty}{z^{k}} = \frac{1}{1 - z}, \ \vert z \vert < 1$$
The sum is only convergent within the circle of radius 1 in the complex plane. But, in this region of convergence, it converges towards a simple algebraic function that is defined everywhere, but $z = 1$, which is a simple pole (and it determines the radius of convergence of the series). What he is calculating is basically $f(2) = -1$. The involved procedure is the one used in evaluating the sum of a geometric series.

 

[tahayassen]

Can anyone clearly explain this video?

 

[Jamma]

What exactly is confusing about it, it seems pretty well explained already to me.

 

[Mute]

There was a previous thread asking about this video. This is the thread: https://www.physicsforums.com/showthread.php?t=532395&highlight=youtube

Copying my explanation from that thread:

Perhaps watching the follow-up video (linked to at the end of the movie, but I'll link here as well) will answer some questions: video link

If you're left with more questions than answers after that (which you likely will be), the 'technique' which physicists use is called "regularization" or "zeta regularization" in some specific instances.

The basic idea is that sometimes when you run into divergent sums in your calculations (in physical problems), they're really not supposed to be divergent sums - they should be something else that's finite, but due to approximations or the theory being incomplete you get this divergent beast. The regularization is a trick to replace the divergent sum with something finite, which is what the sum is "supposed to be".