String theory and zeta function z(-1)

[pcm81]

Over the last couple of years there has been allot of traffic on youtube about the sum of all positive integers being equal to -1/12 as is explained in numberphile video. Some argue that their calculations are wrong and the sum really is infinity. In their original video numberphile shows a string theory book that is referencing this result. My question is: what would be different in string theory if sum of all positive integers and with it the value of Riemann zeta function at s=-1 is not -1/12 but is infinity? Loooooooooong time ago i got am M.Sc. in physics; but they did not teach string theory back then; so feel free to use physics technical lingo in your answer; i simply do not know enough about string theory itself...

 

[pcm81]

I should add that I watched the 10 lecture series from Leonard Susskind posted on youtube by Stanford; but that is basically all that I heard of in regards to string theory.

 

[cosmik debris]

The -1/12 result is an energy level that is supposed to come out at -1, there is a factor of 2 involved as well making it 24 times too small. The reasoning is that 24 dimensions are needed to make this number -1. The 26 dimensions of boson string theory come from these 24 plus 1 from the infinite momentum frame in the direction of travel for the string, plus 1 for time. That's a pretty lay explanation but since you hadn't received anything I thought I'd have a go.

In this case the sum of the infinite series relies on the function having certain properties (Herglotz?), one of which is that the function must got to zero at infinity. All of these properties affect the final sum of the series. Carl Bender does an excellent set of lectures on Youtube explaining how series are summed in Perturbation Theory.

Cheers

 

[Vanadium 50]

The -1/12 result is an energy level that is supposed to come out at -1

Are you serious about this?

 

[cosmik debris]

Are you serious about this?

Sort of. The OP said he had been watching Susskind and this is the kind of reasoning he used in the video, it may not be entirely correct but the OP hadn't had any responses so I thought I would bump it. It is a bit of a vague memory as I watched it some time ago. I'd be interested in a better or correct answer as would the OP.

Cheers

 

[pcm81]

Thank you both for replies. Found Carl Bender videos on youtube, but havn't seen the part about zeta function yet.

 

[Daniel Gallimore]

My question is: what would be different in string theory if sum of all positive integers and with it the value of Riemann zeta function at s=-1 is not -1/12 but is infinity?

Rather than ask what would happen if the series "equaled" infinity, let's think about why it shouldn't equal infinity.

Over the last couple of years there has been allot of traffic on youtube about the sum of all positive integers being equal to -1/12 as is explained in numberphile video. Some argue that their calculations are wrong and the sum really is infinity.

There is a YouTube video by 3Blue1Brown on analytic continuation with regard to the zeta function that I think better explains the -1/12 result. Normally, we say an infinite series equals a finite value $a$ if the sequence of partial sums approaches $a$. Note that you can't use this definition of equality for a diverging series since the sequence of partial sums blows up. However, you can uniquely analytically continue some complex functions, that is, define a new function within the problem region that meshes very nicely with the well-behaved region. In a sense, it is not that the zeta function function evaluated at -1 that equals -1/12, but that the analytically continued zeta function evaluated at -1 that equals -1/12.

There is another way to make sense of this result using a 2-adic number system. Astonishingly, in this system, the series actually does converge. There are even some flavors of quantum mechanics that base their mathematical framework off p-adic number systems since these odd sums appear so often. 3Blue1Brown may touch on that in the video; I can't remember.

 

[DrClaude]

There is a YouTube video by 3Blue1Brown on analytic continuation with regard to the zeta function that I think better explains the -1/12 result.