String theory definitely spits out some strange and unbelievable things. Ten spacetime dimensions. Particles with negative mass. That sort of stuff. But usually there’s some justification for it. There will be a nice little section in the textbook outlining where the result comes from and what we can do to explain it.

However, about halfway through chapter 14 of Barton Zwiebach’s A First Course in String Theory, without any notice or fanfare, the author slips this little gem in on us (equation 14.84):

After some more algebra and problem solving the same thing shows up again, in more formal mathematical notation (equation 14.88):

Let’s just isolate the juicy bit here:

In plain english, the sum of all whole numbers from 1 to infinity equals negative one twelfth. I’ve tried telling some friends this and they all just stare blankly for a second and then say that that’s wrong. No question. It’s just wrong. I say I’m inclined to agree with them, but if it’s in a textbook it has to be right! You don’t just accidentally type “-1/12″ when you meant to type “infinity”.

What makes it worse is that I haven’t been able to find any reference to where this result comes from. There’s no explanation in the textbook, and the prof in class treated it similarly (“…and so that’s equal to the sum from one to infinity, but that’s just negative one over twelve…”), as if it were a regular everyday mathematical fact.

Would anybody please care to explain?

### 8 Responses to “Negative one twelfth”

1. brina says:

hey, buddy. i’ve explained it already. it’s wrong. and don’t get all “you only know about biology blah blah blah” cuz you and i both know it’s wrong. the sum of infinity is infinity. the sum of twice infinity is twice infinity. and all of it divided over whatever is infinity. so don’t make me hit you. cuz you smell. MOOAHAHAHAHAHA.

2. Michel says:

I would say that you are quite right and the “what’s wrong” is pretty obvious. A few opposite views on that topic, first is they try to give you a quickie there and they drank too much computers when they were little, where the biggest number 0xfffffff (in case of an int in c) also means -1, then, the 12 is … odd …. Second is the author made an errata, but then it’s not on his errata’s web site. Third, since I do not own that book, I cannot say if there’s a special universe in which the numbers lives on, that would explain that equation… or if it’s to be calculated in a special mathematical setting. But purely mathematically speaking, this is bullcrap. I would ask your teacher and I’m curious about the answer ;)

On topic of string theory, I do understand the basic principle they are trying to advance, I find it very interesting, however, the sheer amount of calculation involved in _all_ parts of the theory makes me believe it’s quite fishy. Strings, resonance and harmonics are quite unpredictive in nature and incredibly hard to simulate (or else we’d have perfect computer-based pianos and guitars that are not sample-based). Multidimensionality is another topic where maths are hard at work, we are barely getting a knack out of the 3rd, 4th as time, 5th … well, yeeesh, but now it’s getting out of hand. Finally, the straw that did it to me is the lack of precise calculations. How can a theory still stand when only the oscillation and resonance noises would allow to receive any given answer to any given question, geez they can’t even properly model two strings together and their interaction, so let’s forget a whole universe here. Buuuut that’s another topic… and that’s my “smartass” “offtopic” “flamebait” answer ;)

3. GP says:

Higher dimensions is actually a snap—once you get the hang of four-vectors generalising to higher D is almost trivial. You can’t throw out a theory just because you don’t like the calculations invovled. Considering it’s the only serious candidate for unifying the forces, it would be foolish to do so, unless there is something else that has any hint of being able to do better. The three body orbital problem can’t even be solved analytically, but that doesn’t mean the whole model of the solar system is wrong.

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5. Michel says:

Well, to discuss these beliefs, I do not think the string theory is bad, I read about it, got the Cohen book and avidly read it, read about it elsewhere too. But I’m not one to be perturbated by heavy maths, I find it very stimulating. I find the theory very interesting as a first draft, and fun to talk about. However, there are many points I find hard to believe here.

First, the way we can have as many universes as we want, and they are all as plausible as the other one. It’s like saying, well, the unified theory of all things is the alphabet, and here we got a page full of text and you can type everything in it, and hence the theory is the alphabet. Ok… now that’s the lamest answer I ever threw on this. :) But with all the possibilities (what, 10 ^ 300-400-500-600, I don’t remember), and only pick and choose with what we like and hate, well, it’s simply another way to do maths, not a real theory. But it doesn’t disprove. Then, there is no actual way to disprove the theory, experimentally, you cannot say it will not do it, again because it’s totally generalistic in approach, it cannot be tested, it cannot be experimented upon, then it cannot be proven or disproven, or in other words, it’s not falsifiable. then there cannot be tentative theory, there cannot be error elimination, because there are an infinity of solutions (so 1/12 solutions ;) ). Hence the analogy with the alphabet I made. But one has to know the alphabet to be able to describe. So I’m not against. Even better because it gives another WAY of seeing the universe, so it gives novel approaches, so even if it finally get disprovable (or approvable, let’s be optimistic), then it will have given a new headstart.

The problem I was referring to was the string theory in general is all based on approximations to its core because we don’t have the necessary tools to calculate most if not all the techniques involved in string theory, and because it has to go with probabilities, and if one had problems with uncertainty principle, well, a string is a big uncertainty principle in itself ;) What I was referring to as n-dimensionality’s calculation problem is because of the resonnant aspect of strings, as well as the usage of the dimensions themselves. Right now, you can only give that n-dimensionality of strings perturbatively and it’s not a fun prospect mathematically speaking… speaking of which…

About the Riemann Zeta function you described with Ramanujan’s Summation, it is interesting as a side effect, it also describes something that is valid, but it has to be written as such that 1/12 is with a stylized (R), to define it as a Ramanujan summation. I am sorry I haven’t made the 1+1=2 there ;) Number theory is very far in my head, I create games, I create virtual realities, my reality is much easier than the real reality (And if someone reads that sentence alone, I’m out to Louis-H very quickly ;) )

6. GP says:

The curly R just picks out the real part of the argument, it doesn’t indicate a Ramanujan summation.

String theory certaintly does have some falsafiable claims, such as the existence of extra dimensions and new particle states. If we fail to discover supersymmetry with the next generation of accelerators it’ll be a very bad sign (and for other theories as well, since supersymmetry isn’t unique to string theory).

There are problems with finding just the right configuration of extra dimensions to create a universe which looks like ours since there are so many possibilities, like you said. And you’re right, it is very difficult to do that, but not every possibility is valid since most look nothing like our universe. It’s important to realise, though, that this isn’t any more of a problem (conceptually) than the twenty adjustable parameters in the standard model. We have to put in those constants (like the strength of the electric force) by hand the same way we (so far) have to put in the spacetime topology into string theory by hand.

Multiple universes and uncertainties are the realm of quantum physics, they’re not new to string theory.

7. Michel says:

I’ve been able to find in the Riemann Zeta page on Wikipedia the same symbol I’m referring to (do we love Wikipedia ;) ) http://en.wikipedia.org/wiki/Riemann_zeta#Applications

On string theory and falsifiable claims, the supersymmetry is not something that will make string theory falsifiable, actually if supersymmetry would be shown to exist, it would not help string theory, as it is not something that is unique to string theory, all theories I know of require some kind of supersymmetrical structure to properly work, at least for the current assumptions to be valid. On the same topic, it would probably not really discredit string theory if supersymmetry would not exist, as there are many researches being made on string theory that would inhibit its suppersymmetrical requirements, especially at low energy like we have right now (at least when discussing the other particles types). If we look at the current theory, if we go to … what is the name already, tachyonic … or I don’t remember, bosonic, something like that, anyways, if we go with more and more “dimensions”, eventually, it would make it more and more impossible to get the proper energy field to correctly perturbate the dimensions enough to deduce the supersymmetry, hence not making the theory falsifiable by our current technology (nor probably the next 20 years technologies). Also, in the current state of technology was discovered finally a neutral particle, meaning that tachyons are more and more probable, making the assymetrical string theory another theory in its own (slashdot is my friend too http://science.slashdot.org/article.pl?sid=06/12/06/2212250 )

Not saying it’s false, that said… Like I told you, I love the idea, but I am not convinced at this point and time that it’s meant for worldwide consumption and all the hype that associated with it. Yay if it makes people interested in the M-Theory, all for it.

8. GP says:

Well, the dimension of string theory is fixed, it’s not something you can adjust as much as you want.

Really the main problem with string theory right now is that it hasn’t been able to go through the usual cycles of experiment and adjustment of the theory. Hopefully at some point it will. Although, it’s interesting that you mention axions, as they’re consistent with string theory (but also, like supersymmetry, with other theories as well).