Solving the Mystery of -1/12 in String Theory

[zaybu]

In String Theory, one of the most crucial calculations involves summing all the numbers from 1 to infinity, which obviously should be infinite. But not in ST, where the Rieman Zeta Function is used, and gives a value of -1/12?!

The zeta function is given by ζ(s) = Σ 1/n^s, where s is any complex number. The earliest calculation of this function was made by Euler with s = 2

ζ(2) = Σ1/n² = 1/1² + 1/2² + 1/3²+ ... = π²/6.

If one uses the functional theorem:

ζ(s)= 2^s π ^s-1 sin(πs/2) ζ(1-s)Γ(1-s), where Γ is the well-known gamma function.

And let s = -1

ζ(-1)= 2^-1 π ^-2 sin(-π/2) ζ(2)Γ(2)
ζ(-1)= (1/2) (1/π²)(-1) (π²/6)(1) = -1/12

This yields the dreaded sum:

ζ(-1) = Σ 1/n^-1 = Σ n = 1 + 2 + 3 +... = -1/12

What I find strange is that this calculation is done with complex numbers on a complex plane, using analytic continuation. But how does one explain that the initial sum deals with real numbers, not complex numbers? This result is counter common sense. Any help would be greatly appreciated.

 

[atyy]

That means that the original calculation is not strictly correct, and it is the analytic continuation of the function (not the series, which only corresponds to the function when its argument is >1) that has the value -1/12 when its argument is -1. But somehow Euler was able to sniff this out with his great intuition.

BTW, have you seen John Baez's http://math.ucr.edu/home/baez/numbers/24.pdf ? It's the third of 3 great entertainments.

 

[genneth]

I prefer to think of it as another case of "the hardest thing in mathematics is the equals sign". Equals never really mean equality, but always some form of isomorphism. In arithmetic (which is presumably what we're discussing), we mean that the left and right hands behave the same with respect to arithmetic operations. What we find, if we use the usual rules, is that the sum is undefined --- there is no number (remember that infinity is not a number) which correspond to that sum. Therefore, we may ask "what could we set it to, such that no inconsistencies will occur?" Treat it as asking for an extension of the axioms; then the only logical stumbling block is consistency. Turns out, people have thought pretty hard on this: http://en.wikipedia.org/wiki/Divergent_series#Properties_of_summation_methods

 

[atyy]

There is a discussion about this in David Tong's http://www.damtp.cam.ac.uk/user/tong/string.html

He does it the amazing way in section 2.2.2, and then the respectable way in chapter 4, where the complex plane enters right at the start.

I guess it's something like what we actually want is the non-perturbatively defined answer, not the series. The non-perturbative answer has a formal expansion (which in lucky cases may be asymptotic, convergent etc), which we happened to find first.

 

[zaybu]

That means that the original calculation is not strictly correct, and it is the analytic continuation of the function (not the series, which only corresponds to the function when its argument is >1) that has the value -1/12 when its argument is -1. But somehow Euler was able to sniff this out with his great intuition.

BTW, have you seen John Baez's http://math.ucr.edu/home/baez/numbers/24.pdf ? It's the third of 3 great entertainments.

Thanks for the link, it was indeed very interesting. Though he gives an explanation that the strings vibrate with 24 components, and this number 24 is obtained with the zeta function(-1) = -1/12, it does not address my point.

Now you say that it is the analytic continuation of the function that's equal to -1/12, but physicists take that as equal to the sum. I know there is a technical difference, nevertheless, the problem that arises in String theory is the sum of all real integer numbers. But when you go for its actual calculation, it is done on the complex plane, involving residual poles. I really don't know what to make of that.

 

[zaybu]

There is a discussion about this in David Tong's http://www.damtp.cam.ac.uk/user/tong/string.html

He does it the amazing way in section 2.2.2, and then the respectable way in chapter 4, where the complex plane enters right at the start.

I guess it's something like what we actually want is the non-perturbatively defined answer, not the series. The non-perturbative answer has a formal expansion (which in lucky cases may be asymptotic, convergent etc), which we happened to find first.

Thanks immensely for that link. He has

Σn → Σn e^-εn = - ∂/∂ε Σ e^-εn = 1/ε² - 1/12 + O(ε)

He writes that the piece 1/ε² diverges as ε → 0. then claims it would go away under renormalization. Do you know how this would be done?

 

[atyy]

Thanks immensely for that link. He has

Σn → Σn e^-εn = - ∂/∂ε Σ e^-εn = 1/ε² - 1/12 + O(ε)

He writes that the piece 1/ε² diverges as ε → 0. then claims it would go away under renormalization. Do you know how this would be done?

Try section 4.4.1 for the proper calculation. I don't know why this is related to the zeta function method in chapter 2, but the set up in chapter 4 begins with the complex plane.

I would guess that the true calculation is that of chapter 4, and that the function we want has a formal expansion, which historically happened to be found first. As an analogy, N! is a perfectly good expression. This can be approximated for large N by the first few terms of a divergent series (http://en.wikipedia.org/wiki/Stirling's_approximation). The divergent series is not a problem, it's just something we can use to extract properties of N!. But if we discovered the series before we knew that N! was involved, we would be quite mystified. So I think something like that is going on here, except the respectable object involved has a formal expansion of a sort different from asymptotic series.

 

[Hurkyl]

What I find strange is that this calculation is done with complex numbers on a complex plane, using analytic continuation. But how does one explain that the initial sum deals with real numbers, not complex numbers? This result is counter common sense. Any help would be greatly appreciated.

Aside: calculations with real numbers are often more convenient with a detour through the complex plane.

Back on the main topic. The whole problem is that there is something they're not telling you: they aren't computing an infinite sum. Well, more accurately, they aren't computing the sort of infinite sum you learned in your elementary calculus class; they are computing a different kind of infinite sum, a zeta-regularized sum.

There are lots of (partial) linear functionals on the set of infinite sequences, and some of them resemble summation sufficiently to justify calling them an "infinite sum". The properties of Zeta regularization are apparently pretty useful in number theory and string theory, which is why you see it from time to time.

 

[atyy]

I guess there are 2 points of view here.

One says string theory is the fundamental theory, so we can define whatever we want, as long as it is consistent.

However, these bizarre tricks are used even in theories that are not fundamental theories. We would not such a trick in civil engineering, unless it can be justified from Newton's laws, which are the relevant fundamental laws this case.

So in classical statistical physics and quantum field theory, neither of which are fundamental, the use of similar tricks is justified from the Kadanoff-Wilson picture. Generally, things look different at different scales. However, some theories have symmetries so that some of their properties look the same no matter how much you scale out. This fixed point of the renormalization flow is the "true object" that justifies the bizarre tricks, which are just partial practical means of looking at the "true object". Historically, the bizarre tricks in quantum electrodynamics were discovered first (Feynman rules) and the simple picture only much later in the different physical context of condensed matter physics.

The pictures aren't really exclusive, since often there's more than one consistent way of assigning a "true object" to a formal expansion. So picking out one of these consistent ways is saying what the "true object" is. I would guess that the choice of zeta regularization in string theory corresponds conformal field theory being the underlying "true object"?

 

[zaybu]

Try section 4.4.1 for the proper calculation. I don't know why this is related to the zeta function method in chapter 2, but the set up in chapter 4 begins with the complex plane.

I would guess that the true calculation is that of chapter 4, and that the function we want has a formal expansion, which historically happened to be found first. As an analogy, N! is a perfectly good expression. This can be approximated for large N by the first few terms of a divergent series (http://en.wikipedia.org/wiki/Stirling's_approximation). The divergent series is not a problem, it's just something we can use to extract properties of N!. But if we discovered the series before we knew that N! was involved, we would be quite mystified. So I think something like that is going on here, except the respectable object involved has a formal expansion of a sort different from asymptotic series.

You're right, he does develop the same result in section 4, without the funny business of section 2.

I guess I need to learn conformal field theory in order to understand the whole derivation.

Thanks for the help.

 

[negru]

I guess there are 2 points of view here.

One says string theory is the fundamental theory, so we can define whatever we want, as long as it is consistent.

However, these bizarre tricks are used even in theories that are not fundamental theories. We would not such a trick in civil engineering, unless it can be justified from Newton's laws, which are the relevant fundamental laws this case.

So in classical statistical physics and quantum field theory, neither of which are fundamental, the use of similar tricks is justified from the Kadanoff-Wilson picture. Generally, things look different at different scales. However, some theories have symmetries so that some of their properties look the same no matter how much you scale out. This fixed point of the renormalization flow is the "true object" that justifies the bizarre tricks, which are just partial practical means of looking at the "true object". Historically, the bizarre tricks in quantum electrodynamics were discovered first (Feynman rules) and the simple picture only much later in the different physical context of condensed matter physics.

The pictures aren't really exclusive, since often there's more than one consistent way of assigning a "true object" to a formal expansion. So picking out one of these consistent ways is saying what the "true object" is. I would guess that the choice of zeta regularization in string theory corresponds conformal field theory being the underlying "true object"?

This has nothing to do with string theory being fundamental. This zeta regularization is what you do all the time in freshman quantum mechanics when you quantize operators, or qft when you do your usual regularization. I think one should be more shocked of dimensional regularization then this one.
This is also exactly the same as the i\epsilon trick and everything else like it.

 

[atyy]

This has nothing to do with string theory being fundamental. This zeta regularization is what you do all the time in freshman quantum mechanics when you quantize operators, or qft when you do your usual regularization. I think one should be more shocked of dimensional regularization then this one.
This is also exactly the same as the i\epsilon trick and everything else like it.

Yes, that was my point, though I was actually thinking of eg. Borel summation or the epsilon-expansion in statistical physics. I didn't know the epsilon trick in elementary QM was the same as zeta-regularization - how is it so?

Also, has it nothing to do with conformal field theory? The derivation by that route seems so much nicer.

 

[negru]

Yes, that was my point, though I was actually thinking of eg. Borel summation or the epsilon-expansion in statistical physics. I didn't know the epsilon trick in elementary QM was the same as zeta-regularization - how is it so?

Also, has it nothing to do with conformal field theory? The derivation by that route seems so much nicer.

Well it's the same in the sense that you're computing an integral like 1/x or smth like that which diverges on the real line so you move the singularity up or down in the complex plane. Which is cheating but it works so who cares?

 

[MathematicalPhysicist]

Well it's the same in the sense that you're computing an integral like 1/x or smth like that which diverges on the real line so you move the singularity up or down in the complex plane. Which is cheating but it works so who cares?

If your'e using sound mathematical tools, I don't see how you can cheat.

 

[haushofer]

Yes, that was my point, though I was actually thinking of eg. Borel summation or the epsilon-expansion in statistical physics. I didn't know the epsilon trick in elementary QM was the same as zeta-regularization - how is it so?

Also, has it nothing to do with conformal field theory? The derivation by that route seems so much nicer.

I'm also curious about this :)

I've never understood why this argument of regularization and analytic continuation gives precisely the same answer as the more rigorous answer from CFT-point of view. I also don't see a lot of people taking the effort of explaining this carefully.

 

[Thomas Larsson]

If you are interested in this sum because you want to calculate the central charge of the Virasoro algebra, there is an alternative method that does not involve any infinite sums at all. Instead of using the infinite sum

$$L_m = \sum_{n = -\infty}^\infty : a_{m-n} a_n :$$

we can define the representation by its action on the vacuum:

$$L_m|0> = \sum_{n = 1}^m a_{m-n} a_n |0>$$

$$L_0|0> = h|0>$$

$$L_{-m}|0> = 0$$

$$a_{-m}|0> = 0$$

together with

$$[L_m, a_n] = n a_{m+n}$$

$$[a_m, a_n] = m \delta_{m+n}$$

Give or take some signs and perhaps factors of two, this uniquely defines a Virasoro representation. In particular, the central charge follows by calculating $$[L_m, L_{-m}]|0>$$.

 

[arivero]

While we are on this topic... do you know any standard reference doing the same calculations directly for the superstring? I mean, in a way such that they get 1/4 instead of 1/12.

 

[Thomas Larsson]

Sure, just take the usual normal-ordered expressions and apply them to the vacuum. All but finitely many of the bilinears $$a_{m-n}a_n$$ annihilate the vacuum, because either $$n<0$$ or $$m-n<0$$. The same is true for the fermionic oscillators.

 

[Manojg]

Hi zaybu,

In Riemann zeta function,
$$\zeta(s) = \sum^{\infty}_{n=1} n^{-s}$$
isn't s > 1, s != -1?
http://en.wikipedia.org/wiki/Riemann_zeta_function"

 

[negru]

It is, that's why you do the analytic continuation to -1

 

[Manojg]

I am not that much familiar with the topic, but zaybu has used s = -1 to get -1/12.

 

[negru]

With analytic continuation you can extend the domain of a function.

 

[Manojg]

So, the result -1/12 is not right?

 

[negru]

It's just a consistent way of assigning a finite value to a divergent sum.

 

[Manojg]

Ok, if I use this value for some other calculation, (like Euler's cazy calculation mentioned before in this post, actually a link to a talk) does not it give wrong result. Is it just a representation? Sorry, my language is not that much mathematical.

 

[Manojg]

I found couple of papers where authors defined the lower bound for s, which are non negative.
http://iopscience.iop.org/1064-5632/68/6/A05/pdf/1064-5632_68_6_A05.pdf"
http://www.springerlink.com/content/u5042g16t5255505/"

 

[negru]

Look up more on "analytic continuation", that should explain how this works.