We have already found that in the LC gauge the open string mass-shell condition can be written,

[M^{2} = -p_{mu}p^{mu} = 2p^{+}p^{-} – sum_{i=1}^{D-2}p_{i}^{2} = frac{2(N – a)}{l_{s}^{2}} ]

Where, if we were to expand out the $N$ term we have,

[N = sum_{i=1}^{D-2} sum_{n=1}^{infty}alpha_{-n}^{i}alpha_{n}^{i} ]

In the flow of our past discussions up to this point, we of course want to continue working toward a study of the physical spectrum of the bosonic string. As we are going to first study the string spectrum in the LC gauge, we should re-emphasise that all of the string excitations will be generated by the transverse modes $alpha_{n}^{i}$.

Before coming back into direct and explicit contact with Polchinski in his discussion of the open string spectrum, I want to offer a few more remarks about some of the background calculations. In particular, I want to focus more explicitly on the mass of the string and on Polchinski’s reference to the Riemann-zeta function, as these are some of the most inspiring topics early on.

From the final result in the last post, a detailed study would observe that there is room for a bit of trouble. To explain, let’s first write a few things out. I will use slightly different notation in the hope that the power of the ideas are pedagogically motivating.

To start, note that we may set

[ alpha_{n}^{-} ~ L_{n}^{perp} ]

Where, one should recall, $L_{n}^{perp} = frac{1}{2} sum_{p in mathbb{Z}}^{infty} alpha_{n-p}^{I}alpha_{p}^{I}$.

Now, this is where the room for trouble may enter. Look at the above expression, notice we have an infinite sum. It is also not clear at this point in what order we might compute this infinite sum such that, now in the context of QFT, the $alpha$ terms are operators (and therefore do not commute). How to think about this?

Let’s deepen our ongoing study for just a few moments, and work things out in very detailed and explicit way. Pedagogically, a note of importance is that when we go to write the Hamiltonian for the oscillator, the destruction operator should be on the right-hand side: $mathcal{H} = a^{+}a^{-}$. So that is something. But, as we have yet to introduce normal ordering, this assertion may not yet be clear. Such a discussion will come later. So for the reader lost at this part, maintain patience.

Advancing forward, we may also note that in studying the above expression, for $n neq 0$ it is defined. What about for $L_{0}^{perp}$ (i.e., $n=0$)? Let’s check,

[ L_{0}^{perp} = frac{1}{2} sum_{p in mathbb{Z}} alpha_{-p}^{I}alpha_{p}^{I} = frac{1}{2} alpha_{0}^{I}alpha_{0}^{I} + frac{1}{2}sum_{p=1}^{infty}(alpha_{-p}^{I}alpha_{p}^{I} + alpha_{p}^{I}alpha_{-p}^{I}) ]

In the sum on the far right-hand side, the first $alpha$ terms are fine but the second set of $alpha$ terms are not good! So, what if we reorder as,

[ alpha^{prime}p^{I}p^{I} + sum_{p=1}^{infty} alpha_{-p}^{I}alpha_{p}^{I} + frac{1}{2}sum_{p=1}^{infty}[alpha_{p}^{I}, alpha_{p}^{I}] ]

It can be found that $[alpha_{p}^{I}, alpha_{p}^{I}] = pdelta^{IJ} = p(D-2)$. It follows,

[ L_{0}^{perp} = alpha^{prime}p^{I}p^{I} + sum_{p=1}^{infty} p(alpha_{p}^{I})^{dagger}alpha_{p}^{I} + frac{1}{2}(D-2)sum_{p=1}^{infty}p ]

While we have made progress, in truth this expression is still very messy and, thinking about it, it is not entirely clear how one may decide to proceed. Luckily, a slightly different approach is available to us.

As a first recourse, recall $2p^{I}p^{I} stackrel{?}{=} frac{1}{alpha^{prime}}L_{0}^{perp}$ (*). We should also recall that we previously found $M^{2} = -p^{2} = 2p^{+}p^{-} – p^{I}p^{I}$.par

What we do is declare that $L_{0}^{perp} + alpha^{prime} equiv alpha^{prime}p^{I}p^{I} + sum_{p=1}^{infty}p (alpha_{p}^{I})^{dagger}alpha_{p}^{I}$. Upon making this declaration, we then suggest that (*) is the real formula. So we rewrite (*),

[2p^{+}p^{-} = frac{1}{alpha^{prime}}(L_{0}^{perp} + a) ]

Where $a$ is a constant such that we set it equal to the infinite sum that we ended up with on the right-hand side of the final expression in the last post:

[ a = frac{1}{2}(D-2)sum_{p}^{infty}p ]

Of course this is still rather ambiguous. What is this infinite sum? Is it zero? What we come to understand is that it is a constant. And it is upon us to start investigating this constant term. But first, let’s update the mass formula. Notice,

[ M^{2} = frac{1}{alpha} (L_{0}^{perp} + a) – p^{I}p^{I} ]

From $L_{0}^{perp}$ the $p^{I}p^{I}$ terms cancel. We are left with,

[ M^{2} = frac{1}{alpha} (sum_{p=1}^{infty}) p (alpha_{p}^{I})^{dagger}alpha_{p}^{I} + a) ]

It should be fairly clear that given the definition for the $a$ term, this affects the masses of all possible states. In other words, to find the mass of any state we must find some value for the constant $a$.

Here comes one of the great results in string theory. Remember: in the classical theory massless states were a problem and there certainly was no constant $a$ term. It turns out, in our present case, if there was no $a$ term the oscillators would be massive. Needless to say, this would be problematic. So this constant $a$ we’ve introduced turns out to be very important! But it also turns out that it must be negative.

How do we put everything together? Well, let’s recall once more for the sake of completeness that the entire expression for the modes can be written as,

[ frac{1}{2}sum_{i=1}^{D-2}alpha_{n=-infty}^{infty}alpha_{-n}^{i}alpha_{n}^{i} = frac{1}{2}sum_{i=1}^{D-2}sum_{n=-infty}^{infty} alpha_{-n}^{i}alpha_{n}^{i} + frac{1}{2}(D-2)sum_{p=1}^{infty}p ]

Where we have used normal ordering, which, again, will be introduced more explicitly in a future discussion in the context of Conformal Field Theories.

Now, for the piece $a = frac{1}{2}(D-2)sum_{p}^{infty}p$ on the far right-hand side, this sum is divergent. This means that it needs to be regularised. In the background, we’re being careful to ensure this regularisation is Lorentz invariant (Becker, Becker, Schwarz, p.50). But the ‘drum-roll moment’ is that, in order to regularise this sum, we should invoke the Riemann zeta function (see also

Ramanujan summation). In using $zeta$-function regularisation, we first consider the general sum

[zeta(s) = sum_{n=1}^{infty}n^{-s} ]

This sum is defined for any complex number $s$. In the case where $Re(s) > 1$ the sum converges to the Riemann zeta function $zeta(s)$. For the unfamiliar read, there is a lot of detailed literature and study on the Riemann zeta function. Understanding some of its unique properties and characteristics, such as in how using analytic continuation about $s=1$ we obtain the rather awesome result $zeta(-1) = -frac{1}{12}$. It follows that we may state this in terms of how the sum of positive integers, or simply natural numbers, is equal to $-frac{1}{12}$.

In other words, the best way to define the number that is our constant $a$ term is through the Riemann zeta function: $zeta(s) = sum_{n=1}^{infty} frac{1}{n^{s}}$. The sum for $a$ seems precisely to be $zeta(-1)$. And if you take $-1 = s$, you obtain the sum of all natural numbers. That is, $zeta(-1) = -frac{1}{12}$. Therefore, we come to write,

[ sum_{p=1}^{infty} = – frac{1}{12} implies a = – frac{1}{24}(D-2) ]

In the case that the normal ordering constant $a$ must be equal to 1, we may then simply write

[ 1 = frac{D-2}{26} implies D = 26 ]

Which gives us give and immediate insight into the values of $a$ and $D$. And we will find that, in the context of bosonic strings, this value for $D$ proves important for the analysis of negative norm states referenced previously.

But what should be emphasised here is the use of the Riemann zeta function. Polchinski describes this result as “odd” (p.22). It really is curious, to say the least. Moreover, the Riemann hypothesis more generally is one of the most profound conjectures of our time. There is a moral ought for any unfamiliar reader to study the hypothesis, learn about its connection with the primes, and so on. (For the truly uninitiated, Grant of 3blue1brown offers a visual presentation).

What’s more, if a person proves the hypothesis they will obtain a $1 million prize, a fields medal and they will surely also be ascensed to the status of legend. Toward the end of 2018 the late Sir Michael Atiyah, who has deep ties with string theory, presented a lecture claiming to have found a proof (to clarify, Atiyah’s presented proof does not relate to string theory). If his proof holds – or any potential future proof – it will be an immense feat (as it stands, it doesn’t look like Atiyah’s claims will hold).

I will close this post by saying that, in the context of our present discussion in sting theory, the percipient reader will likely already be aware that a stringy inspired proof of the Riemann hypothesis would be predigous and monumental. Indeed, this is the sort of thing that surfaces in a string theorist’s dreams. It’s something I have been thinking about quite a bit, and is a topic that would be fun to write about at some point in the future.

In the next post, we will follow up by discussing the Hilbert space of single string states. And then, following this, in another post we will think about the Virasoro operators. After all of this, we will then finally turn our attention to a study of the open and closed string spectrum!

References

Katrin Becker, Melanie Becker, John H. Schwarz. (2006). “*String Theory and M-Theory: A Modern Introduction*“.

Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.

*Edited for clarity 28/04/2019.