Notes on string theory: Internal excitations of a free open bosonic string

After reviewing some of the particulars that comprise Chapter 1 in Polchinski’s book (I have mostly kept to key concepts in this blog series, with a more thorough review to be found in my lecture notes), we now arrive at the culmination of the first stage in our study.

We will first study the free open string spectrum. Then, in a following post, we will review the closed string spectrum. This will also allow us the opportunity to explore some basic closed string physics. After these two entries, we will then switch focus and engage in a thorough study of Conformal Field Theory, as directed by Polchinski in Chapter 2.

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So far a lot of what we have been constructing, especially in relation to the quantum string, is premised on the idea of wanting to obtain quick and direct access to the study of the physical states of the bosonic string. One may recall that this was the motivation to invoke light cone quantisation in the first place. Ultimately, we will want to work toward path integral quantisation of the string; but for now, let’s see what we find.

Recall the mass-shell condition from previous entries. For the open string spectrum, the mass shell condition becomes

    \[ M^{2} = \frac{1}{\alpha^{\prime}} (N^{\perp} - 1) \]

Where N^{\perp} is the number operator and N^{\perp} = \sum_{k=1}^{\infty} k (\alpha_{k}^{I})^{\dagger}\alpha_{k}^{I}. The commutator can be written as [N^{\perp}, (\alpha_{k}^{I})^{\dagger}] = k(\alpha_{k}^{I})^{\dagger}.

One will recall the mass operator M^{2} from previous discussions. The calculation to get to this point is straightforward, and can be reviewed in many textbooks.

For the state space of the theory, we should take note of: x_{0}^{I}, p^{I}, p^{+}, \alpha_{n}^{I}, (\alpha_{n}^{I})^{\dagger}. P_{T} is the collection of all p^{I}‘s.

We introduce a vacuum – that is, a ground state of the string. This is like the ground state of all harmonic oscillators of every single mode. For each spatial direction there are infinitely many modes.

| p^{+}, p_{T}> is killed by all \alpha_{n}^{I}

And so the important thing is that the oscillators act on this state. Hence, as written a few sections earlier, a general state can be built by acting on the vacuum,

    \[ | N, k> = \prod_{i=2}^{D-1} \prod_{n=1}^{\infty} \frac{(a_{-n}^{i})^{N_{in}}}{(n^{N_{in}N_{in}!})^{\frac{1}{2}}} | 0, k> \]

So, for our catalogue of states, we can now explicitly compute:

For N^{\perp} = 0,

    \[ M^{2} |p^{+}, p_{T}> = \frac{1}{\alpha^{\prime}}(-1 + N^{\perp}) |p^{+}, p_{T}> \]

    \[= -\frac{1}{\alpha^{\prime}} |p^{+}, p_{T}> \]

Notice something important: The ground state in free open bosonic string theory has negative mass squared. This gives us our first encounter with the tachyon.

The tachyon is often described as physically unrealistic, as, in addition to it having negative mass squared, it also travels faster than the speed of light.

For a long time, it was seen as a pathology of bosonic string theory. But fear of the tachyon has diminished over time. Luboš Motl offers a terrific summary of the history behind the study of the tachyon, including some of the sociology as well as a wonderful overview of Ashoke Sen’s timely interventions. A discussion of Sen’s contributions would extend beyond the limits of this particular article; but a single comment would point out that Sen developed the conjecture about the tachyon potential, as he pushed forward the study of brane/anti-brane physics. From his study it emerged that the instability of the tachyon corresponds to the instability of the D-brane to which the ends of the open string are attached. As there is a D-brane present in the physics of the string, the decay or disintegration of the D-brane relates to the potential – that is, the local minimum – of the tachyon.

I will save a deeper and more detailed discussion about this for a separate entry. Meanwhile, in that one will frequently read a description of the tachyon as being unphysical, a more pedagogically focused introduction would emphasise that as the tachyon is highly unstable, this relates to one of the issues with bosonic string theory: namely, the vacuum is unstable. In general, the situation we are left with is that for the consistency condition of bosonic string theory to be satisfied, which requires a = 1, this means the tachyon cannot be simply removed from the theory. That said, and as alluded, removal of the open string tachyon has been studied as it relates to the decay of D-branes into closed string radiation. But the closed string tachyon, which we will meet soon, has not yet been resolved.

Of course, all of this leaves one to say that one of the highly publicised benefits of superstring theory is that, by way of the GSO projection, we are able to construct a consistent theory that does not include tachyons at all. This will be a topic for a later time.

In general, before we get to the other masses, it should be pointed out that the tachyon is interesting for a number of reasons. The prospect of a non-perturbative study raises many intriguing questions. There is also some suspicion and some speculation in the literature, particularly in the case of the closed string case, that the instability of the tachyon – which we have already noted correlates with the instability of the D-brane – could therefore relate to the instability of spacetime itself (i.e., spacetime could disintegrate). That is, a tachyon fluctuation creates the spacetime and even annihilates it, something Zwiebach commented on in his 2012/13 lecture series.

Another interesting point is that, at the time of writing, it does not seem that anyone has been able to calculate the tachyon potential. In general, the study of the tachyon is intriguing and worth exploring, but deeper discussion shall be reserved for another time.

Now, in the case for N^{\perp} = 1 \implies M^{2} = 0, the mass for the first excited state can be found as

    \[(\alpha_{1}^{I})^{\dagger} |p^{+},p_{T}> \]

This is the Maxwell field, or one photon states. Again, we have a massless state with D − 2 components, with transverse spin.

As expected, we can continue up the scale of states. In the case of N^{\perp} = 2, there are a total of 324 states. But they are all one particle states.

In the next post, we will study the closed string spectrum. Here we will go over some basic closed string physics, and, in the study of the spectrum, we will also arrive at our first encounter with the graviton and the dilaton.

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.

Kevin Wray. (2009). “An Introduction to String Theory”.