Now that we understand how to quantise the string, we can summarise by way of Polchinski the construction of the Hilbert space of single string states.

The key realisation is that | k^{+}, k^{i}\rangle is the same as the PP labels (p. 20). So, k^{+}, k^{i} \rightarrow eigenvalues of P^{+}, P^{i}. Moreover, Polchinski writes, “The state | 0, k\rangle, where k = (k^{+}, k^{i}), is defined to be annihilated by the lowering operators and to be an eigenstate of the centre-of-mass momenta”.

    \[ P^{+} | k^{+}, k^{i} \rangle = k^{+} |k^{+}, k^{i} \rangle \]

    \[ P^{i} | k^{+}, k^{i}\rangle = k^{i} | k^{+}, k^{i} \rangle \]

    \[ a_{m}^{i} | 0, k \rangle = 0 \ \ \ m \geq 0, \ \ i = 2, ..., D-1\]

And so, as we would expect, we come to the fact that a general state can be built by acting on | 0, k \rangle,

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

Where N_{in} are the occupation numbers of i,n modes.

The picture being built is really quite beautiful. Indeed, one of the things we are realising is that ST can be incredibly complicated. Like ripples along a jump rope, the string can have internal excitations. As these waves travel along the string, they can interact with one another. In other words, we a have sort of internal interaction. But this is only one type of interaction. It is also the case that in building our picture we have yet to actually consider this interaction, because our theory so far is based on constructing a set of harmonic oscillators. But the key emphasis here is that in principle these interactions can and do exist.

As Polchinski describes (p.21), when we therefore construct the Hilbert space as we’ve done, we did so in terms of the internal excitations of a single string.

Thus, another subtlety should be emphasised: there is another type of type of interaction in addition to the one just described, which we have not yet even glimpsed in our ongoing study: namely, how multiple strings can also interact. One may sometimes be delivered an intuitive picture of such interaction in the form of strings splitting and joining. We’ll come to this later.

For now, one way I like to think of this increasingly complex picture is like a bubbling soup or stew made of strings and string interactions. Or, think, for example, of some sort of thick mucilage, like hair gel, and the sort of stringy geometries that form when covering your hands in the gel, pressing them together, and then slowly pulling them apart. But then substitute the mucilage strings and maybe think of something like stringy plasma chains, then add internal vibrations, rotations, and so on. Though not accurate, it may serve some intuition for the idea often conveyed in the form of a bubbling spaghetti soup.

In any case, the essence of Polchinski’s commentary on p.21 – that if we turn off all interactions, as we might be inclined to do at this point, we can study the Hilbert space and observe what it looks like:

    \[ \mathcal{H} = H_{0} + H_{1} + H_{2} + ... \]

Which is a Dirac sum. And so we read,

    \[ H_n = (\otimes H_{1})^{n} \]

This tells us that as long as string interactions are ‘switched off’, we are allowed to take the product of single string Hilbert spaces. What this means is that we also arrive at (1.3.30) in Polchinski’s book,

    \[ \mathcal{H} = \frac{P^{i} P^{i}}{2P^{+}} + \frac{1}{2P^{+}\alpha^{\prime}} (\sum_{n=1}^{\infty} a_{-n}^{i} a_{n}^{i} + a) \]

Here, again, \mathcal{H} is X^{i}(\tau, \sigma), P^{i}(\tau, \sigma). We can see that we have our mode expansions, and, as we have already constructed the SHO algebra, the key idea is that we substitute the mode expansions and integrate over the string. As for the a term, this is the normal ordering constant which we defined earlier.

References

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