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Something has been bothering me about string theory. From what I can tell, string theory is a first quantized theory of the position and momenta along a string, and that the fields of QFT are not really fundamentally needed anymore. However, we allowed for creation of particles in QFT by "exciting" a fundamental field. If there is no corresponding "string field" which we can disturb to create new strings, how would strings get created or destroyed? If they are not created or destroyed, how does first quantized string theory handle processes where particle number is seen to change (e.g. decays, etc)?

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  • $\begingroup$ That's right, the fundamental field of aether that we dare to disturb! $\endgroup$ – safesphere Aug 25 '17 at 17:48
  • $\begingroup$ There is a string field, more precisely, a string functional $\Psi[X^\mu]$ which is the whole point of string field theory. $\endgroup$ – JamalS Aug 28 '17 at 10:47
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First- and second-quantization

In quantum theories there are two possible formulations: first- and second quantized. This is true irrespective of the type of the theory under concerned (this can be particles - i.e. standard QFT -, strings, etc.). The difference between the two is the following:

  • First-quantized: all the space(time) positions $x^\mu$ of the classical theory are treated as the dynamical variables. They are mapped to quantum operators: $x^\mu \to \hat X^\mu$. In order to define "time" evolution one needs to define a time, that can be the proper-time $\tau$ (or more generally proper-positions), the "real" time $x^0$, etc. The variables are thus interpreted as describing the "target spacetime", the parameters describe the worldvolume (worldline for a particle, worldsheet for a string, etc.). Since the positions are given in the action as variables, only the objects at these positions exist: there is no annihilation or creation (even though this can be introduced via the path integral).
  • Second-quantized: all the spacetime positions are mapped to labels of a field. The classical fields $\phi(x^\mu)$ are mapped to quantum fields $\hat\Phi(x^\mu)$. The fields contain creation and annihilation operators, each associated with a first-quantized state.

Which perspective you choose is a matter of choice, of computability and of the problem you want to address. Formally it is always possible to get a field theory from a first-quantized one.

String theory

In standard worldsheet string theory one uses the first-quantized approach because it is technically simpler: there is not much freedom in the interactions (compared to the ones possible for a particle) and conformal field theory techniques allow to perform many complications. For a particle sometimes it is also simpler to work with a first-quantized approach (to compute anomalies à la Bastianelli…).

But there is also a second-quantized approach, called string field theory. In this framework one can describe the creation and annihilation of strings (there is not much easy introduction to the topic, see for example the section 4 of arXiv:hep-th/9411028). Some problems of great importances cannot be treated in a first-quantized approach (off-shell amplitudes and renormalization, background independence, some non-perturbative effects…). Unfortunately there are huge technical problems in building a useful string field theory yet (even if a lot of progresses have been achieved, especially the previous years), but in my opinion it is important to achieve such a construction.

So to conclude both approaches exist for particles, strings (and formally for any objects), the question is which is one is simpler to use. By chance most of the computations in string theory can be performed in a first-quantized approach, so it is logical to stick with it. For a particle the converse holds, fields are much simpler to handle.

Computing scattering amplitudes

I am not an expert in the wordline approach so I may be not be fully correct. First one should specify the theory: it is defined by the following two structures:

  1. the possible interaction vertices (eg. cubic, quartic, etc.) and the corresponding couplings;
  2. an action (often free) for a single particle from which the propagator is derived.

The fact that one specifies by hand the interactions explain why this approach is not much useful in for particles if you don't know already the interactions from a QFT: there is too much possibilities.

Then a scattering amplitudes is computed from the following data:

  1. in- and out-states (which can be in different numbers);
  2. the interaction graph (and in particular its topology), i.e. what is the full graph constructed from the fundamental vertices (for example $12 → X → 3Y, Y → 45$ for a process $12 → 345$ with only cubic interactions).

Then the path integral will connect the various in- and out-states together with propagator, the path being prescribed by the diagram, which serves as a kind of boundary condition. The point is that at each step one is specifying what are the states which are created/annihilated, so you can handle these processes, but only by hand and not "dynamically".

For a reference I would suggest to look at the article on nLab, which links to more references.

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  • $\begingroup$ So you're saying that first quantized string theory (which is as far as I know the one most everyone uses) can not handle creation or annihilation of strings? Doesn't that mean that it can't describe any decays or scattering cross sections where particle numbers are seen to change? $\endgroup$ – David Santo Pietro Aug 27 '17 at 14:55
  • $\begingroup$ I have added a section to explain more this point. To summarize: in scattering amplitudes everything is specified by hand, so you can create/annihilate strings in an amplitude, but only because there are rules that you give for that. This is not automatic as in QFT. $\endgroup$ – Harold Aug 28 '17 at 9:55
  • $\begingroup$ Thanks for adding the section, this is helping. So how might one do string theory and get the scattering amplitudes without putting things in by hand? How might one ever let some form of string theory do this automatically? $\endgroup$ – David Santo Pietro Aug 28 '17 at 15:25
  • $\begingroup$ This is all the point of creating a string field theory (SFT). Since one does not know how to do a priori, one starts from the scattering amplitudes and split them in contributions, each interpreted as a Feynman graph. Then one reads the Feynman rules and from that write the action. This general procedure is explained in a recent review (that I have co-authored) arxiv:1703.06410. There are some problems with this approach which is not totally explicit: a series of works (1703.10563, 1706.07366, 1708.04977) tries to address this problem. See also the question #206985 for more references. $\endgroup$ – Harold Aug 29 '17 at 13:09
  • $\begingroup$ I should add that what I describe first is a backward procedure (scattering amplitudes define the QFT, instead of the opposite), and the work by Moosavian-Pius aims to provide a more constructive procedure. In the case of the bosonic string theory a constructive (but still quite abstract) procedure – called the minimal area construction – was provided by Zwiebach and collaborators (unfortunately it could not be generalized to the superstring). $\endgroup$ – Harold Aug 29 '17 at 13:12

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