Graduate String Vacua and Particle Interactions

[Urs Schreiber]

At present, if one uses the perturbative approach to calculate the S-matrix, incorporating the higher order (virtual string) processes, the solution diverges, is this correct?

Yes, the perturbation series of every non-toy QFT diverges (Dyson 52).

The modern perspective is that these series are to be regarded as "asymptotic series".

Presumably (hopefully!) then, a non-perturbative approach would suppress contributions from virtual processes (loops), leading to a finite result for string amplitudes. Is this essentially correct?

Concepts like "virtual loops" only exist in perturbation theory. The renormalized Feynman perturbation series is finite at each loop order , and so is the string perturbation series (not proven rigorously though, i suppose) but both still diverge when summing up all loop orders.

See also the string theory FAQ at Isn't it fatal that the string perturbation theory does not converge?

 

[Urs Schreiber]

In a sense, quantum string theory is just a third quantization, namely the second quantization of a 2D conformal field theory describing a single quantum string, and thus follows all the rules of QFT.

Just to amplify that this is not special to string theory and that the same statement applies also to QFT: The Feynman amplitudes in QFT may be understood as coming from the 1d worldline field theory of a quantum particle in direct analogy to how the string scattering amplitudes come form a 2d worldsheet field theory of a quantum string.

This fact (or insight) is called worldline formalism of QFT, due to Bern-Kosower 92, Strassler 92. It makes manifest how perturbative string theory is a straightforward/natural variant of perturbative QFT.

 

[Urs Schreiber]

if I do an electron-electron scattering experiment, would I expect the 'electron' strings to actually join and then split?

Yes.

Standard Feynman diagram are not representative of interactions in this way, so I just wanted to be sure...

On the contrary, it's exactly as in standard Feynman diagrams, just with 1-dimensional graphs replaced by 2-dimensional surfaces. That's the very definition of perturbative string theory: Replace the formula for the S-matrix, originally given by a sum over Feynman graphs, by a corresponding sum over 2-dimensional surfaces.

What is observable about this (in both cases) is the end result of the sum, which is (an asymptotic series of) the probability amplitude for given states to come in from the asymptotic past and for other given states to emerge in the asymptotic future.

A priori nothing tells you that each single term in the sum has a corresponding physical interpretation. What you keep asking is what the physical interpretation is for each single term in this series. Generally the answer is: It has none.

But of course if you look at these terms, it appears extremely suggestive, intuitively, to assign physical meaning to them, in terms of "virtual processes". But since this is not what the maths tells you, but just what your intuition tells you, the rule to proceed is the following:

As long as you find it helpful to think of single summands in the perturbation series as "virtual processes of particle/string interactions" run with it, but as soon as you find yourself bogged down in trying to make concrete sense of this intution, let go of it. Because, it's just that: an intuitive picture that only carries so far.