I am new to string theory and I hope that the following question, if not actually deep, may provoke answers that might not only be useful to me but also to other newcomers:
I believe to have understood that, when first quantizing a free string, the resulting Hilbert space is that of different "one particle states" of different "particles" (e.g. Zwiebach, 2nd ed., 2009, section 12.6, page 267: "Each state |Lambda> of the quantum string represents a one-particle state of fixed momentum. Thus, the ... are one-photon states, and the a ... are one-particle tensor states not two-photon states", or alternatively see also Tong chapters 2 - 3, http://www.damtp.cam.ac.uk/user/tong/string.html ).
Now, the most natural way for me to construct a two particle state |1,2> out of known one particle states |1> and |2> would be to form |1>|2> + |2>|1> (or the antisymmetric counterpart), i.e. to ultimately build a new Fock space of free multiparticle states out of free one particle states and to formulate creation/annihilation operators in that space, and to try to find a Hamiltonian acting in that space such as to formulate interacting theories. I guess this is what string field theory is trying to do, but I am not entirely sure.
Instead, this route is apparently not followed in any textbook known to me, and rather two or more particles are later on handled by the usual scattering formalism that considers a sum over worldsheets of more complicated topology, e.g. Zwiebach, section 25.3 "Two strings, with energies E1 and E2, can come together to form a single open string, of energy E1 + E2", or Tong chapter 6.
I have two questions:
1) why is the Fock space formalism suggested by me above not introduced (not even as a counterexample) or followed in the textbooks? The idea seems natural and not very deep to me. So why not? Is it wrong, or a dead end? If so, why and where does it fail? Is it eventually getting too complicated? Why?
2) assuming the suggested Fock space formalism was sound in principle: can one show its equivalence to the usual multiparticle treatment in string scattering amplitudes (possibly the same question: is Tong chapter 6 "derivable" from a string field theory? If so, what Hamiltonian is used for deriving first quantized string theory?)
I would be truly grateful for explanations, clarifications, and especially also for pointers to the literature (preferably, if at all possible, at a beginner's level).