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).

  • $\begingroup$ I think this paper is what you want. app.box.com/s/j4cc7jmdlhlwvs6he2sfx40jqxijawk2 $\endgroup$
    – user73352
    Commented Nov 29, 2015 at 16:51
  • $\begingroup$ Welcome to Physics! It is preferred if you can post separate questions instead of combining your questions into one. That way, it helps the people answering your question and also others hunting for at least one of your questions. Thanks! $\endgroup$
    – Kyle Kanos
    Commented Nov 29, 2015 at 19:25
  • $\begingroup$ Thanks for the welcome & true, its a couple of questions in one. Maybe one can restrict to the first two only? i.e. slighlty reworded but identical in spirit to the above: 1) why not building a multiparticle state simply as a product state of single particle states (and eventually Fock space(s)) ? 2) how does such a multiparticle product state description relate to the multiparticle descriptions assumed by standard string theory scattering amplitudes? Thanks again. $\endgroup$ Commented Nov 30, 2015 at 12:08

1 Answer 1


Thanks for all replies/comments to my post. By using them as a starting point, I believe to have eventually found what I have been asking for, i.e. is it possible to use one string product states to define multi-string states and to obtain the "usual" scattering amplitudes also from this viewpoint. The answer seems to be "yes" (as expected) and the papers one might want to read are indeed relating to the operator formalism for String Field Theory, including, for example:

Charles B. Thorn, String field theory, Physics Reports, Volume 175, Issue 1, 1989, Pages 1-101, ISSN 0370-1573, http://dx.doi.org/10.1016/0370-1573(89)90015-X

Charles B. Thorn, Perturbation theory for quantized string fields, Nuclear Physics B, Volume 287, 1987, Pages 61-92, ISSN 0550-3213, http://dx.doi.org/10.1016/0550-3213(87)90096-4

David J. Gross, Antal Jevicki, Operator formulation of interacting string field theory (I), Nuclear Physics B, Volume 283, 1987, Pages 1-49, ISSN 0550-3213.

These papers use special "product" definitions (e.g. eq. (2.19) in the Gross, Jevicki paper), which is not surprising since different string excitations correspond to different types of particles: thus the need for a special Fock space construction was anyway to be expected.

PS: "Hamiltonian Formulation Of String Theory And Multiloop Amplitudes In The Operator Context", by Adrian R. Lugo, Jorge G. Russo, SISSA, Trieste, Nov 1988 as cited above by Jake Lebovic brought me on the right track (so special thanks to him), even though, if I am not mistaken, that paper is not constructing multi-string product states, but rather uses an operator approach to rewrite the sum over higher genus topologies within the standard scattering formalism.


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