# What is the Hilbert space of string theory?

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I looked at the String theory FAQ referred to in this comment by @UrsSchreiber. I miss an entry with a heading such as: What is the Hilbert space of string theory?

Somehow one should base the theory in the standard framework of quantum mechanics, which requires an answer to this question. What are the states, what are the observables?

Maybe later I find time to write a decent answer. But briefly the answer (as to most questions about the nature of string theory) is that perturbative string theory is an S-matrix theory, and the space of asymptotic states (the vector space on which the S-matrix is a "matrix", ,namely a multilinear map) is the BRST complex of the single (super-)string. (Namely the "BRST complex", that's just the graded vector space of all excitations states of the single free string which we think of as entering into the scattering process from an asymptoticdistance.) This is made nicely manifest in most accounts of string field theory.

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Briefly the answer (as to most questions about the nature of string theory) is that perturbative string theory is an S-matrix theory, and the space of asymptotic states (the vector space on which the S-matrix is a "matrix", namely a multilinear map) is the BRST complex of the single (super-)string. (Namely the "BRST complex", that's just the graded vector space of all excitations states of the single free string which we think of as entering into the scattering process from an asymptotic distance.) This is made nicely manifest in most accounts of string field theory. (Maybe later I find the time to write a more comprehensive reply, sorry.)

answered Jan 13, 2017 by (6,095 points)
edited Jan 13, 2017

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