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Is the S-matrix the only observable in string theory? What about time varying spacetime backgrounds, or thermal states then?

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2 Answers 2

There are backgrounds like FRW with thermal uniform matter which can't be modelled by S-matrices. See http://arxiv.org/abs/arXiv:0908.3844 and http://arxiv.org/abs/arXiv:0710.1129 for more details.

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Thermal states don't make sense in infinite volume in a theory of gravity, they collapse on themselves. So you can't really talk about thermal string theory on flat space. The closest you can come to a gravitational thermal state is a deSitter space, and this is not describable by the exact formulations of string theory as yet. But in my opinion, it should be the analog of a thermal state in a quantum field theory.

Time varying backgrounds can be thought of as split up into two different things--- classical time varying backgrounds at infinity, corresponding to the massless modes and their classical dynamics, plus quantum stuff on top of that, a finite number of particles which are doing scattering in this background. The split is sort of arbitrary, like in the infrared problem in QED. But in string theory, you make this decomposition explicitly--- you have equations of motion for the classical backgrounds, and given a solution for the background, you have the quantum scattering of the remaining quantum particles, with infrared divergences if you screwed up the classical behavior of the background. The division is consistent and analogous infrared QED.

This hokey thing looks suspicious at first, but it is really the right way to formulate a quantum gravity theory. It differs from the S-matrix theory of the 1960s only in that it includes massless field backgrounds, otherwise, it is the same philosophical leap. It is very difficult to swallow, especially after the decades of anti-S-matrix propaganda one is brought up with now.

So to answer the question, the only observable is the S-matrix and the infrared background, which is defined in terms of the boundary values of the massless fields. This is used often without comment by Witten in his AdS/CFT papers, and makes his reasoning hard to follow sometimes.

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