How do we reproduce QFT's predictions about local field observables, in the low energy limit of predictions of string theory?

Question

I was reading this answer about how string theory may reproduce QFT. It talks about how, in the low energy limit, both theories may have the same scattering amplitudes.

This means that string theory and QFT can agree on scattering predictions.

But we also know that the fundamental observables in QFT are local observables tied to spacetime. Now (e.g. in David Tong's String Theory notes), the idea of local observables doesn't make sense in string theory because it's a theory of quantum gravity which makes the notion of spacetime fuzzy.

But, in the low energy limit of string theory, we have to reproduce the local observables of QFT, and we have to reproduce the predictions about the expectation values of these observables (with respect to an arbitrary state). How is this achieved in string theory?

More precisely, how is the following step carried out:

Define the states and field observables of QFT as approximations of states and observables of strings theory, after taking the low energy limit, and show that QFT's predictions are approximated by string theory's predictions

The particles states of string theory can easily be identified with the particle states of QFT.

If we then identify local field observables with observables of strings theory, in the low energy limit, we establish a correspondence between the predictions of the two theories.

So what is the identification relationship between the observables of two theories?

A scheme that several people have suggested is to simply defined local field observables $\phi (x,0)$ in string theory, in terms of linear combinations of string creation and annihilation operators.

But this scheme does not reproduce time dependent predictions that QFT makes about local observables $\phi(x,t)$. In QFT, these observables are arrived at using the Heisenberg picture evolution of the observables $\phi(x,0)$ But in string theory, this method doesn't work as we don't have finite time unitary evolution. We only have the S-matrix which evolves the asymptotic past to the asymptotic future.

In a nutshell, I would require this from the answer :

In QFT, I prepare a field in a state $\rho$, I evolve it for a time $t$ to get $\rho (t)$, I perform a measurement of a local observable A, and I get an expected value prediction of $Tr(\rho(t)A)$

1. What is the string theoretic description of this experiment?

and

2. Why is prediction of the string theory close to the QFT's prediction for this experiment?

Answer 1

The string theory operators constructed from creation/annihilation operators for the low-energy states have to be the quantum fields of the corresponding supergravity QFTs simply because their scattering amplitudes match. The question makes the common mistake of thinking that the scattering amplitudes are just one part of the behaviour of a QFT, when in fact it already contains the entire QFT.

The five string theories (I, IIa, IIb, heterotic O and heterotic E) match to the five consistent 10d supergravity QFTs in the following sense: They have the correct spectrum, i.e. the low-energy string states are populated by states that can be organized into the same supermultiplets as the states of the supergravity QFTs, and the low-energy limit of the string scattering amplitudes between those states matches the scattering amplitudes of these quantum field theories.

The general argument for deriving this match is laid out in the standard textbooks on the subject such as Becker, Becker and Schwarz or Green, Schwarz and Witten, see also this answer by pho for a brief summary.

There is no need to explicitly exhibit operators in string theory that would become the quantum fields of the supergravity QFTs or to "match observables", and this is not what string theory texts do. By the Wightman reconstruction theorem, the low-energy scattering functions define a QFT, i.e. we can construct the appropriate quantum fields (including their time evolution!) simply from the scattering functions alone, if the matching of the amplitudes alone is not argument enough.

This is not specific to string theory: People often wonder how the focus on scattering amplitudes/Wightman functions in high-energy QFT is related to physical phenomena that are not scattering. Wightman reconstruction is the rigorous answer in axiomatic QFT to this: The scattering functions in principle already encode all information about a quantum field theory, there is no additional information "hidden" in other properties of the fields.

While it may be surprising that a scattering function we usually conceive of as relating things in the infinite part to things in the infinite future encodes details of what might happen at finite times, the Wightman reconstruction theorem shows this is nevertheless the case - the structure of a QFT is rigid enough that there is no freedom of change any part of its behaviour without changing at least one scattering amplitude.

Answer 2

String theory is formulated on a background spacetime, just like QFT. So we can have local observables that are parameterized by spacetime coordinates just like in QFT. However, in QFT, operating on the quantum state with a "local" observable at one spacetime point affects the local observables at other spacetime points (even though they commute at spacelike separation). This sort of nonlocality just becomes greater in string theory, which is what makes the spacetime "fuzzier".

Answer 3

The CFT partition function on the worldsheet at first order is equal to effective background action. That is the quantum CFT is dual to the effective background theory. Another perspective is that the physical states of the BRST quantized CFT are seen as the classical fields of string field theory. By the operator state correspondense every state is also an operator of the CFT and a classical field on the background. Take such a state with ghost-number one and insert it into the string field theory action and you get the background action of that field/operator which defines some QFT.