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Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert space, and transforms appropriately under the Poincare symmetries.

Is it possible to reconstruct the QFT from its $S$-matrix? By reconstructing the QFT I mean obtaining the Wightman functions, or equivalently, the operator distributions corresponding to the quantum fields.

I'm not asking about a general reconstruction algorithm – I'm aware that it isn't known presently. I'm asking about the possibility of such reconstruction in general. For example, if there is a counterexample of two different Wightmanian QFTs having the same $S$-matrix, that would answer my question in the negative.

It is frequently stated that onshell amplitudes contain less information than offshell correlation functions, which is not obvious to me. It would be nice to have a concrete example demonstrating this.

Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert space, and transforms appropriately under the Poincare symmetries.

Is it possible to reconstruct the QFT from its $S$-matrix? By reconstructing the QFT I mean obtaining the Wightman functions, or equivalently, the operator distributions corresponding to the quantum fields.

I'm not asking for a reconstruction algorithm – I'm aware that it isn't known presently. I'm asking about the possibility of such reconstruction in general. For example, if there is a counterexample of two different Wightmanian QFTs having the same $S$-matrix, that would answer my question in the negative.

It is frequently stated that onshell amplitudes contain less information than offshell correlation functions, which is not obvious to me. It would be nice to have a concrete example demonstrating this.

Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert space, and transforms appropriately under the Poincare symmetries.

Is it possible to reconstruct the QFT from its $S$-matrix? By reconstructing the QFT I mean obtaining the Wightman functions, or equivalently, the operator distributions corresponding to the quantum fields.

I'm not asking about a general reconstruction algorithm – I'm aware that it isn't known presently. I'm asking about the possibility of such reconstruction in general. For example, if there is a counterexample of two different Wightmanian QFTs having the same $S$-matrix, that would answer my question in the negative. It is frequently stated that onshell amplitudes contain less information than offshell correlation functions, it would be nice to have a concrete example demonstrating this.

Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert space, and transforms appropriately under the Poincare symmetries.

Is it possible to reconstruct the QFT from its $S$-matrix? By reconstructing the QFT I mean obtaining the Wightman functions, or equivalently, the operator distributions corresponding to the quantum fields.

I'm not asking about a general reconstruction algorithm – I'm aware that it isn't known presently. I'm asking about the possibility of such reconstruction in general. For example, if there is a counterexample of two different Wightmanian QFTs having the same $S$-matrix, that would answer my question in the negative. 

It is frequently stated that onshell amplitudes contain less information than offshell correlation functions, which is not obvious to me. It would be nice to have a concrete example demonstrating this.

Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert space, and transforms appropriately under the Poincare symmetries.

Is it possible to reconstruct the QFT from its $S$-matrix? By reconstructing the QFT I mean obtaining the Wightman functions, or equivalently, the operator distributions corresponding to the quantum fields.

I'm not asking about a general reconstruction algorithm – I'm aware that it isn't known presently. I'm asking about the possibility of such reconstruction in general. For example, if there is a counterexample of two different Wightmanian QFTs having the same $S$-matrix, that would answer my question in the negative.

Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert space, and transforms appropriately under the Poincare symmetries.

Is it possible to reconstruct the QFT from its $S$-matrix? By reconstructing the QFT I mean obtaining the Wightman functions, or equivalently, the operator distributions corresponding to the quantum fields.

I'm not asking about a general reconstruction algorithm – I'm aware that it isn't known presently. I'm asking about the possibility of such reconstruction in general. For example, if there is a counterexample of two different Wightmanian QFTs having the same $S$-matrix, that would answer my question in the negative. It is frequently stated that onshell amplitudes contain less information than offshell correlation functions, it would be nice to have a concrete example demonstrating this.