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Consider a state $|\psi\rangle$ in a quantum field theory and a local operator $\mathcal{O}(x)$. It's known that the $n$-point function

$\langle \psi | \mathcal{O}(x_1) \cdots \mathcal{O}(x_n) | \psi\rangle$

is singular for certain $x_i$ (for example, when two of the $x_i$ are coincident or null separated). My question is: given two states $|\psi\rangle$ and $|\phi\rangle$ (which for simplicity we can take to be orthogonal), can anything be said about the divergences of the matrix element

$\langle \psi | \mathcal{O}(x_1) \cdots \mathcal{O}(x_n) | \phi\rangle$?

For instance, does the above object have any singularities? If it does, are they related in any known way to singularities of the corresponding correlator in the states $|\psi\rangle$ and $|\phi\rangle$? (Also note that I'm only interested in Hadamard states.)

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The statement that the coinciding point singularities do not depend on the choice of states is the statement that Wilson's operator product expansion holds.

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Probably not what you're looking for, but these off-diagonal matrix elements (the so-called "strange correlator") has been studied in detail in https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.247202.

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