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.)


2 Answers 2


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.


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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