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In quantum mechanics and many-body theory, one often encounters correlators like $$\langle A(t)B \rangle, \quad \langle [A(t),B] \rangle, \quad \langle \{A(t),B\} \rangle,$$ where $A$ and $B$ are two operators.

Question: What is the physical meaning of these correlators?

My partial answer: If $A=\psi(\mathbf r)$ and $B=\psi^\dagger(\mathbf r')$, then the above correlators become the well-known 1-particle Green's function (commutator for boson, anticommutator for fermion). The physical meaning of these functions is well-known. For example, the Greater Green's function $\langle \psi(\mathbf r,t) \psi^\dagger(\mathbf r',0)\rangle$ can be interpreted as the overlap between between the following two states:

(1) one add a hole at $\mathbf r'$ and evolve to time $t$,

(2) one add a hole at $\mathbf r$.

This can be again interpreted as the "propagation amplitude of a particle during time $t$". For general $A,B$, commutator $\langle [A(t),B] \rangle$ also has a nice interpretation in the linear response theory. This represents the susceptibility function.

Then, how about anticommutators $\langle \{A(t),B\} \rangle$ for general case? Another correlators that I don't have a physical interpretation is the current noise $$\langle I(t)I(0)\rangle,$$ where $I$ is the current operator, and even its Fourier transform $$\int dt e^{i\omega t} \langle I(t)I(0)\rangle.$$ What is the physical meaning of these operators?

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  • $\begingroup$ $\langle[A(t),B]\rangle$ appears in the Robertson-Schrödinger uncertainty relation. If $A(t)$ and $B$ commute, then $\langle A(t)B\rangle=\langle A(t)\rangle\langle B\rangle$. If they do not commute, there might exist a result from statistics that let you express the expectation value of the product of two random variables as the product of their expected values minus some correlation. $\endgroup$
    – bodokaiser
    Nov 30, 2021 at 11:31
  • $\begingroup$ Ah yes, you might want to check en.wikipedia.org/wiki/… $\endgroup$
    – bodokaiser
    Nov 30, 2021 at 11:32
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    $\begingroup$ If one often encounters them in problem X then part of the physical meaning is that they contain information about problem X. $\endgroup$ Nov 30, 2021 at 12:09

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Imho there is no deeper meaning in these correlators; In the book from Mahan, there was a single interpretation for the definition of the time-ordered Green-function given. That was "The probability of creating an electron at some time and annihilating it at some other time." These different correlation-functions that you wrote down (there are also more) just turned out to be useful when one wants to derive Lehmann-identities and compute observables.

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