# Meaning of correlators $\langle A(t)B \rangle$, $\langle [A(t),B] \rangle$, $\langle \{A(t),B\} \rangle$, etc

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?

• $\langle[A(t),B]\rangle$ 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. Nov 30, 2021 at 11:31
• Ah yes, you might want to check en.wikipedia.org/wiki/… Nov 30, 2021 at 11:32
• If one often encounters them in problem X then part of the physical meaning is that they contain information about problem X. Nov 30, 2021 at 12:09