The real time linear response function is given by Kubo's formula $$\chi_{AB}(t-t')=-i\Theta(t-t')\langle [A,B]\rangle.$$ This can also be obtained by analytically continuing the imaginary time ordered correlator function $-\langle T_{\tau}AB\rangle $
A key difference between these two function is that the Kubo's formula is invariant under the substitution $A\rightarrow A+C_1$ and $B\rightarrow B+C_2$ where $C_1$ and $C_2$ are constant operators while the the time ordered correlator is not invariant and changes by a constant.
However we can look at the response function at zero frequency two ways, the first by taking the fourier transform of the real time Kubo formula and the other by taking the 0th frequency matsubara term from the time ordered correlator. However because of the previous paragraph, the zero frequency response ends up not being unique. What's mathematically gone wrong here?
