The Bethe Ansatz

Besides the retarded correlation function, it is also possible to define the following functions: the advanced correlation function which (contrary to the retarded function) is only nonvanishing for negative time arguments,

\[ {\cal C}^{\hat{O}, \hat{P}}_{adv, \psi} (t- t') \equiv i \theta (t' - t) \langle \psi | [ \hat{O}^I (t), \hat{P}^I (t') ] | \psi \rangle \tag{Cadv}\label{Cadv} \]

or (taking again \(t' = 0\) without loss of generality)

\[ {\cal C}^{\hat{O}, \hat{P}}_{adv, \psi_\alpha} (t) = i \theta (-t) \sum_{\alpha'} \left(O_{\alpha \alpha'} P_{\alpha' \alpha} e^{i (E_\alpha - E_{\alpha'}) t} - P_{\alpha \alpha'} O_{\alpha' \alpha} e^{-i(E_\alpha - E_{\alpha'}) t} \right). \]

Under the (time) Fourier transform (using the same conventions as for the retarded function), this becomes

\[ {\cal C}^{\hat{O}, \hat{P}}_{adv, \psi_\alpha} (\omega) = \sum_{\alpha'} \left( \frac{O_{\alpha \alpha'} P_{\alpha' \alpha}}{\omega + E_\alpha - E_{\alpha'} - i\eta} - \frac{P_{\alpha \alpha'} O_{\alpha' \alpha}}{\omega - (E_\alpha - E_{\alpha'}) - i\eta} \right) \]

so the advanced function is analytic in complex \(\omega\) in the entire lower-half plane.

We can also define the real-time correlation function involving the time-ordered product of the operators,

\[ {\cal C}^{\hat{O}, \hat{P}}_{\psi} (t- t') \equiv -i \langle \psi | T_t (\hat{O}^I (t) \hat{P}^I (t')) | \psi \rangle \]

or equivalently

\[ {\cal C}^{\hat{O}, \hat{P}}_{\psi_\alpha} (t) = -i \sum_{\alpha'} \left(O_{\alpha \alpha'} P_{\alpha' \alpha} ~\theta (t) e^{i (E_\alpha - E_{\alpha'}) t} + P_{\alpha \alpha'} O_{\alpha' \alpha} ~\theta (-t) e^{-i(E_\alpha - E_{\alpha'}) t} \right). \]

The (time) Fourier transform of the real-time correlation function can thus be written

\[ {\cal C}^{\hat{O}, \hat{P}}_{\psi_\alpha} (\omega) = \sum_{\alpha'} \left( \frac{O_{\alpha \alpha'} P_{\alpha' \alpha}}{\omega + E_\alpha - E_{\alpha'} + i\eta} - \frac{P_{\alpha \alpha'} O_{\alpha' \alpha}}{\omega - (E_\alpha - E_{\alpha'}) - i\eta} \right) \]

and is not analytic either in the lower or upper half-plane of \(\omega\).