# The Bethe Ansatz

#### Correlators; Lehmann representationd.b.c

For a given specific state $$\alpha$$ and (Hermitian) operator $${\cal O}$$, we will define the dynamical correlator

\begin{equation*} S_\alpha (j, j'; t, t') \equiv \langle \alpha | O_j (t) O_{j'} (t') | \alpha \rangle. \end{equation*}

If the system exhibits translational invariance, and if the Hamiltonian is time-independent, then the correlator becomes a function of the position and time differences only,

\begin{equation*} S_\alpha (j, j'; t, t') = S_\alpha (j - j', t - t'). \end{equation*}

Let us introduce the Fourier representation

\begin{equation*} S_\alpha (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k (j-j')} \int_{-\infty}^\infty dt e^{i \omega t} S_\alpha (j-j', t) \end{equation*}

for the dynamical correlation, and

\begin{equation*} {\cal O}_j = \frac{1}{N} \sum_k e^{ikj} {\cal O}_k, \hspace{5mm} {\cal O}_k = \sum_j e^{-ikj} {\cal O}_j \end{equation*}

for the operator. Introducing a resolution of the identity $${\bf 1} = \sum_{\alpha} | \alpha \rangle \langle \alpha |$$ in terms of eigenstates of the Hamiltonian, we have

\begin{align*} S_\alpha (k, \omega) &= \frac{1}{N} \int_{-\infty}^\infty dt e^{i \omega t} \langle \alpha | e^{iHt} \sum_j e^{-ikj} {\cal O}_j e^{-iHt} \sum_{\alpha'} | \alpha' \rangle \langle \alpha' | \sum_{j'} e^{i k j'} {\cal O}_{j'} | \alpha \rangle \\ &= \frac{1}{N} \sum_{\alpha'} \int_{-\infty}^\infty dt e^{i (\omega + E_\alpha - E_{\alpha'}) t} \langle \alpha | {\cal O}_k | \alpha' \rangle \langle \alpha' | O_{-k} | \alpha \rangle. \end{align*}

This can be rewritten using the identity $$\int_{-\infty}^\infty dt e^{i (\omega - \omega') t} = 2\pi \delta (\omega - \omega')$$ (and $$(O_k)^\dagger = O_{-k}$$ for a Hermitian operator), yielding the Lehmann representation

$$S_\alpha (k, \omega) = \frac{2\pi}{N} \sum_{\alpha'} |\langle \alpha | O_k | \alpha' \rangle|^2 \delta(\omega - (E_{\alpha'} - E_\alpha)). \tag{Lr}\label{Lr}$$

The physical content of the Lehmann representation is made clear by Fermi's Golden rule Fgr: starting from the state $$\alpha$$, one entry of the operator induces a transition to the excited state $$\alpha'$$ which lives at momentum/energy $$k, \omega$$ above $$\alpha$$, and acts as a mediator of correlations. The full correlator is given by the sum over all accessible intermediate states $$\alpha'$$.

Note the very important fact that through the Lehmann representation, the dynamical correlator is given by a sum of strictly non-negative terms. This is of immense importance for the practical utility of sum rules.