The Bethe Ansatz

To define a measure in our Hilbert space, we also introduce the dual pseudovacuum \(\langle 0 | = | 0 \rangle^{\dagger}\), with properties

\begin{equation} \langle 0 | 0 \rangle = 1, \hspace{1cm} \langle 0 | A(\lambda) = a(\lambda) \langle 0 |, \hspace{1cm} \langle 0 | D(\lambda) = d(\lambda) \langle 0 |, \hspace{1cm} \langle 0 | B(\lambda) = 0. \tag{dpv}\label{dpv} \end{equation}

Similarly to states prodBpv, the dual states

\begin{equation} \langle \{ \lambda_j \}_M | \equiv \langle 0 | \prod_{j=1}^M C(\lambda_j) \tag{dpvprodC}\label{dpvprodC} \end{equation}

are eigenstates of the transfer matrix tau with eigenvalue tauev if the set \(\{ \lambda_j \}_M\) satisfies the Bethe equations BER1. This allows us to easily prove the orthogonality condition

\begin{equation} \langle \{ \lambda_j^C \}_M | \{ \lambda_k^B \}_M \rangle = \langle 0 | \prod_{j=1}^M C(\lambda_j^C) \prod_{k=1}^M B(\lambda_k^B) | 0 \rangle = 0, \hspace{5mm} \{ \lambda_j^C \}_M \neq \{ \lambda_k^B \}_M \tag{lClBo}\label{lClBo} \end{equation}

from the fact that

\begin{equation*} \langle \{ \lambda_j^C \}_M | \tau (\lambda) | \{ \lambda_k^B \}_M \rangle = \tau (\lambda | \{ \lambda_j^C \}_M) \langle \{ \lambda_j^C \}_M | \{ \lambda_k^B \}_M \rangle = \tau (\lambda | \{ \lambda_k^B \}_M ) \langle \{ \lambda_j^C \}_M | \{ \lambda_k^B \}_M \rangle \end{equation*}

and that

\begin{equation*} \tau (\lambda | \{ \lambda_j^C \}_M) \neq \tau (\lambda | \{ \lambda_k^B \}_M ), ~~\{ \lambda_j^C \}_M \neq \{ \lambda_k^B \}_M. \end{equation*}

We will consider the normalization of eigenstates explicitly later on, after proving an extremely important theorem due to N. A. Slavnov.