= |\Theta + \xi
\rangle. \label{transla}$$

It is also known that $(|\Theta \rangle)^* \neq \langle \Theta|$, [@wein; @imelda]. The dual Fock space ${\cal F}^*_F$ is constructed from the dual vacuum state $\langle 0|$ as follows $$\begin{aligned}
\langle \Theta | &=& \langle 0 | \prod_{\vec{x},\alpha}
\widehat{\Theta}_\alpha (\vec{x}) \exp \left\{-{i\over\hbar} \int
d^d x \Theta
(\vec{x}) \cdot \widehat{\pi}_\Theta (\vec{x}) \right\}\nonumber \\
&=& \langle 0 | \prod_{\vec{x},\alpha} \widehat{\Theta}_\alpha
(\vec{x}) \exp \left\{{i\over\hbar} \int d^d x
\widehat{\pi}_\Theta (\vec{x}) \cdot \Theta (\vec{x}) \right\}.
\label{bra}\end{aligned}$$ This bra satisfies $\langle \Theta | \widehat{\Theta}_\alpha
(\vec{x}) = \langle \Theta | \Theta_\alpha (\vec{x})$ and $$\langle \Theta' | \Theta \rangle = \prod_{\vec{x},\alpha}
\bigg(\Theta_\alpha (\vec{x}) - \Theta'_\alpha (\vec{x})\bigg) =:
\delta (\Theta - \Theta').$$ Analogously for the momentum representation we define $$|\pi_\Theta \rangle = \exp \left\{ -{i\over\hbar} \int d^d x
\widehat{\Theta} (\vec{x}) \cdot \pi_\Theta (\vec{x}) \right\}
\prod_{\vec{x},\alpha} \widehat{\pi}_{\Theta_\alpha} (\vec{x}) |0
\rangle, \ \ \ \ \ \ \langle \pi_\Theta| = \langle 0 | \exp
\left\{ {i\over\hbar} \int d^d x \widehat{\Theta} (\vec{x}) \cdot
\pi_\Theta (\vec{x}) \right\}.$$ Then one gets $$\langle \pi'_\Theta | \pi_\Theta \rangle = \prod_{\vec{x},\alpha}
\bigg(\pi'_{\Theta_\alpha} (\vec{x}) - \pi_{\Theta_\alpha}
(\vec{x})\bigg) =: \delta (\pi'_{\Theta} - \pi_\Theta).$$ From the above relations it is easy to compute $$\langle \pi_\Theta | \Theta \rangle = \exp \left\{ -{i\over\hbar}
\int d^d 