as was outlined in Section \[sec\_intro\], for coherent states on complex Hilbert spaces. Indeed, let $V^L_H$ be a left quaternionic Hilbert space whose dimension could be finite or countably infinite and let $\phi_m , \;\; m =0,1,2, \ldots$, be an orthonormal basis of this space. Let $X$ be a locally compact space and $\mu$ a (Radon) measure on it. Consider a set of functions $\Phi_m : X \longrightarrow H , \;\; m =0,1,2, \ldots ,$ of the same cardinality as the dimension of $V^L_H$, and which satisfy the two conditions,

-   $ 0 < \mathcal N (x) := \sum_m\vert\Phi_m (x)\vert^2 < \infty$, for all $x \in X$.

-   $\int_X \Phi_m (x)\overline{\Phi_n (x)}\; d\mu (x) = \delta_{m n}$, for all $m$ and $n$.

A family of coherent states $\{ \eta_x \mid x \in X\}\subset V^L_H$ can now be defined to be the vectors, $$\eta_x = \mathcal N (x)^{- \frac 12} \sum_m \Phi_m (x) \phi_m\; .
\label{quat-CS}$$

By construction, these coherent states are seen to be normalized, i.e., $\Vert \eta_x \Vert^2 =1$, for all $x\in X$, and to satisfy the resolution of the identity, $$\int_X \langle f \mid \eta_x \rangle \langle \eta_x \mid g \rangle \; \mathcal N (x)d\mu (x)\;
= \langle 