physical meaning to $\mathfrak{L}_H$. Further investigation of this point, quantization of the type done in [@Nic; @Ga] and a study of the modular structures along the lines of [@AFG], using quaternionic coherent states, are left for future work.

Mathematical preleminaries
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In order to make the paper self-contained, we recall a few facts about quaternions which may not be well-known. In particular, we revisit the $2\times 2$ complex matrix representations of quaternions, quaternionic Hilbert spaces, their duals, the Cullen derivative and the definition of regularity of a function of a quaternionic variable.

Quaternions
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Let $H$ denote the field of quaternions. Its elements are of the form ${\mathbf{q}}=x_0+x_1i+x_2j+x_3k$ where $x_0,x_1,x_2$ and $x_3$ are real numbers, and $i,j,k$ are imaginary units such that $i^2=j^2=k^2=-1$, $ij=-ji=k$, $jk=-kj=i$ and $ki=-ik=j$. The quaternionic conjugate of ${\mathbf{q}}$ is defined to be $\overline{{\mathbf{q}}} = x_0 - x_1i - x_2j - x_3k$. We shall find it convenient to use the representation of quaternions by $2\times 2$ complex matrices: $${\mathbf{q}}= x_0 \sigma_{0} + i \underline{x} \cdot \underline{\sigma},$$ with $x_0 \in \mathbb R , \quad \underline{x} = (x_1, x_2, x_3)
\in \mathbb R^3$, $\sigma_0 = \mathbb{I}_2$, the $2\times 2$ identity matrix, and $\underline{\sigma} = (\sigma_1, -\sigma_2, \sigma_3)$, where the 