\phi, \psi)= d\tau(r) d\theta d\Omega(\phi ,\psi ), \quad \text{ with} \quad
d\Omega(\phi ,\psi) = \displaystyle{\frac{1}{4\pi}} \sin{\phi}
d\phi d\psi .$$

Quaternionic Hilbert spaces
---------------------------

In this subsection we define left and right quaternionic Hilbert spaces. For details we refer the reader to [@Ad]. We also define the Hilbert space of square integrable functions on quaternions based on [@Vis].

### Right Quaternionic Hilbert Space

Let $V_{H}^{R}$ be a linear vector space under right multiplication by quaternionic scalars (again $H$ standing for the field of quaternions). For $f,g,h\in V_{H}^{R}$ and ${\mathbf{q}}\in H$, the inner product $$\langle\cdot\mid\cdot\rangle:V_{H}^{R}\times V_{H}^{R}\longrightarrow H$$ satisfies the following properties

1.  $\overline{\langle f\mid g\rangle}=\langle g\mid f\rangle$

2.  $\|f\|^{2}=\langle f\mid f\rangle>0$ unless $f=0$, a real norm

3.  $\langle f\mid g+h\rangle=\langle f\mid g\rangle+\langle f\mid h\rangle$

4.  $\langle f\mid g{\mathbf{q}}\rangle=\langle f\mid g\rangle{\mathbf{q}}$

5.  $\langle f{\mathbf{q}}\mid g\rangle=\overline{{\mathbf{q}}}\langle f\mid g\rangle$

where $\overline{{\mathbf{q}}}$ stands for the quaternionic conjugate. We assume that the space $V_{H}^{R}$ is complete under the norm given above. Then, together with $\langle\cdot\mid\cdot\rangle$ this defines a right quaternionic Hilbert space, which we shall assume to be separable. Quaternionic Hilbert spaces share most of the standard properties of complex Hilbert spaces. In particular, the Cauchy-Schwartz inequality holds on quaternionic Hilbert spaces as well as the Riesz 