I read about Fractional Quantum Mechanics and it seemed interesting. But are there any justifications for this concept, such as some connection to reality, or other physical motivations, apart from the pure mathematical insight?
If there are none, why did anyone even bother to invent it?
It seems the goal here is to be able to explain all kind of phenomena considering complex situations, in which nonlinearity could be infeasible to handle as it happens in non-quantic systems. According to this reference, the fractional Schrödinger equation
$$i\hbar\dfrac{\partial\Psi(\vec{x},t)}{\partial t}=-[D_{\alpha}(\hbar\nabla)^{\alpha}+V(\vec{x},t)]\Psi(\vec{x},t)$$
where $(\hbar\nabla)^{\alpha}$ is the quantum Riesz fractional derivative
$$(-\hbar ^2\Delta )^{\alpha /2}\Psi (\vec{x},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i\frac{\vec{p}\cdot\vec{x}}\hbar }|\mathbf{p}|^\alpha \varphi ( \vec{p},t)$$
still corresponds/represents quantic systems. For instance, Laskin shows that uncertainty (fractal) it does exist, because
$$\langle|\Delta x|^\mu\rangle^{1/{\mu}}\cdot\langle|\Delta p|^\mu\rangle^{1/{\mu}}>\dfrac{\hbar}{(2\alpha)^{1/{\mu}}}$$
for $\mu<\alpha$ and $1<\alpha\leq 2$.
