In the quantum mechanics, one asked if the complex number was necessary? A typical answer was that it was not, or that it's simple direct product of real numbers.
However, consider rational number to real number, with additional uncountable irrational numbers. One then think, just because rational number to real number added infinite irrational number, was that necessarily for quantum mechanics?
For a simple argument, without the imaginary number, the solution to $x^2=-1$ was unsolvable, but then do we necessarily need the solution to $x^2=2$, since $$\prod_{k=0}^{100} \left(1-\frac{1}{(4k+2)^2} \right)$$ was precipice enough? or even volume computation with $\pi$ with similar procedures.
However, from rational number to real number and from real number to complex number, in each case an uncountable number of elements was added into the set, while rational number was countable, thus there might be some thing there in the view of number theorists.
But after all, quantum mechanics probably had failed at $10^{-100}$ m, the total range of number that quantum mechanics worked was a finite set!
Do we necessarily need real number for quantum?
Math Manifestly Rational Quantum Mechanics:
Derivatives: $\displaystyle y'(t_0)=\lim_{t\rightarrow 0} \frac{y(t_0+t)-y(t)}{t} \Rightarrow \lim_{n\rightarrow \infty,n\in \mathbb{Z}} \frac{y(t_0+t_n)-y(t_n)}{t_n}$ where $t_n\in \mathbb{Q}$ and $\lim_{n\rightarrow \infty} t_n=0$. (In case any question about $\lim_{n\rightarrow \infty} t_n=0$ from analysis, set $\epsilon_n\in \mathbb{Q}$).
If $y'(t_0)=A\in \mathbb{Q}$ case resolved, if $A$ being a rational number, one make the following argument with $-i\frac{\hbar^2}{m} \frac{\partial^2}{\partial x^2} \Psi(x,t)$.
From uncertainty principle, $\Delta x\Delta p\geq\frac{\hbar}{2}$, one adopt the idea from statistical manifold, that the space had fuzziness, that a minimum length was discrete as somewhat ontic. This should be consistent with the current view of physics, for example, pick $\min(x)=l$ such that the $\frac{1}{l}$ was comparable with all the energy in the observable universe, and $\min(p)=m$ to be such that such that a change in $\frac{1}{m}$ would be much larger than the size of the universe. This was achievable since $\mathbb{Q}$ was dense in $[0,1]$.
Now that there's $l,m$, the next was to argue on how to use them.
Let $y'(t_0)=A$ be in real domain. One recognize that, because of the fuzziness in physical space, $A$ was not actually the irrational $A$. Rather, one make the following equivalence class $[x_n]$ where $\alpha\sim x_n$ if $\alpha \in [n\cdot l,(n+1)\cdot l)$. Claim $y'(t_0)=[A]\in \mathbb{Q}$. Just in case if someone worry about the decimal places, change the previous $l$ with $l/10^{82}$, i.e. divided by all the atoms there was in the universe. Also $\hbar^2/m\sim [\hbar^2/m] \in \mathbb{Q}$ by similiar argument, since the rational field was closed.
First, quantum was "scaled back" from $\mathbb{C}$ to $\mathbb{R}^2$ though vectors and an obvious bilinear operation.("enlarged" in another point of view, actually, since a specific bilinear operation was required to capture the analytic(cauchy) condition) Thus, in the component form $-i\frac{\hbar^2}{m} \frac{\partial^2}{\partial x^2} \Psi(x,t)\Rightarrow [\frac{\hbar^2}{m}]\frac{\partial^2}{\partial x^2} \Psi(x,t)$.
Consider $\displaystyle \frac{\partial^2}{\partial x^2} \Psi(x,t)=\lim_{n\rightarrow,n\in \mathbb{Z}} \frac{\Psi(x+t_n,t)-2\Psi(x,t)+\Psi(x-t_n,t) }{t_n^2}=[p_n]^2$
Here's something interesting $[n m,(n+1) m)$ taking second order power was sent to $[n^2m^2,(n^2+2n+1)m)$ the length of the interval changed!! from $[m]$ to $[(2n+1)m^2]$ with $m$ fixed. (Now if one took a transformation to make $m,l$ large number and stare back to the uncertainty principle and think... That's not the point here, the point was $(2n+1)m$ was in comparable with the size of an integer.) That's why I mentioned before to divide the original $l,m$ by a factor of $10^{82}$, so that the $(2n+1)m$ was still much smaller than the original minimal length $\min(x)$.
Now, the configuration space was labeled by integers, so does all the physical meaningful quantities such as energy, momentum, etc. Not only this was a rational quantum mechanics, it's practically an integer quantum mechanics. The precision should match whatever the calculation one had in mind with the quantum. If not, divide $l,m$ by extra $10^{1000...}$ power. This was just one of such attempt to directly scale back the usual quantum mechanics to rational or even integer domain.
It's interesting to see what was gained or what was lost during the procedure.
