Alright I'm not sure if this question is too speculative but here goes:

So in my graduate days I remember wishing I could put some kind of bounds on the S-matrix. I came to understanding that one of the problems was there was no other way to talk about the creation and annihilation operator in than the standard method in bra-ket notation. Due to this I decided to focus on creating my own formulation of them in Quantum Mechanics. I did (partially) succeed: https://mathoverflow.net/questions/301699/can-one-calculate-the-following-operator

One can define a creation operator:

$$ A^\dagger | n \rangle = | n+1 \rangle$$

In fact,

$$ A^\dagger = |2 \rangle \langle 1 | + |3 \rangle \langle 2| + |4 \rangle \langle 3 | + \dots$$

The quantum mechanical creation operator is $\hat A^{\dagger} \leq \hat a^{\dagger}$

Taken from the answer (but edited due notation consistency error):

... we interpret $|x\rangle \langle y|$ as a rank 1 operator. In particular, $\hat{n}$ is a coisometry which takes the orthonormal set $\{|n\rangle, |2n\rangle, |3n\rangle, \ldots\}$ to the standard basis $\{|1\rangle, |2\rangle, |3\rangle, \ldots\}$. The "rational operator" $\frac{\hat{m}}{n}$ takes the basis vector $|kn\rangle$ to $|km\rangle$ when $kn$ is an integer...

Now we make our reformulation:

$$ (\hat 1 - A^\dagger)^{-1} = ( \sum_{0 < R \leq 1} \hat R)^\dagger $$

where $( \sum_{0 < R \leq 1} \hat R)$ represents the sum of all rational operators whose corresponding rational number is less than or equal to $1$.

Assuming the math somehow works itself out (?) Would a reformulation of the creation and annihilation operator help me in the endeavour of "bounding the S-matrix"? Have other people tried similar games for the similar purposes?