# Creation and Annnihilator Operators: generality and meaning

I am studying my fisrst course in quantum mechanichs where we treated the example of the Harmonic Oscillator through the Weyl Heisenberg Spectrum Generating Algebra Method. In that context we defined creation adn annihilation operators as $$a^+=\sqrt{\frac{m\omega}{2h}}(x-i\frac{p}{2m})\ \ \ \ \ \ \ a=\sqrt{\frac{m\omega}{2h}}(x+i\frac{p}{2m})$$

Moreover we searched minimum uncertainty states (coherent states) for this system as the eigenstates of $$a$$. Searching around I found that this is not a case, since one can charachterize states minimizing uncertainty for a couple of operators $$X,P$$ in the eigenstates of an operator of the form $$(X+i\delta P)$$ with $$\delta\in\mathbb{R}$$. An operator showing a structure analogous to the annihilation one.

Finally, I glared by chance at Wick quantization, where one, instead of quantizing operators $$X,P$$ quantizes polynomial expressions of the form $$(X+i\delta P)^i(X-i\delta P)^j$$.

Then I wonder:

• Is the definition of creation annihilation operators 'canonically' extendable to any couple of operators?
• Are the two instances where I encountered somehow related?
• What is their deep meaning in the theory?