Would anyone know what the relationship between the Poisson bracket of classical mechanics and the commutator of quantum mechanics is?

When you quantize, Poisson brackets become commutators. Does that help? Sorry, I’m not really sure what question you’re asking.

The classical and QM equations for the time rate of change of an observable are almost identical except the classical equation uses the Poisson bracket and the QM equation uses the commutator. So I was wondering why that was. May I ask what you mean by “when you quantize”? Do you mean replacing a function by an operator?

You can use the process of ‘canonical quantization’ (a procedure that helps you turn a classical field theory into a quantum theory, which is a lot more diffcult as it sounds as canonical quantum gravity still has many flaws) to derive commutators from the Poisson bracket. Remebre that quantumphysics is essientially new physics though and therefore cannot be solely derived from classical theories making such processes inductive.