Deformation quantization

Deformation quantization is a mathematical framework that smoothly transitions classical physics, which describes large-scale phenomena, into quantum physics, which governs the microscopic world. Instead of replacing classical equations abruptly, it "deforms" or modifies them gradually by introducing a small parameter (like Planck’s constant) that measures quantum effects. This process alters the algebraic rules of classical observables (like position and momentum) into non-commutative versions, capturing quantum behavior while maintaining a connection to classical structures. It provides a rigorous way to understand how classical mechanics naturally evolves into quantum mechanics through a continuous deformation.