A D-module is an algebraic structure with several differential operators acting on it.
Observables are represented by operators acting on a Hilbert space of such quantum states.
Working in standard spherical coordinates, we can define a particular operator acting on a function as:
's are assumed to be positive operators acting on appropriate state space and .
The key element of the operational calculus is to consider differentiation as an operator acting on functions.
In this theory, the kernel is understood to be a compact operator acting on a Banach space of functions.
Let be an operator acting on an -dimensional Hilbert space .
The spin raising and lowering operators acting on these eigenvectors give:
Quantization converts classical fields into operators acting on quantum states of the field theory.
Likewise, the operator acting on a state produces and .
