Quantization (physics) attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in. One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space.
Heisenberg group of differential operators on ℝ with polynomial coefficients, since any such operator has a unique representation in the form This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra "W" is a quotient of formula_79. However, this is also easy to see directly from the above representations; viz. by the mapping The application that led Hermann Weyl to an explicit realization of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly, the reason is the Stone–von Neumann theorem: there is a unique unitary representation with