Groenewold’s theorem, also known as the Groenewold-van Hove theorem, is a fundamental result in the theory of quantization, which demonstrates the impossibility of a consistent mapping from classical observables to quantum operators that preserves the Poisson bracket structure.

It shows that it is impossible to consistently quantize all classical observables while maintaining the correspondence between Poisson brackets and quantum commutators. The theorem has significant implications for the foundations of quantum mechanics, as it highlights the limitations of canonical quantization and the challenges of formulating a consistent quantum theory from classical mechanics.

Groenewold’s theorem also has connections to the Wigner-Weyl transform, which provides a way to map between phase-space functions and quantum operators. However, Groenewold’s theorem asserts that no such map can have all the ideal properties one would desire for a consistent quantization scheme. Despite this, the Wigner-Weyl transform remains a valuable tool for understanding quantum mechanics in phase space.