Post #4050997

Alpöge's new counterexample to the Jacobian conjecture has consequences for quantum mechanics in 3 dimensions! You can find operators obeying the usual position-momentum commutation relations that generate a smaller algebra than the usual ones do. Some observables become unreachable. Here's the setup: the Weyl algebra W₃ is generated by q₁,q₂,q₃ and p₁,p₂,p₃ obeying [pⱼ,qₖ] = −iℏδⱼₖ, all other commutators zero. Concretely, it consists of polynomial differential operators on ℝ³ - roughly the observables of a quantum particle in 3 dimensions that are polynomial in position and momentum. In 1968, the famous mathematician Jacques Dixmier conjectured that every endomorphism of W₃ is an automorphism: basically, you can't squoosh the algebra into smaller piece of itself. But the new counterexample to the Jacobian conjecture shows you CAN! (1/n) https://en.wikipedia.org/wiki/Weyl_algebra