Valuations and orderings on the real Weyl algebra

Abstract

The first Weyl algebra A 1 (k) over a field k is the k-algebra with two generators x, y subject to [y, x] = 1 and was first introduced during the development of quantum mechanics. In this article, we classify all valuations on the real Weyl algebra A 1 (R) whose residue field is R. We then use a noncommutative version of the Baer-Krull theorem to classify all orderings on A 1 (R). As a byproduct of our studies, we settle two open problems in real algebraic geometry. First, we show that not all orderings on A 1 (R) extend to an ordering on a larger ring R[y; δ], where R is the ring of Puiseux series, introduced by Marshall and Zhang in [16], and characterize the orderings that do have such an extension. Second, we show that for valuations on noncommutative division rings, Kaplansky's theorem that extensions by limits of pseudo-Cauchy sequences are immediate fails in general.