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$\newcommand{\R}{\mathbb R} \newcommand{\rweyl}{\mathcal{A}_1(\R)}$ The firstWeyl algebra $\mathcal{A}_1(k)$ over a field $k$ is the $k$-algebra with twogenerators $x, y$ subject to $[y, x] = 1$ and was first introduced during thedevelopment of quantum mechanics. In this article, we classify all valuationson the real Weyl algebra $\mathcal{A}_1(\mathbb{R})$ whose residue field is$\mathbb{R}$. We then use a noncommutative version of the Baer-Krull theoremfrom real algebraic geometry to classify all orderings on$\mathcal{A}_1(\mathbb{R})$. As a byproduct of our studies, we settle two openproblems in noncommutative valuation theory. First, we show that not allvaluations on $\mathcal{A}_1(\mathbb{R})$ with residue field $\R$ extend to avaluation on a larger ring $R[y ; \delta]$, where $R$ is the ring of Puiseuxseries, introduced by Marshall and Zhang, with the same residue field, andcharacterize the valuations that do have such an extension. Second, we showthat for valuations on noncommutative division rings, Kaplansky's theorem thatextensions by limits of pseudo-Cauchy sequences are immediate fails in general.

Valuatuions and orderings on the real Weyl algebra