  We study particular classes of states on the Weyl algebra $\mathcal{W}$
associated with a symplectic vector space $S$ and on the von Neumann algebras
generated in representations of $\mathcal{W}$. Applications in quantum physics
require an implementation of constraint equations, e.g., due to gauge
conditions, and can be based on so-called Dirac states. The states can be
characterized by nonlinear functions on $S$ and it turns out that those
corresponding to non-trivial Dirac states are typically discontinuous. We
discuss general aspects of this interplay between functions on $S$ and states,
but also develop an analysis for a particular example class of non-trivial
Dirac states. In the last part, we focus on the specific situation with $S =
L^2(\mathbb{R}^n)$ or test functions on $\mathbb{R}^n$ and relate properties of
states on $\mathcal{W}$ with those of generalized functions on $\mathbb{R}^n$
or with harmonic analysis aspects of corresponding Borel measures on Schwartz
functions and on temperate distributions.
