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 the 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.

我们研究状态的上外尔代数特定类\（\ mathcal【W} \）与辛向量空间相关联的小号和冯·诺依曼的的表示代数生成\（\ mathcal【W} \） 。量子物理学中的应用例如由于规范条件而要求实现约束方程式，并且可以基于所谓的狄拉克状态。可以通过S上的非线性函数来表征这些状态，事实证明，与非平凡的Dirac状态相对应的状态通常是不连续的。我们讨论了S上函数之间相互作用的一般方面和状态，还可以对非平凡的狄拉克状态的特定示例类别进行分析。在最后一部分中，我们将重点放在\（S = L ^ 2（\ mathbb {R} ^ n）\）的特定情况下或在\（\ mathbb {R} ^ n \）上测试函数，并关联状态的属性在\（\ mathcal {W} \）上使用\（\ mathbb {R} ^ n \）上的广义函数，或者在Schwartz函数和温度分布上进行相应的Borel测度的调和分析。