This paper discusses several functional analytic issues relevant for field theories in the context of the Hamiltonian formulation for a free, massless, scalar field defined on a closed interval of the real line. The fields that we use belong to a Sobolev space with a scalar product. As we show this choice is useful because it leads to an explicit representation of the points in the fibers of the phase space (the cotangent bundle of the configuration space). The dynamical role of the boundary of the spatial manifold where the fields are defined is analyzed.

中文翻译：

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有界区间上 Sobolev 空间的自由标量场理论

本文在实线闭区间上定义的自由、无质量标量场的哈密顿公式的背景下讨论了与场论相关的几个泛函分析问题。我们使用的字段属于具有标量积的 Sobolev 空间。正如我们所表明的，这种选择是有用的，因为它导致相空间纤维（配置空间的余切丛）中的点的显式表示。分析了定义场的空间流形边界的动力学作用。