I develop a theory of symplectic reduction that applies to bounded regions in electromagnetism and Yang-Mills theory. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang-Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call ``flux rotations,'' generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang-Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka ``edge modes.'' However, I argue that a commonly used phase space extension by edge modes is inherently ambiguous and gauge-breaking.

中文翻译：

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带边界的杨米尔斯理论的辛约简：从超选择扇区到边缘模式，再返回

我开发了一种辛归约理论，适用于电磁学和杨米尔斯理论中的有界区域。在这个理论中，通过区域边界的电通量的规范协变超选择扇区起着核心作用：在这些扇区内，存在一个自然的、规范定义的、用于简化杨米尔斯理论的辛结构。这种辛结构不需要包含任何新的自由度。在非阿贝尔情况下，它还支持一族哈密顿矢量场，我称之为“通量旋转”，由模糊的、泊松非交换的电通量产生。由于通量旋转的作用影响系统的总能量，我认为通量旋转不能是杨米尔斯理论的动力学对称性限制在一个区域。我还考虑了在所有超级选择部门的联合上定义辛结构的可能性。这反过来又需要包括额外的边界自由度，即“边缘模式”。然而，我认为边缘模式的常用相空间扩展本质上是模糊的和规范的。