Abstract Local gauge theories are in a complicated relationship with boundaries. Whereas fixing the gauge can often shave off unwanted redundancies, the coupling of different bounded regions requires the use of gauge-variant elements. Therefore, coupling is inimical to gauge-fixing, as usually understood. This resistance to gauge-fixing has led some to declare the coupling of subsystems to be the raison d’etre of gauge (Rovelli, 2014). Indeed, while gauge-fixing is entirely unproblematic for a single region without boundary, it introduces arbitrary boundary conditions on the gauge degrees of freedom themselves—these conditions lack a physical interpretation when they are not functionals of the original fields. Such arbitrary boundary choices creep into the calculation of charges through Noether's second theorem (Noether, 1971), muddling the assignment of physical charges to local gauge symmetries. The confusion brewn by gauge at boundaries is well-known, and must be contended with both conceptually and technically. It may seem natural to replace the arbitrary boundary choice with new degrees of freedom, for using such a device we resolve some of these confusions while leaving no gauge-dependence on the computation of Noether charges (Donnelly & Freidel, 2016). But, concretely, such boundary degrees of freedom are rather arbitrary—they have no relation to the original field-content of the field theory. How should we conceive of them? Here I will explicate the problems mentioned above and illustrate a possible resolution: in a recent series of papers (Gomes, Hopfmller,& Riello, 2018; Gomes & Riello, 2017, 2018) the notion of a connection-form was put forward and implemented in the field-space of gauge theories. Using this tool, a modified version of symplectic geometry—here called ‘horizontal’—is possible. Independently of boundary conditions, this formalism bestows to each region a physically salient, relational notion of charge: the horizontal Noether charge. Meanwhile, as required, the connection-form mediates a composition of regions, one compatible with the attribution of horizontal Noether charges to each region. The aims of this paper are to highlight the boundary issues in the treatment of gauge, and to discuss how gauge theory may be conceptually clarified in light of a resolution to these issues.

中文翻译：

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测量场空间的边界

摘要局部规范理论与边界有着复杂的关系。固定仪表通常可以消除不必要的冗余，而不同边界区域的耦合需要使用仪表可变元素。因此，如通常所理解的，耦合对于量规固定是有害的。这种对量规固定的抵制导致一些人宣称子系统的耦合是量规的存在理由（Rovelli，2014）。的确，尽管对于没有边界的单个区域，轨距固定完全没有问题，但它会在轨距自由度本身上引入任意边界条件-当这些条件不是原始场的功能时，这些条件就缺乏物理解释。这种任意的边界选择会通过Noether的第二定理（Noether，1971年）渗入电荷的计算中，混淆将物理电荷分配给局部规范对称性。边界处的量规引起的混乱是众所周知的，并且必须在概念和技术上加以解决。用新的自由度替换任意边界选择似乎是很自然的，因为使用这样的设备，我们解决了其中的一些困惑，同时又不依赖于Noether电荷的计算而依赖于尺度（Donnelly＆Freidel，2016）。但是，具体而言，这种边界自由度是相当随意的-它们与场论的原始场内容无关。我们应该如何构思它们？在这里，我将说明上述问题并举例说明可能的解决方案：在最近的一系列论文中（Gomes，Hopfmller和Riley，2018年； Gomes和Riello，2017年，2018年）提出了一种连接形式的概念，并在规范理论的领域中得以实施。使用此工具，可以实现辛几何的修改版本（此处称为“水平”）。与边界条件无关，这种形式主义赋予每个区域一种物理上显着的关系电荷概念：水平Noether电荷。同时，根据需要，连接形式介导区域的组成，该区域与水平Noether电荷对每个区域的归属兼容。本文的目的是强调仪表测量的边界问题，并讨论如何根据这些问题的解决方案在概念上澄清仪表理论。可以修改辛几何的版本（此处称为“水平”）。与边界条件无关，这种形式主义赋予每个区域一种物理上显着的关系电荷概念：水平Noether电荷。同时，根据需要，连接形式介导区域的组成，该区域与水平Noether电荷对每个区域的归属兼容。本文的目的是强调仪表测量的边界问题，并讨论如何根据这些问题的解决方案在概念上澄清仪表理论。可以修改辛几何的版本（此处称为“水平”）。与边界条件无关，这种形式主义赋予每个区域一种物理上显着的关系电荷概念：水平Noether电荷。同时，根据需要，连接形式介导区域的组成，该区域与水平Noether电荷对每个区域的归属兼容。本文的目的是强调仪表测量的边界问题，并讨论如何根据这些问题的解决方案在概念上澄清仪表理论。一个与水平Noether电荷归属于每个区域兼容。本文的目的是强调仪表测量的边界问题，并讨论如何根据这些问题的解决方案在概念上澄清仪表理论。一个与水平Noether电荷归属于每个区域兼容。本文的目的是强调仪表测量的边界问题，并讨论如何根据这些问题的解决方案在概念上澄清仪表理论。