标题： 规范对称性破缺格点正则化及其连续极限 显示英文标题

标题： Gauge Symmetry Breaking Lattice Regularizations and their Continuum Limit

摘要： 格点正则化在规范场论的非微扰量子化中起着关键作用。 威尔逊提出使用群值链接场，简化了主纤维丛中规范场的正则化，同时在离散化的格点理论中保持规范对称性。 保持规范对称性是可取的，因为其破坏可能会引入不需要的自由度。 然而，并非所有具有规范对称性的理论都允许规范不变的格点正则化，如广义相对论中微分同胚群作为规范对称性时所观察到的。 在这些情况下，需要采用规范对称性破坏的正则化方法。 在本文中，我们认为只要在连续极限中规范对称性得以恢复，格点规范对称性的破坏是可以接受的。 我们提出了一种方法，为一类格点正则化的哈密顿场论构建连续极限，其中正则化破坏了第一类约束的李代数。 此外，我们还提供了一种在连续理论的希尔伯特空间上表示精确规范群的方法。 所考虑的理论类别仅限于那些第一类约束线性于动量的理论，排除了广义相对论的整个规范群，但包含了其空间微分同胚的子群。 我们讨论了将这种量子化扩展到完整规范群的潜在技术。 显示英文摘要

摘要： Lattice regularizations are pivotal in the non-perturbative quantization of gauge field theories. Wilson's proposal to employ group-valued link fields simplifies the regularization of gauge fields in principal fiber bundles, preserving gauge symmetry within the discretized lattice theory. Maintaining gauge symmetry is desirable as its violation can introduce unwanted degrees of freedom. However, not all theories with gauge symmetries admit gauge-invariant lattice regularizations, as observed in general relativity where the diffeomorphism group serves as the gauge symmetry. In such cases, gauge symmetry-breaking regularizations become necessary. In this paper, we argue that a broken lattice gauge symmetry is acceptable as long as gauge symmetry is restored in the continuum limit. We propose a method to construct the continuum limit for a class of lattice-regularized Hamiltonian field theories, where the regularization breaks the Lie algebra of first-class constraints. Additionally, we offer an approach to represent the exact gauge group on the Hilbert space of the continuum theory. The considered class of theories is limited to those with first-class constraints linear in momenta, excluding the entire gauge group of general relativity but encompassing its subgroup of spatial diffeomorphisms. We discuss potential techniques for extending this quantization to the full gauge group.