二阶主约束是否生成规范变换？电磁学与引力，有质量与无质量情形
Does a Second-Class Primary Constraint Generate a Gauge Transformation? Electromagnetisms and Gravities, Massless and Massive
- University of Cambridge（剑桥大学）
- University of Lincoln（林肯大学）
- University of South Carolina（南卡罗来纳大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明在约束哈密顿动力学中，二阶主约束（如Proca有质量电磁学和有质量自旋2引力理论中的）也能像一阶主约束一样生成规范变换，从而挑战了传统观点。
AI中文摘要：
在约束哈密顿动力学中，关于一阶约束如何生成规范变换存在两种观点：一种是各自独立生成，另一种是仅通过特定组合生成，即 Rosenfeld-Anderson-Bergmann-Castellani 规范生成元。该规范生成元 $G$ 保持哈密顿方程不变，并且至多通过边界项改变正则作用量；哈密顿方程是正则作用量的欧拉-拉格朗日方程。因此，正则形式体系等价于拉格朗日形式体系，并且在重要例子中确实被后者所包含，其中许多正则动量充当辅助场。$G$ 生成的变换基本上等价于通常的四维拉格朗日表达式，例如电磁学中的四维梯度或广义相对论中的时空坐标变换（在壳上）。最近的研究表明，在 Proca 非规范与 Stueckelberg 规范有质量电磁学之间，各自独立的一阶约束会导致不等价的观测量。然而，普遍认同的是，二阶约束不生成规范变换。本文表明，在麦克斯韦理论中一阶主约束生成规范变换的同样意义上，Proca 有质量电磁学中的二阶主约束也生成规范变换。同样地，广义相对论的各种有质量自旋 $2$ 相关理论中的二阶主约束所生成的规范变换，与广义相对论中相应的一阶主约束所生成的规范变换一样多。因此，一阶约束通常各自独立生成规范变换的观点面临一个难题，而规范生成元观点则不会面临该难题。
英文摘要：
In constrained Hamiltonian dynamics there are two views regarding how first-class constraints generate gauge transformations: individually or only in a certain combination, the Rosenfeld-Anderson-Bergmann-Castellani gauge generator. This gauge generator $G$ preserves Hamilton's equations and changes the canonical action at most by a boundary term; Hamiltonian's equations are the Euler-Lagrange equations for the canonical action. Hence the canonical formalism is equivalent to the Lagrangian formalism, and indeed subsumed within it (in important examples) with many canonical momenta serving as auxiliary fields. $G$ generates transformations basically equivalent to the usual 4-dimensional Lagrangian expressions, such as a 4-gradient in electromagnetism or a space-time coordinate transformation (on-shell) in General Relativity. It has been shown recently that separate first-class constraints lead to inequivalent observables between Proca non-gauge and Stueckelberg gauge massive electromagnetism. There is, however, widespread agreement that second-class constraints do not generate gauge transformations. Here it is shown that in such a sense as the first-class primary constraint in Maxwell's theory generates a gauge transformation, the second-class primary constraint in Proca's massive electromagnetism also generates a gauge transformation. Likewise the second-class primary constraints in various massive spin $2$ relatives of General Relativity generate as much of a gauge transformation as do the corresponding first-class primary constraints in GR. Hence the view that first-class constraints typically generate gauge transformations _individually_ faces a puzzle not faced by the gauge generator view.
