Faculty of Philosophy, University of Cambridge, United Kingdom; Programme in Philosophy, University of Lincoln, United Kingdom; Department of Philosophy, University of South Carolina, United States.
Stud Hist Philos Sci. 2022 Oct;95:1-27. doi: 10.1016/j.shpsa.2022.06.012. Epub 2022 Aug 2.
The problem of observables and their supposed lack of change has been significant in Hamiltonian quantum gravity since the 1950s. This paper considers the unrecognized variety of ideas about observables in the thought of constrained Hamiltonian dynamics co-founder Peter Bergmann, who trained many students at Syracuse and invented observables. Whereas initially Bergmann required a constrained Hamiltonian formalism to be mathematically equivalent to the Lagrangian, in 1953 Bergmann and Schiller introduced a novel postulate, motivated by facilitating quantum gravity. This postulate held that observables were invariant under transformations generated by each individual first-class constraint. While modern works rely on Bergmann's authority and sometimes speak of "Bergmann observables," he had much to say about observables, generally interesting and plausible but not all mutually consistent and much of it neglected. On occasion he required observables to be locally defined (not changeless and global); at times he wanted observables to be independent of the Hamiltonian formalism (implicitly contrary to a definition involving separate first-class constraints). But typically he took observables to have vanishing Poisson bracket with each first-class constraint and took this result to be justified by the example of electrodynamics. He expected observables to be analogous to the transverse true degrees of freedom of electromagnetism. Given these premises, there is no coherent concept of observables which he reliably endorsed, much less established. A revised definition of observables that satisfies the requirement that equivalent theories should have equivalent observables using the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first-class constraints that changes the canonical action ∫dt(pq̇-H) by a boundary term. Bootstrapping from theory formulations with no first-class constraints, one finds that the "external" coordinate gauge symmetry of GR calls for covariance (a transformation rule and hence a 4-dimensional Lie derivative for the Poisson bracket), not invariance (0 Poisson bracket), under G (not each first-class constraint separately).
自 20 世纪 50 年代以来，可观性及其所谓的不变性问题一直是哈密顿量子引力中的一个重要问题。本文考虑了约束哈密顿动力学的共同创始人彼得·伯格曼（Peter Bergmann）的思想中未被认识到的各种可观性观念。伯格曼在锡拉丘兹（Syracuse）培养了许多学生，并发明了可观性，他最初要求约束哈密顿形式主义在数学上与拉格朗日形式等价，而在 1953 年，伯格曼（Bergmann）和席勒（Schiller）提出了一个新的假定，这一假定是为了促进量子引力而提出的。这一假定认为，可观性在由每个第一类约束生成的变换下是不变的。虽然现代作品依赖于伯格曼的权威，有时会说“伯格曼可观性”，但他对可观性有很多话要说，通常很有趣，也很合理，但并不完全一致，而且很多都被忽视了。有时他要求可观性在局部上是定义的（不是不变的和全局的）；有时他希望可观性独立于哈密顿形式主义（隐含地违背了涉及单独的第一类约束的定义）。但通常他认为可观性与第一类约束的泊松括号为零，并认为这一结果可以用电动力学的例子来证明。他期望可观性类似于电磁学的横向真实自由度。在这些前提下，没有一个他可靠认可的连贯的可观性概念，更不用说建立了。一个经过修订的可观性定义，该定义满足要求，即等价理论应该具有等价的可观性，使用罗森菲尔德-安德森-伯格曼-卡斯特拉尼规范生成器 G，这是第一类约束的调谐和，通过边界项改变正则作用量∫dt(pq̇-H)。从没有第一类约束的理论公式开始，人们发现 GR 的“外部”坐标规范对称性要求协变性（一个变换规则，因此对于泊松括号来说是一个 4 维李导数），而不是不变性（0 泊松括号），在 G 下（不是每个第一类约束单独）。
