Equivalent theories redefine Hamiltonian observables to exhibit change in general relativity
Equivalent theories redefine Hamiltonian observables to exhibit change in general relativity
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等效理论重新定义哈密顿可观测量以展示广义相对论的变化
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发表时间：
2016
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通讯作者：
Brian Pitts
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作者：
Brian Pitts
通常定义的规范广义相对论的可观测值中缺少变化和局部空间变化，这是时间问题的一个方面。可以使用理论、非规范和规范的等效公式来测试定义，因为它们必须具有等效的可观测值，并且在非规范公式中一切都是可观测的。从非规范公式中获取可观测量并在规范公式中找到等价物，要求等价物是可观测量，从而限制了定义。对于大质量光子，德布罗意-普罗卡非规范公式可观测量 Aμ 相当于 Stueckelberg-Utiyama 规范公式量 Aμ+∂μφ，因此它必须是可观测量。为了实现这一结果，可观测量必须具有 0 泊松括号，而不是每个第一类约束，而是使用 Rosenfeld-Anderson-Bergmann-Castellani 规范生成器 G，即第一类约束的调整总和，符合可观测量的 Pons-Salisbury-Sundermeyer 定义。外部规范对称性的定义可以使用大重力进行测试，其中可以通过时钟场 XA 的参数化来安装规范自由度。非规范可观测量 gμν 具有规范当量 XA,μgμνXB,ν。 XA,μgμνXB,ν 与 G 的泊松括号结果不是 0，而是李导数。这个非零泊松括号完善并系统化了 Kuchař 的利用哈密顿约束放宽 0 泊松括号条件的提议。因此，可观测量需要相对于外部规范对称性的协变性，而不是不变性。大质量引力的拉格朗日和哈密顿量是广义相对论 + Λ + 4标量的拉格朗日和哈密顿量，因此相同的可观测量定义也适用于广义相对论。诸如 gμν 之类的局部场是可观测量的。因此可观察到的变化。要求等价理论有等价可观测量也恢复了哈密顿-拉格朗日等价。
Change and local spatial variation are missing in canonical General Relativity’s observables as usually defined, an aspect of the problem of time. Definitions can be tested using equivalent formulations of a theory, non-gauge and gauge, because they must have equivalent observables and everything is observable in the non-gauge formulation. Taking an observable from the non-gauge formulation and finding the equivalent in the gauge formulation, one requires that the equivalent be an observable, thus constraining definitions. For massive photons, the de Broglie–Proca non-gauge formulation observable Aμ is equivalent to the Stueckelberg–Utiyama gauge formulation quantity Aμ+∂μϕ, which must therefore be an observable. To achieve that result, observables must have 0 Poisson bracket not with each first-class constraint, but with the Rosenfeld–Anderson–Bergmann–Castellani gauge generator G, a tuned sum of first-class constraints, in accord with the Pons–Salisbury–Sundermeyer definition of observables. The definition for external gauge symmetries can be tested using massive gravity, where one can install gauge freedom by parametrization with clock fields XA. The non-gauge observable gμν has the gauge equivalent XA,μgμνXB,ν. The Poisson bracket of XA,μgμνXB,ν with G turns out to be not 0 but a Lie derivative. This non-zero Poisson bracket refines and systematizes Kuchař’s proposal to relax the 0 Poisson bracket condition with the Hamiltonian constraint. Thus observables need covariance, not invariance, in relation to external gauge symmetries. The Lagrangian and Hamiltonian for massive gravity are those of General Relativity + Λ + 4 scalars, so the same definition of observables applies to General Relativity. Local fields such as gμν are observables. Thus observables change. Requiring equivalent observables for equivalent theories also recovers Hamiltonian–Lagrangian equivalence.