布兰斯-迪克理论中的规范时钟扇区与关系框架等价性
Canonical Clock Sectors and Relational Frame Equivalence in Brans--Dicke Theory
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中文总结 AI 辅助
研究布兰斯-迪克理论中约旦和爱因斯坦框架等价性，以布兰斯-迪克标量为关系时钟，发现约化哈密顿量差异源于时钟扇区规范嵌入不匹配，构建规范图表恢复等价性，确定规范时钟扇区为基本结构并提供理解框架依赖性的通用框架。
中文摘要 AI 辅助
我们研究了布兰斯-迪克理论中约束哈密顿系统关系去参数化前后约旦框架和爱因斯坦框架的等价性。尽管这两种共形表述在协变层面等价且通过标准ADM相空间上的正则变换相关，但相对于内部时钟约化后的等价性并不平凡。以布兰斯-迪克标量作为关系时钟，我们表明约化哈密顿量之间的明显差异并不表明两个框架在物理上不等价，而是源于约化前时钟扇区的规范嵌入不匹配。虽然标量配置变量在共形变换下保持不变，但其共轭动量因涉及引力动量迹的贡献而发生偏移。因此，关系动力学并非仅由时钟变量\(T\)决定，而是由完整的规范时钟对\((T,P_T)\)决定。我们构建了一个适应框架的规范图表，其中时钟扇区在去参数化之前被一致地变换。由此得到的约化哈密顿量与爱因斯坦框架的一致，恢复了约化关系动力学的等价性。我们的结果将规范时钟扇区确定为布兰斯-迪克理论中支配关系演化的基本结构，并为理解标量-张量引力中的框架依赖性和约化相空间量子化提供了一个通用框架。
英文摘要
We investigate the equivalence of the Jordan and Einstein frames of the Brans--Dicke theory before and after relational deparametrization of the constrained Hamiltonian system. Although the two conformal formulations are equivalent at the covariant level and related by a canonical transformation on the standard ADM phase space, their equivalence after reduction with respect to an internal clock is nontrivial. Using the Brans--Dicke scalar as a relational clock, we show that the apparent discrepancy between the reduced Hamiltonians does not indicate a physical inequivalence of the two frames. Instead, it arises from a mismatch in the canonical embedding of the clock sector prior to reduction. While the scalar configuration variable is preserved under the conformal transformation, its conjugate momentum is shifted by a contribution involving the gravitational momentum trace. Consequently, relational dynamics are determined not by the clock variable $T$ alone, but by the complete canonical clock pair $(T,P_T)$. We construct a frame-adapted canonical chart in which the clock sector is consistently transformed prior to deparameterization. The resulting reduced Hamiltonian coincides with that of the Einstein frame, restoring the equivalence of the reduced relational dynamics. Our results identify the canonical clock sector as the fundamental structure governing relational evolution in Brans--Dicke theory and provide a general framework for understanding frame dependence in scalar--tensor gravity and reduced phase-space quantization.
