概率测度的因果演化与连续性方程
Causal evolution of probability measures and continuity equation
AI总结：
本文研究全局双曲时空中守恒非局域量的因果演化，证明其可用测度上的因果优先关系、因果曲线空间上的概率测度以及满足分布意义下连续性方程的因果向量场三种等价方式刻画，并揭示其广义协变性。
摘要：
我们研究全局双曲时空 $\mathcal{M}$ 中守恒非局域物理量的因果时间演化概念。这里“全局时间”的角色由一个选定的柯西时间函数 $\mathcal{T}$ 来扮演，而非局域量的瞬时构型则由支撑在相应时间切片 $\mathcal{T}^{-1}(t)$ 上的概率测度 $μ_t$ 来建模。我们证明，这种演化的因果性可以用三种等价方式表达：（i）通过扩展到概率测度上的因果优先关系 $\preceq$，（ii）借助未来定向连续因果曲线空间上赋予紧开拓扑的一个概率测度 $σ$，以及（iii）通过一个具有 $L^\infty_{\textrm{loc}}$-正则性的因果向量场 $X$，使得映射 $t \mapsto μ_t$ 在分布意义下满足连续性方程。在证明过程中，我们发现紧开拓扑对因果曲线的微分性质是敏感的，它等于由一个合适的 $H^1_{\textrm{loc}}$-Sobolev 空间诱导出的拓扑。这使我们能够将 $X$ 构造为在某种意义下“切于” $σ$ 的向量场。此外，我们讨论了描述（i）-（iii）的广义协变性，揭示出它们背后与观测者无关的几何概念。
英文摘要：
We study the notion of a causal time-evolution of a conserved nonlocal physical quantity in a globally hyperbolic spacetime $\mathcal{M}$. The role of the `global time' is played by a chosen Cauchy temporal function $\mathcal{T}$, whereas the instantaneous configurations of the nonlocal quantity are modeled by probability measures $μ_t$ supported on the corresponding time slices $\mathcal{T}^{-1}(t)$. We show that the causality of such an evolution can be expressed in three equivalent ways: (i) via the causal precedence relation $\preceq$ extended to probability measures, (ii) with the help of a probability measure $σ$ on the space of future-directed continuous causal curves endowed with the compact-open topology and (iii) through a causal vector field $X$ of $L^\infty_{\textrm{loc}}$-regularity, with which the map $t \mapsto μ_t$ satisfies the continuity equation in the distributional sense. In the course of the proof we find that the compact-open topology is sensitive to the differential properties of the causal curves, being equal to the topology induced from a suitable $H^1_{\textrm{loc}}$-Sobolev space. This enables us to construct $X$ as a vector field in a sense `tangent' to $σ$. In addition, we discuss the general covariance of descriptions (i)-(iii), unraveling the geometrical, observer-independent notions behind them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
