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 $\mu_t$ 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 $\sigma$ on the space of future-directed continuous causal curves endowed with the compact-open topology and (iii) through a causal vector field $X$ of $L^2_{\textrm{loc}}$-regularity, with which the map $t \mapsto \mu_t$ 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 a 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 $\sigma$. In addition, we discuss the general covariance of descriptions (i)-(iii), unraveling the geometrical, observer-independent notions behind them.

Tomasz Miller

Advances in Theoretical and Mathematical Physics

ADVANCES IN THEORETICAL AND MATHEMATICAL PHYSICS 是国际出版社的双月刊，发表关于理论物理和数学相互作用的所有领域的论文。

Advances in Theoretical and Mathematical Physics is a bimonthly journal published by an international publisher, focusing on research papers in all areas of the interaction between theoretical physics and mathematics.

《Advances in Theoretical and Mathematical Physics》是国际出版社出版的双月刊，专注于发表理论物理与数学交叉领域的研究论文。

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物理与天体物理	4
物理与天体物理, 物理数学物理	4
物理与天体物理, 物理粒子与场物理	4

Causal evolution of probability measures and continuity equation

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Journal Info

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PHYSICS, PARTICLES & FIELDS	4