Tomasz Miller - Causal evolution of measures and continuity equation

This talk was part of the Workshop on "Non-regular Spacetime Geometry" held at the ESI March 13 to 24, 2023.

Let M be a globally hyperbolic spacetime and T be a fixed Cauchy temporal function providing the notion of a global time. Consider a 'time-evolving probability measure', i.e. a one-parameter family of probability measures {μ_t} with each μ_t supported on its own Cauchy hypersurface T = t. There is a natural way of defining what it means for such an evolution to be causal, which employs the extension of the standard causal relation J+ onto the space P(M) of probability measures on M. Do there exist, however, some equivalent, manifestly covariant descriptions of causal time-evolution of measures, which do not require fixing the temporal function T in the first place? The answer is positive and I shall present two such alternative descriptions. The first one involves a probability measure on the space of causal curves endowed with a suitable topology. The second one employs a causal L2_loc-vector field, which is in a sense ‘tangent’ to the probability flow. The latter can be regarded as a 4-velocity field, with which μ_t satisfies the continuity equation in the distributional sense. The continuity equation thus rigorously encapsulates the causality of the time-evolution of measures. I will present how the three above descriptions are related with each other and discuss how they transform under the change of T. Finally, I will mention how the developed formalism extends to the setting of the N-particle causality theory.

Видео Tomasz Miller - Causal evolution of measures and continuity equation канала Erwin Schrödinger International Institute for Mathematics and Physics (ESI)