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Causality and time order -- relativistic and probabilistic aspects

We investigate temporal and causal threads in the fabric of contemporary physical theories with an emphasis on empirical and operationalistic aspects. Building on the axiomatization of general relativity proposed by J. Ehlers, F. Pirani and A. Schild and the global space-time structure elaborated by R. Penrose, S.W. Hawking, B. Carter and others, we argue that the current way of doing relativistic physics presupposes treating time and causality as primitive concepts, neither of them being `more primitive' than the other. The decision regarding which concepts to assume as primitive and which statements to regard as axioms depends on the choice of the angle at which we contemplate the whole. This standard approach is based on the presupposition that the concept of a point-like particle is a viable approximation. However, this assumption is not supported by a realistic approach to doing physics and, in particular, by quantum theory. We remove this assumption by analysing the recent works by M. Eckstein and T. Miller. They consider the space $P(M)$ of probability measures on space-time $M$ such that, for an element $μ\in P(M)$, the number $μ(K)$ specifies the probability of the occurrence of some event associated with the space-time region $K$ and the measure $μ$. In this way, $M$ is not to be regarded as a collection of space-time events, but rather as a support for corresponding probability measures. As shown by Eckstein and Miller, the space $P(M)$ inherits the causal order from the underlying space-time and facilitates a rigorous notion of a `causal evolution of probability measures'. We look at the deductive chains creating temporal and causal structures analysed in these works, in order to highlight their operational (or quasi-operational) aspect. This is impossible without taking into account the relative frequencies and correlations observed in relevant experiments.