Generally covariant $N$-particle dynamics

Abstract

A simultaneous description of the dynamics of multiple particles requires a configuration space approach with an external time parameter. This is in stark contrast with the relativistic paradigm, where time is but a coordinate chosen by an observer. Here we show, however, that the two attitudes toward modelling $N$-particle dynamics can be conciliated within a generally covariant framework. To this end we construct an '$N$-particle configuration spacetime' ${\mathcal{M}}_{(N)}$, starting from a globally hyperbolic spacetime $\mathcal{M}$ with a chosen smooth splitting into time and space components. The dynamics of multi-particle systems is modelled at the level of Borel probability measures over ${\mathcal{M}}_{(N)}$ with the help of the global time parameter. We prove that with any time-evolution of measures, which respects the $N$-particle causal structure of ${\mathcal{M}}_{(N)}$, one can associate a single measure on the Polish space of '$N$-particle wordlines'. The latter is a splitting-independent object, from which one can extract the evolution of measures for any other global observer on $\mathcal{M}$. An additional asset of the adopted measure-theoretic framework is the possibility to model the dynamics of indistinguishable entities, such as quantum particles. As an application we show that the multi-photon and multi-fermion Schrödinger equations, although explicitly dependent on the choice of an external time-parameter, are in fact fully compatible with the causal structure of the Minkowski spacetime.

33 pages