Generally covariant N-particle dynamics

• Published in: Journal of geometry and physics 2021-02, Vol.160, p.103990, Article 103990
• Main Authors: Miller, Tomasz, Eckstein, Michał, Horodecki, Paweł, Horodecki, Ryszard
• Format: Article
• Language: English
• Subjects: General covariance; Multiparticle quantum systems; Optimal transport; Relativistic causality
• ISSN: 0393-0440; 1879-1662
• DOI: 10.1016/j.geomphys.2020.103990

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 towards modelling N-particle dynamics can be conciliated within a generally covariant framework. To this end we construct an ‘N-particle configuration spacetime’ M(N), starting from a globally hyperbolic spacetime 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 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 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 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.