Generally covariant $N$-particle dynamics

• URL: http://arxiv.org/abs/2004.07015v2
• Date: Fri, 6 Nov 2020 13:41:33 GMT
• Title: Generally covariant $N$-particle dynamics
• Authors: Tomasz Miller, Micha{\l} Eckstein, Pawe{\l} Horodecki, Ryszard Horodecki
• Abstract summary: A simultaneous description of the dynamics of multiple particles requires a configuration space approach with an external time parameter. Here we show, however, that the two attitudes toward modelling $N$-particle dynamics can be conciliated within a generally covariant framework. The dynamics of multi-particle systems is modelled at the level of Borel probability measures over $mathcalM_scriptscriptstyle (N)$ with the help of the global time parameter.
• Score: 0.10427337206896375
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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' $\mathcal{M}_{\scriptscriptstyle (N)}$, 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 $\mathcal{M}_{\scriptscriptstyle (N)}$ 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 $\mathcal{M}_{\scriptscriptstyle (N)}$, 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 Schr\"odinger 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.

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