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.

arXiv comment33 pages

References in corpus (27)

• Quantum Information and Relativity Theory
• Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes
• Photon wave functions, wave-packet quantization of light, and coherence theory
• Recent developments in bimetric theory
• Reconstruction of the early Universe as a convex optimization problem
• A reconstruction of the initial conditions of the Universe by optimal mass transportation
• Photon wave function
• The causal hierarchy of spacetimes
• Two-photon wave mechanics
• A Causal Order for Spacetimes with $C^0$ Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves
• An algebraic formulation of causality for noncommutative geometry
• An optimal transport formulation of the Einstein equations of general relativity
• Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
• Finsler metrics and relativistic spacetimes
• K-causality coincides with stable causality
• A noncommutative view on topology and order
• Causality for nonlocal phenomena
• Time functions as utilities
• Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes
• Causal evolution of wave packets
• Polish spaces of causal curves
• Theory of optimal transport for Lorentzian cost functions
• Operational causality in spacetime
• Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity
• Optimal transport by omni-potential flow and cosmological reconstruction
• Time functions and $K$-causality between measures
• Solutions of the multiconfiguration Dirac-Fock equations

Cited by in corpus (5)

• Quantum particle localization observables on Cauchy surfaces of Minkowski spacetime and their causal properties
• The Differentiation Lemma and the Reynolds Transport Theorem for Submanifolds with Corners
• Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime
• Does Quantum Information Require Additional Structure?
• Causal evolution of probability measures and continuity equation