On the relativistic micro-canonical ensemble and relativistic kinetic theory for N relativistic particles in inertial and non-inertial rest frames

Abstract

A new formulation of relativistic classical mechanics allows a reconsideration of old unsolved problems in relativistic kinetic theory and in relativistic statistical mechanics. In particular a definition of the relativistic micro-canonical partition function is given strictly in terms of the Poincaré generators of an interacting N -particle system both in the inertial and non-inertial rest frames. The non-relativistic limit allows a definition of both the inertial and non-inertial micro-canonical ensemble in terms of the Galilei generators.a The same happens in the parametrized Galilei theories defined in [11]. This allows us to define non-relativistic quantum mechanics in Galilean non-inertial frames. 1550049-2On the relativistic micro-canonical ensemble b With particle systems this can be done only in rotating systems till now [10]. Instead the problem is completely open for the massive Klein-Gordon field due to the no-go theorem of [12] forbidding unitary evolution in non-inertial frames. c Even if non-physical we must give the Cauchy data on a space-like 3-space, to be able to use the existence and unicity theorem for the solution of partial differential equations to predict the future. The subsequent evolution is consistent with special relativity even in the case of actionat-a-distance interactions.

From Relativistic Mechanics towards Relativistic Statistical Mechanics

“…ρ r (τ), while the standard [22][23][24][25]27] and Section IIB of [35] for the definition of the relative variables of N free particles.…”

“…In the two-body model of [24] and in the N-body model of Section 4 of [35], the invariant mass has the form…”

“…See Section 4 of [35] for the Poincaré generators of an N-body system with action-at-a-distance interaction and Appendix A of [35] for the case of N charged particles plus Coulomb and Darwin potentials [27,28].…”

“…This approach allowed one for the first time to define the relativistic micro-canonical ensemble in the Hamiltonian framework for systems of N particles interacting through either short-or long-range forces both in inertial frames and in a special family of non-inertial ones (the non-inertial rest frames) of Minkowski space-time [34] (see [35] for an extended version with extra results, which will be used in this paper) and a Lorentz scalar micro-canonical temperature T (mc) (see Endnote [36]-which refers to References [37][38][39][40][41][42]) extending the non-relativistic results of [43] for the extended distribution function of the micro-canonical ensemble (see Endnote [44]-which refers to Reference [45]) in the presence of long-range forces (see Endnote [46]-which refers to References [47][48][49][50]). When the forces in inertial frames are short range and the thermodynamical limit (N, V → ∞ with N/V = const.)…”

From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect

“…Moreover this framework allows us to define the inertial and non-inertial rest frames of the isolated systems, where to develop the rest-frame instant form of the dynamics and to build the explicit form of the Lorentz boosts for interacting systems. This makes possible to study the problem of the relativistic center of mass [26], relativistic bound states [27,28,29], relativistic kinetic theory and relativistic micro-canonical ensemble [30] and various other systems [31,32]. Moreover a Wigner-covariant relativistic quantum mechanics [33], with a solution of all the known problems introduced by SR, has been developed after some preliminary work done in Ref.…”