Poincare invariant Hamiltonian dynamics: Modelling multi-hadronic interactions in a phase space approach

Stimulated by experimental progress in high energy physics and astrophysics, the unification of relativistic and stochastic concepts has re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description of relativistic diffusion processes. After a brief historical overview, we will summarize basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are followed by a detailed discussion of relativistic Langevin equations in phase space. We address the choice of time parameters, discretization rules, relativistic fluctuation-dissipation theorems, and Lorentz transformations of stochastic differential equations. The general theory is illustrated through analytical and numerical results for the diffusion of free relativistic Brownian particles. Subsequently, we discuss how Langevin-type equations can be obtained as approximations to microscopic models. The final part of the article is dedicated to relativistic diffusion processes in Minkowski spacetime. Due to the finiteness of velocities in relativity, nontrivial relativistic Markov processes in spacetime do not exist; i.e., relativistic generalizations of the nonrelativistic diffusion equation and its Gaussian solutions must necessarily be non-Markovian. We compare different proposals that were made in the literature and discuss their respective benefits and drawbacks. The review concludes with a summary of open questions, which may serve as a starting point for future investigations and extensions of the theory.

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Relativistic Brownian Motion

Relativistic Brownian motion

Starting from the standard one‐time dynamics of n nonrelativistic particles, the n‐time equations of motion are inferred, and a variational principle is formulated. A suitable generalization of the classical Lie–Konig theorem is demonstrated, which allows the determination of all the associated presymplectic structures. The conditions under which the action of an invariance group is canonical are studied, and a corresponding Noether theorem is deduced. A formulation of the theory in terms of n first‐class constraints is recovered by means of coisotropic imbeddings. The proposed approach also provides for a better understanding of the relativistic particle dynamics, since it shows that the different roles of the physical positions and the canonical variables is not peculiar to special relativity, but rather to any n‐time approach: indeed a nonrelativistic no‐interaction theorem is deduced.

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On the many‐time formulation of classical particle dynamics

Relativistic Brownian Motion and Diffusion Processes

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World-line condition and the noninteraction theorem.

A generalization of the predictive relativistic mechanics is studied where the initial conditions are taken on a general hypersurface of M4. The induced realizations of the Poincare group are obtained. The same procedure is used for the Galileo group. Noninteraction theorems are derived for both groups.

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A. Montoto

Papers

World-line condition and the noninteraction theorem.

Realization of Poincaré group induced by a second‐order ordinary differential system. Noninteraction theorem