Summary
An explicit construction is given of some classes of Hamiltonians invariant under the Poincaré group and describing the evolution of relativistic dynamical systems composed of a single free point, of a point in external fields or of two interacting points. The no-interaction theorem is circumvented by making a clear distinction between evolution parameter and time co-ordinates of each point. The way to introduce the interaction potentials is suggested by the forms of the Hamiltonians of a single point in the different external fields. The possibility of completely using the symplectic structure of the Hamiltonian mechanics allows a classification of the interactions according to which characteristics of the Poincaré groups associated to each particle are conserved by the interaction itself. Various models are thus obtained including, as particular cases, interesting results as the «relativistic rigid rotator» and some generalizations of the classical mechanics with a correct Galileian limit. The equations of motion are analysed in the light of some requirements of physical coherence. Finally the quantization of the most meaningful models is discussed.
Riassunto
Si costruiscono esplicitamente alcune classi di hamiltoniane invarianti di Poincaré che descrivono l'evoluzione di sistemi dinamici relativistici composti da un solo punto sia libero che in campi esterni oppure da due punti interagenti. Il teorema di non interazione è evitato distinguendo accuratamente tra parametro di evoluzione e coordinate temporali dei punti. Il modo di introdurre i potenziali di interazione tra i due punti è suggerito dalla forma delle hamiltoniane di un singolo punto nei diversi campi esterni. La possibilità di usare pienamente la struttura simplettica della meccanica hamiltoniana permette di classificare le interazioni a seconda delle caratteristiche dei gruppi di Poincaré associati ad ogni particella che sono conservate dall'interazione stessa. Si ottengono così vari modelli che contemplano, come casi particolari, risultati interessanti quali il «rotatore rigido relativistico» ed alcune generalizzazioni della meccanica classica con un corretto limite galileiano. Le equazioni del moto sono analizzate alla luce di alcune richieste di coerenza fisica. Si discute infine la quantizzazione dei modelli più significativi.
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Giachetti, R., Sorace, E. Relativistic two-body interactions: a Hamiltonian formulation. Nuov Cim B 56, 263–301 (1980). https://doi.org/10.1007/BF02729264
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DOI: https://doi.org/10.1007/BF02729264