Summary
A consideration of the fundamental requirements of relativistic invariance for the global observables of relativistic quantum systems leads to the postulate that these observables should be treated as geometric objects on the manifold of spacelike hyperplanes in Minkowski space. The problem of determining the most general possible (in the sense of being consistent with the demands of relativistic covariance) coupling of the Poincaré Lie algebra with the Lie algebra of an « internal » symmetry is then dealt with in terms of connections (in the sense of differential geometry) on fields of Lie algebras over the manifold of hyperplanes. It is found that the possible couplings are considerably more restricted than one might have expected.
Riassunto
Considerando le esigenze fondamentali dell’invarianza relativistica per le osservabili globali dei sistemi quantici relativistici si giunge al postulato che queste osservabili dovrebbero essere trattate come oggetti geometrici nell’insieme degli iperpiani spaziali nello spazio di Minkowski. Si tratta successivamente il problema della determinazione dell’accoppiamento più generale possibile (nel senso della consistenza con le condizioni della covarianza relativistica) delle algebre di Lie di Poincaré con l’algebra di Lie di una simmetria « interna » in termini di connessioni (nel senso della geometria differenziale) su campi dell’algebra di Lie sull’insieme degli iperpiani. Si trova che gli accoppiamenti possibili sono più ristretti di quanto ci si potrebbe aspettare.
Реэюме
Рассмотрение фундаментальных требований релятивистской инвариантности для рассматриваемых в целом наблюдаемых релятивистских квантовых систем приводит к постулированию, что зти наблюдаемые должны рассматриваться, как геометрические общекты на множестве пространственно-п одобных гиперплоскостей в пространстве Минковского. Затем рассматривается проблема определения наиболее обшей воэможной (в смысле соответствия требованиям релятивистской ковариантности) свяэи Ли-алгебры Пуанкаре с Ли-алгеброй «внутренней» симметрии на основе соотнощений (в смысле дифференциальной геометрии) на поля Ли-алгебр на множестве гиперплоскостей. Окаэывается, что воэможные свяэи являются более ограниченными, чем ожидалось.
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References
For a comprehensive review of this work, seeG. C. Hegerfeldt andJ. Hennig:Fortschr. d. Phys.,16, 491 (1968).
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Like most of the terminology that we shall introduce, this is taken from differential geometry. For the analogous concepts in the differential geometry of affine connections see, for instanceR. L. Bishop andS. I. Goldberg:Tensor Analysis on Manifolds (New York, 1968);D. Laugwitz:Differential and Riemannian Geometry (New York, 1965);T. J. Willmore:An Introduction to Differential Geometry (Oxford, 1959).
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Yodzis, P. The geometry of broken symmetries. Nuovo Cimento B (1965-1970) 68, 153–164 (1970). https://doi.org/10.1007/BF02710410
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DOI: https://doi.org/10.1007/BF02710410