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Il Nuovo Cimento A (1965-1970)

, Volume 89, Issue 2, pp 105–125 | Cite as

A fibre bundle formulation of quantum geometry

  • E. Prugovečki
Article

Summary

Quantum geometries whose points are stochastic and serve as seats for quantum space-time excitons are formulated as fibre bundles over base spaces of mean values with a Minkowski or general relativistic structure. The fibres contain the proper wave functions of all exciton states in a given model. The notion of covariance and propagation in quantum space-times constituting such fibre bundles is investigated. Maxwell and Yang-Mills gauge degrees of freedom are introduced by appropriately enlarging the structure group, which in all cases contains phase-space representations of the Poincaré group corresponding to the exciton wave function sample space specific to a given model. It is shown that these formulations give rise in a natural manner to certain realizations of the relativistic canonical commutation relations in terms of covariant derivatives involving internal as well as external degrees of freedom of space-time excitons.

PACS. 12.90

Miscellaneous theoretical ideas and models 

Формулировка квантовой геометрии в виде семейства нитей

Резюме

Формулируются квантовые геомстрии, точки которых являются стохастическими и служат как положения для квантовых пространственно-временных экситонов, в виде семейства нитей на базовых пространствах для средних величин со структурой Минковского или более общей релятивистской структурой. Нити содержат собственные волновые функции для всех экситонных состояний в заданной модели. Исследуются понятия ковариантности и распространения в квантовых пространстве-времени, определяемых такими семействами нитей. Вводятся калибровочные степени свободы Максвелла и Янга-Миллса, посредством расширения структурной группы, которая во всех случаях содержит представления фазового пространства группы Пуанкаре, соответствующей экситонной волновой функции. Показывается, что эти фомулировки приводят естественным образом к некоторым реализациям канонических коммутационных соотношений в терминах ковариантных производных, включая внутренние и внешние степени свободы пространственновременных экситонов.

Riassunto

Si formulano geometrie quantiche, i cui punti sono stocastici e servono come siti degli eccitoni dello spazio-tempo quantistico, come fasci di fibre su spazi di base dei valori medi con struttura di Minkowski o relativistica generale. Le fibre contengono le funzioni d'onda proprie di tutti gli stati eccitonici in un dato modello. Si ricerca la nozione di covarianza e propagazione negli spazi-tempi quantici che costituiscono questi fasci di fibre. Si introducono gradi di libertà di gauge di Maxwell e Yang-Mills allargando opportunamente il gruppo di struttura, che in tutti i casi contiene le rappresentazioni dello spazio delle fasi del gruppo di Poincaré che corrisponde allo spazio campionario delle funzioni d'onda eccitoniche specifiche di un dato modello. Si mostra che queste formulazioni danno origine naturalmente a certe realizzazioni delle relazioni di commutazione canonica relativistiche in termini delle derivate covarianti che comprendono sia i gradi di libertà interni che quelli esterni degli eccitoni dello spazio-tempo.

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Copyright information

© Società Italiana di Fisica 1985

Authors and Affiliations

  • E. Prugovečki
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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