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Il Nuovo Cimento A (1965-1970)

, Volume 64, Issue 3, pp 669–714 | Cite as

Classical and quantum fields in de Sitter space

  • G. Börner
  • H. P. Dürr
Article

Summary

In the (1+4) de Sitter space the equations for free fields of spin 0, spin 1/2 and spin 1 are derived as eigenvalue equations of the Casimir operators. Completeness relations for the solutions are given in the case of spin 0 and spin 1/2, and with these the different Green functions and commutation functions are obtained. It is possible to construct causal commutation functions only for a certain part of the spectrum. This gives rise to a spectrum condition, by which,e.g., the solution satisfying the conformal invariant equation for spin 0 is excluded from the physical state space. All functions are shown to permit an expansion in powers of the de Sitter curvature 1/R, where the first term is in each case the corresponding invariant function of Minkowski space.

Keywords

Green Function Minkowski Space Discrete Spectrum Casimir Operator Completeness Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Классические и квантовые поля в пространстве де Ситтера

Резюме

В пространстве де Ситтера (1+4) выводятся уравнения для свобддных полей спина 0, спина 1/2 и спина, 1, как уравнения для собственных значений операторов Казимира. Приводятся соотношения полноты для решений в случае спино 0 и спина 1/2, и с их помощью получаются различные функции Грина и коммутационные функции. Возможно сконструировать причинные коммутационные функции только для некоторой части спектра. Это дает спектральное условие, благодаря которому, например, решение, удовлетворяющее конформному инвариантному уравнению для спина 0, исключается из пространства физических состояний. Показывается, что все функции допускают разложения по степеням кривизны пространства де Ситтера 1/R, в которых первый член представляет в каждом случае соответствующую инвариантную функцию пространства Минковского.

Riassunto

Nello spazio di de Sitter (1+4) si ricavano le equazioni per campi liberi di spin 0, 1/2 e 1 come equazioni agli autovalori degli operatori di Casimir. Si forniscono le relazioni di completezza per le soluzioni nel caso di spin 0 e 1/2, e con queste si ottengono le differenti funzioni di Green e le funzioni di commutazione. È possibile costruire funzioni di commutazione causali solo per una certa parte dello spettro. Ciò dà origine ad una condizione spettrale per cui, per esempio, la soluzione soddisfacente l’equazione invariante conforme per spin 0 è esclusa dallo spazio degli stati fisici. Si dimostra che tutte le funzioni permettono uno sviluppo in potenze della curvatura di de Sitter 1/R, dove il primo termine è in ogni caso la corrispondente funzione invariante dello spazio di Minkowski.

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Copyright information

© Società Italiana di Fisica 1969

Authors and Affiliations

  • G. Börner
    • 1
  • H. P. Dürr
    • 1
  1. 1.Max-Planck-Institut für Physik und AstrophysikMunich

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