Summary
It is shown that generalized Bose-like and Fermi-like numbers, which are useful in formulating the problem of supersymmetry, can be treated in a unified manner to a considerable extent, and hence they are collectively called generalized numbers. In the present paper the generalized numbersx i (i=1, 2, …,n) are defined as those satisfying the commutation relations [x i ,x j ]−(ij)=0 together with some other constraints, where (ij)=+ or −. Elementary mathematical properties as well as possible physical applications are studied, with emphasis upon the connection with the problems of Fermi-Bose similarity and of supersymmetry.
Riassunto
Si mostra che i numeri generalizzati di tipo Bose e di tipo Fermi, utili nella formulazione del problema della supersimmetria, possono essere trattati fino a un considerevole limite in modo unificato, e quindi sono chiamati collettivamente numeri generalizzati. In questo lavoro i numeri generalizzatix i (i=1, 2, …,n) sono definiti come quelli che soddisfano le relazioni di commutazione [x i ,x j ]−(ij)=0 insieme ad alcuni altri vincoli, dove (ij)=+ o −. Si studiano le proprietè matematiche elementari cosí come possibili applicazioni fisiche, sottolineando la connessione con i problemi della somiglianza fra numeri di Bose e di Fermi e della supersimmetria.
Резюме
Показывается, что обобщенные Бозе-подобные и Ферми-подобные числа, которые являются полезными при формулировке проблеме суперсимметрии, могут рассматриваться единым образом и называются обобщенными числами. В этой работе обобщенные числаx i (i=1, 2, …,n) определяются, как числа, удовлетворяющие коммутационным соотношениям [x i ,x j ]−(ij)=0 вместе с другими ограничениями, причем (ij)=+ или −. Исследуются элементарные математические свойства, а также возможные физческие применения обращая особое внимание на связь с проблемами ферми-Бозе подобия и суперсимметрии.
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An erratum to this article is available at http://dx.doi.org/10.1007/BF02734675.
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Ohnuki, Y., Kamefuchi, S. Fermi-Bose similarity, supersymmetry and generalized numbers. Nuov Cim A 70, 435–459 (1982). https://doi.org/10.1007/BF02902265
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DOI: https://doi.org/10.1007/BF02902265