Summary
We are investigating a class of representations of the conformal group which are physically interesting because there exists a position-time 4-vector operator within them and, related to this, because they are used in field transformation laws. We show that in these representations the three Casimir operators of the group have a very simple interpretation; we discuss when these representations are irreducible and we show that this 4-position operator can be expressed in terms of the generators of the group.
Riassunto
Si stanno facendo ricerche su una classe di rappresentazioni del gruppo conforme, che sono fisicamente interessanti perché all’interno di esse esiste un operatore quadrivettoriale spazio-temporale e, in relazione con ciò, perché sono usate nelle leggi di trasformazione dei campi. Si mette in luce come in queste rappresentazioni i tre operatori di Casimir del gruppo abbiano un’interpretazione molto semplice; si discute quando queste rappresentazioni sono irriducibili; e si fa vedere che questo operatore quadriposizionale può essere espresso per mezzo dei generatori del gruppo.
Реэюме
Мы исследуем класс представлений конформной группы, которые представляют фиэический интерес, потому что сушествует оператор, определяюший четырех-вектор положения-времени, в зтих представлениях, и которые свяэаны с зтой группой, так как они испольэуются в эаконах преобраэования полей. Мы покаэываем, что в зтих представлениях три оператора Каэимира зтой группы имеют очень простую интерпретацию. Мы обсуждаем вопрос, когда зти представления являются неприводимыми. Мы покаэываем, что зтот оператор, определяюший четырех-вектор положения-времени, может быть выражен в терминах генераторов группы.
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References
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This ansatz is related to the most general transformation law under conformal transformations of fields obtained in ref. (5). The expressions obtained there appear to be slightly more general than the ansatz (4) because of the presence of an additional operatorx μ to be added to the expression (4d) forK μ . This additionalx μ , however, if unequal to zero, leads to a reducible representation of the conformal Lie algebra, at least for finite-dimensionalx μ . This is a consequence of Lemma 1 of Sect.4 in ref. (5). Therefore, we takex μ to be zero here.
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When writing up this paper, we noticed thatT. Yao:Journ. Math. Phys.,12, 315 (1971) has also calculated the Casimir operators ofSO 4,2 for certain representations of this group. His calculation was done in a basis whose elements correspond to a decomposition of the given representation into irreducible representations of an iso-Poincaré subgroup. This approach is different from ours, and, therefore, his results although similar in character cannot easily be compared to ours nor do they seem to lead to an immediate physical interpretation.
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See ref. (7).
For similar considerations, see the investigation of the question of self-adjointness of the operatori(d/dx) inN. L. Achieser andI. M. Glasmann:Theorie der linearen Operatoren in Hilbert-Raum, Sect.49 (Berlin, 1954).
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Research supported by the NRC.
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Lagu, S., Laue, H. The conformal group, its casimir operators, and a four-position operator. Nuov Cim A 20, 217–231 (1974). https://doi.org/10.1007/BF02727449
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DOI: https://doi.org/10.1007/BF02727449