Summary
A recently proposed algebraic model for relativistic hadrons that contains dilatations in an interesting substructure is realized in terms of differential operators acting on a bilocal field. Wave equations are found for both 0 and 1/2 spin. It is shown that, in spite of dilatation invariance, particles can have nonvanishing rest mass. This result is a consequence of having multicomponent wave functions, where the components belong to different values of the dilatation quantum number. A canonical formalism is developed and the currents, related to various symmetries, are computed. It is possible to represent the currents of interest as various first moments of two energy-momentum tensors.
Riassunto
Si realizza, in termini di operatori differenziali agenti su un campo bilocale, un modello algebrico proposto di recente per gli adroni relativistici, il quale contiene dilatazioni in una interessante sottostruttura. Si trovano le equazioni d’onda sia per spin 0 che 1/2. Si fa vedere che, malgrado l’invarianza di dilatazione, le particelle possono avere una massa a riposo che non tende a zero. Questo risultato è una conseguenza della presenza di funzioni d’onda a più componenti, in cui le componenti appartengono a valori diversi del numero quantico di dilatazione. Si sviluppa un formalismo canonico e si calcolano le correnti relative alle varie simmetrie. Si possono rappresentare le correnti che interessano come vari momenti primi di due tensori energia-impulso.
Реэюме
Недавно предложенная алгебраическая модель для релятивистских адронов, которая содержит расщирения представляюшей интерес субструктуры, реалиэуется в зтой работе в терминах дифференциальных операторов, действуюших на билокальное поле. Получаются волновые уравнения для 0 и 1/2 спинов. Покаэы-вается, что, несмотря на инвариантность относительно расщирения, частицы могут иметь ненулевую массу покоя. Этот реэультат является следствием многокомпо-нентных волновых функций, где раэличные компоненты принадлежат раэличным эначениям квантового числа расщирения. Раэвивается канонический формалиэм и вычисляются токи, свяэанные с раэличными симметриями. Отмечается воэможность представить рассматриваемые токи как раэличные первые моменты двух тенэоров знергии-импульса.
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Literatur
P. Roman, J. J. Aghassi andP. L. Huddleston:Journ. Math. Phys.,13, 1852 (1972).
A similar picture of hadronic matter was first suggested by Japanese physicists, cf.S. Ishida:Progr. Theor. Phys.,46, 1570, 1905 (1971), and subsequent papers in the same journal. An algebraic model with bilocal co-ordinates has been used recently byF. Gürsey andS. Orfanidis:Nuovo Cimento,11 A, 225 (1972). An interesting connection between bilocal wave equations and the chain model has been studied, for example, byT. Takabayashi:Progr. Theor. Phys.,48, 1718 (1972).
In ref. (1) this was called Class III. It is characterized by the facts thatp≢0,π≢0 and thatp andπ are not parallel.
There is an oversight in ref. (1) where we wrote ℛ2 ≠ 0. We realized our mistake later when, using the method of induced representations, we explicitly constructed the Class III representations.
V. Bargmann:Ann. of Math.,48, 568 (1947), see pp. 629, 630.
V. Bargmann andE. P. Wigner:Proc. Nat. Acad. Sci.,34, 211 (1948).
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On leave of absence from the University of Madrid, sponsored by G.I.F.T.
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Roman, P., Lorente, M. & Huddleston, P.L. A bilocal scale-invariant field theory for massive particles. Nuov Cim A 21, 592–610 (1974). https://doi.org/10.1007/BF02731359
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DOI: https://doi.org/10.1007/BF02731359