Summary
A class of currents not covariant under space-time translations is considered. The assumptions about the members of such class are deduced from some general properties of frequently used symmetry transformations,e.g. gauge transformations of second kind and dilatation transformations. In the framework of Wightman field theory Goldstone’s theorem is proved, without use of Lorentz covariance and positivity for the metric.
Riassunto
Si studia una classe di correnti che non sono covarianti per trasformazioni spaziotemporali. Le ipotesi sulle correnti appartenenti a tale classe sono dedotte da proprietà generali di trasformazioni di simmetria spesso usate in teoria dei campi, per esempio le trasformazioni di gauge di seconda specie e le dilatazioni. Si dimostra il teorema di Goldstone per tali correnti nello schema di una teoria di campo alla Wightman, Senza far uso della covarianza di Lorentz e della positività della metrica.
Реэюме
Рассматривается класс токов, не ковариантных относительно про-стр анственно-временных трансляций. Иэ некоторых обших свойств часто испольэуемых преобраэований симметри, т.е. калибровочных преобраэований второго рода и преобраэований расщирения, выводятся предположения относительно членов такого класса токов. В рамках теории поля Вайтмана докаэывается теорема Голдстоуна беэ испольэования Лорентц-ковариан тности и положительности для метрики.
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See ref. (14), Chap. VII, and ref. (15), Sect. 7.1.2.
It will be clear from eq. (17), to come, that the Goldstone theorem is still valid if II) is weakened in the form: It exists aΛ for which ∂μJμ[Λ](x)=0 andJμ[□Λ](x) generates an exact symmetry. This is the case, for instance, of the conformal transformations (seeH. Reeh:Lectures at the International Seminar on Statistical Mechanics and Field Theory, Haifa ref. (9)).
See ref. (15)), Sect.11.2.
See ref. (14), Chap. 4, Sect.5.
See ref. (14), Chap. 3, Sect.3 (Theorem XIII) and Chap. 7, Sect.6 (theorem XIII).
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See ref. (21), Sect.137.
See ref. (14), Chap. 1, Sect.2.
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Ferrari, R. On goldstone’s theorem for a class of currents not covariant under translations. Nuov Cim A 14, 386–402 (1973). https://doi.org/10.1007/BF02728960
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DOI: https://doi.org/10.1007/BF02728960