Summary
We give renormalization group representations for the propagator and four-point vertex function of thegϕ 4-model. We state in a precise way the properties that are necessary for the existence of these representations in a global nonperturbative sense. Special attention is given to the invertibility postulate for the coupling constant which is explicitly discussed. We remark that in fact this property, as well as the renormalization group representations, just expresses the possibility of changing the parameters used to describe the theory (herem 2 andg). The relation to the Callan-Symanzik equation is exhibited in view of a global discussion of the validity of this equation. An important case in which the global functionβ(g) vanishes identically is pointed out.
Riassunto
Si espongono le rappresentazioni del gruppo di rinormalizzazione per il propagatore e la funzione di vertice a 4 punti del modellogϕ 4. Si stabiliscono in modo preciso le proprietà necessarie per l’esistenza di queste rappresentazioni in senso globale non perturbativo. Si presta particolare attenzione al postulato di invertibilità per la costante di accoppiamento, che si discute esplicitamente. Si sottolinea che nei fatti questa proprietà, così come le rappresentazioni del gruppo di rinormalizzazione, esprime proprio la possibilità di cambiare i parametri che si usano per descrivere la teoria (quim 2 eg). Per una discussione globale della validità di questa equazione, si mette in evidenza il rapporto con l’equazione di Callan-Symanzik. Si espone un caso importante in cui la funzione globaleβ(g) si annulla identicamente.
Реэюме
Мы приводим представления группы перенормировки для пропагатора и четырехточечной верщинной функции вgϕ 4 модели. Мы формулируем свойства, которые необходимы для сушествования зтих представлений в глобальном непертурбационном смысле. Особое внимание уделяется постулату обратимости для константы свяэи, который обьуждается в явной форме. Мы отмечаем, что зто свойство, а также представления группы перенормировки, непосредственно выражают веэможность иэменения параметров, испольэуемых для описания теории (в рас-сматриваемом случаеm 2 иg). Устанавливается свяэь с уравнением ЧелленаСиманэика при обсуждении справедливости зтого уравнения. Отмечается важный случай, в котором функцияβ(g) тождественно обрашается в нуль.
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Tirapegui, E. Renormalization group and thegϕ 4-model. Nuov Cim A 25, 75–90 (1975). https://doi.org/10.1007/BF02735611
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DOI: https://doi.org/10.1007/BF02735611