Summary
In this paper transformation properties of Green’s functions are studied when, under rather general hypotheses, the local fields are subjected to finite transformations of a continuous group of soft broken symmetries. A perturbative series is found and its convergence conditions are discussed. Also the limiting case of spontaneously broken symmetries is considered to complete the picture. As an application transformation properties of Green’s functions under the dilatation group are considered. Finally a method is given to represent a general Green’s function as an expansion in series of dilatation-invariant terms; the related mathematical difficulties are also discussed.
Riassunto
In questo lavoro si studiano le proprietà di trasformazione delle funzioni di Green quando, entro ipotesi piuttosto generali, i campi locali sono soggetti a trasformazioni finite di un gruppo continuo di simmetrie rotte «mollemente». Si trova una serie perturbativa di cui si discutono le condizioni di convergenza. Per completare il quadro si tratta pure il caso delle rotture spontanee di simmetrie. Come applicazione si considerano le proprietà di trasformazione delle funzioni di Green rispetto al gruppo delle dilatazioni. Infine si dà un metodo per rappresentare una funzione di Green generale come sviluppo in serie di termini invarianti di scala; se ne discutono le difficoltà matematiche.
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References
H. Weyl:The Theory of Groups and Quantum Mechanics (New York).
M. Gell-Mann andY. Ne’eman:The Eightfold Way (New York, 1964);F. Y. Dyson:Symmetry Groups in Nuclear and Particle Physics (New York, 1965).
T. W. B. Kibble, D. Kastler andH.-P. Dürr inProceedings of the Rochester Conference (1967).
G. G. Amch:Algebraic Methods in Statistical Mechanics and Quantum Field Theory (New York, 1972);J. Dixmier:Les C *-algèbres et leurs représentation (Paris, 1969).
D. W. Robinson:Algebraic Aspects of Relativistic Quantum Field Theory, Proceedings of the Brandeis University of Summer School (New York, 1965), p. 389.
R. F. Streater andA. S. Wightman:P.C.T., Spin, Statistics, and All That (New York, 1964).
G. Mack andA. Salam:Ann. of Phys.,53, 174 (1969).
S. Fubini, G. Furlan andC. Rossetti:Nuovo Cimento,40 A, 1171 (1965).
L. P. Eisenhart:Continuous Groups of Transformations (New York).
H. J. Borchers, R. Hag andR. Schroer:Nuovo Cimento,29, 148 (1963).
R. F. Streater:Broken symmetries and the Goldstone theorem, inHigh-Energy Physics and Elementary Particles (Vienna, 1965).
J. Goldstone:Nuovo Cimento,19, 154 (1961).
S. Ciccariello, R. Gatto, G. Sartori andM. Tonin:Ann. of Phys.,65, 265 (1971).
B. W. Lee:Nucl. Phys.,9 B, 649 (1969).
G. S. Guralnick:Phys. Rev.,136, B 1404, B 1417 (1964).
D. Kastler:Proceedings of the Rochester Conference (New York, 1967).
C. G. Callan, S. Coleman andR. Jackiw:Ann. of Phys.,59, 42 (1970).
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Nobili, R. Properties of the Green’s functions under finite transformations of a soft-broken-symmetry group. Nuov Cim A 17, 171–188 (1973). https://doi.org/10.1007/BF02790293
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DOI: https://doi.org/10.1007/BF02790293