Summary
We construct the asymptotic expansions in powers of the coupling constant λ for the asymptotic fields and the scattering operatorS for self-coupled boson fields with space cut-off polynomial interaction in two space-time dimensions. These asymptotic expansions are then used to prove thatS*S=SS*=1 in the sense of asymptotic power series inλ on a dense set of states. The results apply also, under the additional assumption of an ultraviolet cut-off, to large classes of boson-boson, fermion-boson and fermion-fermion interactions as well as to boson nonpolynomial interactions (in all space-time dimensions).
Riassunto
Si costruiscono gli sviluppi asintotici, in serie di potenze della costante di accoppiamentoλ, sia per i campi asintotici che per l’operatore di diffusioneS di campi bosonici interagenti mediante autoaccoppiamenti polinomiali con taglio spaziale, in due dimensioni spazio-temporali. Questi sviluppi asintotici vengono poi adoperati per dimostrare le relazioniS*S=SS*=1, intese nel senso delle serie asintotiche inλ e su un dominio denso nell’insieme di tutti gli stati. Gli stessi risultati valgono anche, con l’ulteriore ipotesi di un taglio ultravioletto, per un’ampia classe di interazioni polinomiali tra soli bosoni, tra bosoni e fermioni e tra soli fermioni, come pure per interazioni bosoniche non polinomiali (in ogni dimensione di spazio-tempo).
Реэюме
Мы конструируем асимптотические раэложения по степеням константы свяэиλ для асимптотических полей и оператора рассеянияS для боэонных полей с пространственным обреэанным полиномиальным вэаимодействием в дьух прост-ранственно-вре менных иэмерениях. Затем зти асимптотические раэложения испольэу-ются для докаэательства того, чтоS*S=SS*=1 в смысле асимптотического степенного ряда поλ, на плотной системе состояний. Испояьэуя дополнительное предположение об ультрафиолетовом обреэании, полученные реэультаты применя-ются также к больщим классам боэон-боэонных, фермион-боэонных и фермион-фермионных вэаимодействий, а также к боэонным неполиномиальным вэаимо-действиям (во всех пространственно-вре менных иэмерениях).
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References
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S. Albeverio:Scattering theory for some models of quantum fields, I, Princeton University preprint (1971) (to appear inJourn. Math. Phys.); II,Helv. Phys. Acta (Fierz Festschrift),45, 303 (1972).
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References for models somewhat inbetween the two mentioned classes, likee.g. external-field models and Lee-type models are mentionede.g. in (1b).
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Related results for the special caseP(ϕ)=ϕ 4 have been obtained also in (9).
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Such asymptotic expansions have been derived in (10) for space and ultraviolet cut-off relativistic fermion interactions. The case of Nelson’s type models is treated in (1b) and the case of nonpolynomial boson models in (11c).
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For the proof that the bottom of the spectrum ofH is a simple, isolated eigenvalue see,e.g., (7).
See,e.g., (7).
For definitions, see,e.g., (7,13).
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It is even known (14) that the Rayleigh-Schrödinger series forE is an asymptotic power series inλ, uniquely Borel summable to its sumE. Also the asymptotic expansion forΩ is known (14). This could also be inserted forΩ in the expression (3.22) ofR N+1 (Sa*(h 1) ...a*(h m))Ω.
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K. Hepp:Renormalization theory, inStatistical Mechanics and Quantum Field Theory, Les Houches Summer School, 1970, edited byC. De Witt andR. Stora (New York, 1971).
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Albeverio, S., Høegh-Krohn, R. Asymptotic series for the scattering operator and asymptotic unitarity of the space cut-off interactions. Nuov Cim A 18, 285–307 (1973). https://doi.org/10.1007/BF02722829
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DOI: https://doi.org/10.1007/BF02722829