Summary
Quantized field theories of the second kind, the so-called non-renormalizable theories, are investigated. It has recently been shown that a number of those theories actually can be renormalized. We show that field theories of the second kind, pertaining to derivative couplings, possess non-local structure. The full recoil neutral PS-PV theory in its non-linear form is discussed. The non-localizability of both the unrenormalized and the renormalized theories is due to the fact that the infinite set of topologically independent direct interactions, which characterizes PS-PV theory, is equivalent to a new type of indirect interaction. The non-localizability manifests itself in an indeterminacy in space-time of the light-cone. This indeterminacy can be interpreted in terms of a statistical spread in space-time of the positions of two interacting point-nucleons over a space-time volume the size of which is determined by the coupling constant |g| of the linear version of PS-PV theory. The length |g| loses its meaning as a coupling in the non-linear formalism in favour of the nucleon mass and plays now the role of a fundamental structure constant. In particular, |g| determines the region in which the equivalence theorem and the 1/r 3-potential become invalid and repulsive-core potentials begin to act. In virtue of the oscillatory behaviour of the non-tempered momentum space operators in the high energy region, positive-definiteness conditions can be established only for regions |k 2v |≳|1/g 2| and |x 2v |≳|1/g 2|. The non-localizability is concentrated in these domains and |g| seems to play the role of the meson Compton wave length. A new interpretation of the non-linear PS-PV theory is suggested. It is conjectured that also β interactions give rise to non-local structures. Multiple propagators are introduced to render the formal coupling constant expansions in theories of the second kind mathematically meaningful without producing non-renormalizable infinities. Arguments are advanced against the mathematical consistency of the concept of creation (annihilation) and localizability of more than one point-particle at a single space-time point.
Riassunto
Si esaminano le teorie quantizzate di campo di seconda specie, le cosiddette teorie non rinormalizzabili. Si è dimostrato recentemente che alcune di queste teorie possono in realità essere rinormalizzate. Dimostriamo che le teorie di campo di seconda specie, appartenenti agli accoppiamenti per derivazione, hanno struttura non locale. Si discute nella sua forma non lineare la teoria PS-PV neutra del rinculo completo. La non localizzabilità delle teorie sia non rinormalizzate che rinormalizzate è dovuta al fatto che il gruppo infinito delle interazioni dirette topologicamente indipendenti che caratterizza la teoria PS-PV è equivalente a un nuovo tipo d’interazione diretta. La non localizzabilità si manifesta in un’indeterminazione del cono di luce nello spazio-tempo. Tale indeterminazione si può interpretare in termini di una diffusione statistica nello spaziotempo delle posizioni di due nucleoni puntiformi interagenti in un volume di spaziotempo la cui estensione è determinata dalla costante d’accoppiamento |g| della versione lineare della teoria PS-PV. La lunghezza |g| perde il suo significato di accoppiamento nel formalismo non lineare in favore della massa del nucleone, ed acquista ora l’ufficio di una costante fondamentale di struttura. In particolare |g| determina la regione in cui il teorema di equivalenza e il potenziale 1/r 3 perdono la loro validità e cominciano ad agire potenziali di core repulsivi. In virtù del comportamento oscillatorio nella regione di alta energia degli operatori dello spazio non temperato dei momenti si possono imporre condizioni di positività definita solo per regioni |k 2v | ≲ |1/g 2| e |x 2v | ≳ |g 2|. La non localizzabilità è concentrata in questi domini e |g| sembra acquistare l’ufficio della lunghezza d’onda Compton del mesone. Si suggerisce una nuova interpretazione della teoria PS-PV non lineare. Si congettura che anche le interazioni diano vita a strutture non locali. Si introducono propagatori multipli per rendere matematicamente significativi nelle teorie di seconda specie gli sviluppi formali delle costanti d’accoppiamento senza produrre grandezze infinite non rinormalizzabili. Si dannargomenti contro la fondatezza matematica del concetto di creazione (distruzione) e di localizzabilità di più di una particella puntiforme in un singolo punto dello spaziotempo.
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Güttinger, W. Non-local structure of field theories with non-renormalizable interaction. Nuovo Cim 10, 1–36 (1958). https://doi.org/10.1007/BF02859602
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DOI: https://doi.org/10.1007/BF02859602