Soluzioni elementari causali dell'operatore di Klein—Gordon iterato

Susana Elena Trione

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti (1972)

  • Volume: 52, Issue: 5, page 607-610
  • ISSN: 0392-7881

Abstract

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The distributions G α ⁢ { P ± i ⁢ 0 , m , n } (formula (1)) share many properties with the Bessel kernel, of which they are "causal" ("anticausal") analogues. In particular (Theorem 1), G α ∗ G - 2 ⁢ k = G α - 2 ⁢ k , Λ ⁢ α ∈ 𝐂 , Λ ⁢ k = 0 , 1 , 2 , ⋯ . The essential tool for the proof of this formula is the multiplication formula (4), namely { m 2 + Q ⁢ ( y ) ∓ i ⁢ 0 } α ⁢ { m 2 + Q ⁢ ( y ) ∓ i ⁢ 0 } β = { m 2 + Q ⁢ ( y ) ∓ i ⁢ 0 } α + β , which is valid for every α , β ∈ 𝐂 . It follows from Theorem (1) that G 2 ⁢ k ⁢ { P ± i ⁢ 0 , m , n } , is, for n ≥ 2 , k = 1 , 2 , ⋯ , a causal (anticausal) elementary solution of the n-dimensional Klein-Gordon operator, iterated k times (Theorem 2). The particular case n = 4 , k = 1 is important in the quantum theory of fields, since G 2 ⁢ { P ± i ⁢ 0 , m , 4 } embodies a useful expression of the causal propagator of Feynman. It may be observed that the elementary solutions G 2 ⁢ k ⁢ { P ± i ⁢ 0 , m , n } have the same form for every n ≥ 2 . This does not happen for other elementary solutions, whose form depends essentially on the parity of n .

How to cite

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Trione, Susana Elena. "Soluzioni elementari causali dell'operatore di Klein—Gordon iterato." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 52.5 (1972): 607-610. <http://eudml.org/doc/295664>.

@article{Trione1972,
author = {Trione, Susana Elena},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
language = {ita},
month = {5},
number = {5},
pages = {607-610},
publisher = {Accademia Nazionale dei Lincei},
title = {Soluzioni elementari causali dell'operatore di Klein—Gordon iterato},
url = {http://eudml.org/doc/295664},
volume = {52},
year = {1972},
}

TY - JOUR
AU - Trione, Susana Elena
TI - Soluzioni elementari causali dell'operatore di Klein—Gordon iterato
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1972/5//
PB - Accademia Nazionale dei Lincei
VL - 52
IS - 5
SP - 607
EP - 610
LA - ita
UR - http://eudml.org/doc/295664
ER -

References

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