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 Serie 8 52 (1972), fasc. n.5, p. 607-610, (Italian)
pdf (361 Kb), djvu (360 Kb). | MR 0390762 | Zbl 0282.46037
Sunto
The distributions $G_{\alpha} \{ P \pm i0, m, n \}$ (formula (1)) share many properties with the Bessel kernel, of which they are "causal" ("anticausal") analogues. In particular (Theorem 1), $G_{\alpha} \ast G_{-2k} = G_{\alpha-2k}$, $\Lambda \alpha \in \mathbf{C}$, $\Lambda k = 0,1,2,\cdots$. The essential tool for the proof of this formula is the multiplication formula (4), namely $$\{ m^{2}+Q(y) \mp i0 \}^{\alpha} \{ m^{2}+ Q(y) \mp i0 \}^{\beta} = \{ m^{2}+ Q(y) \mp i0 \}^{\alpha + \beta},$$ which is valid for every $\alpha,\beta \in \mathbf{C}$. It follows from Theorem (1) that $G_{2k} \{ P \pm i0,m,n \}$, is, for $n \ge 2$, $k = 1, 2, \cdots$, 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 $G_{2} \{ P \pm i0,m,4 \}$ embodies a useful expression of the causal propagator of Feynman. It may be observed that the elementary solutions $G_{2k} \{ P \pm i0,m,n \}$ have the same form for every $n \ge 2$. This does not happen for other elementary solutions, whose form depends essentially on the parity of $n$.
Referenze Bibliografiche
[1]
BATEMAN MANUSCRIPT,
Higher transcendental functions, vol.
II.
Mc Graw Hill, New York,
1953. |
MR 58756[2]
I. M. GELFAND e
G. E. SHILOV,
Generalized functions, vol.
I.
Academic Press, New York,
1964. |
MR 166596[7]
R. P. FEYNMAN,
The theory of positrons, «
Phys. Rev.»,
16, 749-759 (
1949). |
Zbl 0037.12406[8]
J. J. BOWMAN e
J. D. HARRIS,
Green's distributions and the Cauchy problem for the iterated Klein-Gordon operator, «
J. Mathematical Physics»,
3, 396-404 (
1962). |
fulltext (doi) |
MR 149120 |
Zbl 0114.29902