Soluzioni elementari causali dell'operatore di Klein—Gordon iterato

Trione, Susana Elena

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 52 (1972) no. 5, pp. 607-610

Résumé

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 $n$.