International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 69, Pages 3775-3781
doi:10.1155/S0161171204406565
The second-order Klein-Gordon field equation
1Departamento de Matemática, Universidade Federal de Santa Maria, Santa Maria 97119-900, Rio Grande do Sul, Brazil
2Grupo de Física-Matemática, Faculdade de Ciências, Universidade de Lisboa, Lisboa 1649-003, Brazil
3Departamento de Matemática Aplicada, Universidade Estadual de Campinas, Campinas 13083-970, São Paulo, Brazil
Received 22 June 2004
Copyright © 2004 D. Gomes and E. Capelas De Oliveira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We introduce and discuss the generalized Klein-Gordon second-order
partial differential equation in the Robertson-Walker space-time,
using the Casimir second-order invariant operator written in
hyperspherical coordinates. The de Sitter and anti-de Sitter
space-times are recovered by means of a convenient choice of the
parameter associated to the space-time curvature. As an
application, we discuss a few properties of the solutions. We also
discuss the case where we have positive frequency exponentials and
the creation and annihilation operators of particles with known
quantum numbers. Finally, we recover the Minkowskian case, that
is, the case of null curvature.