International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 69, Pages 3775-3781

The second-order Klein-Gordon field equation

D. Gomes1 and E. Capelas De Oliveira2,3

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.


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.