On anti-de Sitter oscillators and de Sitter anti-oscillators

We revisit the principal arguments for interpreting the free quantum fields on anti-de Sitter or de Sitter spacetimes of any dimensions as oscillators or respectively anti-oscillators. In addition, we point out that there exists a chart on the de Sitter background where the free Dirac field becomes a genuine anti-oscillator in the non-relativistic limit in the sense of special relativity.