Self-adjointness-based quantum field theory in de Sitter and anti-de Sitter space-time

As an alternative to existing quantization schemes, the self-adjointness of the classical field equations is employed as the basic principle for the quantization of a scalar field of arbitrary mass $m$. The results thus obtained in de Sitter space differ markedly from previous ones based on the so-called Euclidean approach to quantum field theory. It is argued that if $m^2>{\left(4\mathrm{}{R}^2\right)}^{-1\mathrm{}}$ ($R$ is the de Sitter radius) a geodesic observer will detect a constant flux of particles coming from the cosmological event horizon. In contrast to the result of Gibbons and Hawking, however, the spectrum is nonthermal. For $m^2<{\left(4\mathrm{}{R}^2\right)}^{-1\mathrm{}}$, but except for a discrete mass spectrum containing $m^2=0\mathrm{}$, there is a catastrophic vacuum instability related to the Schiff-Snyder-Weinberg effect in electrostatics. As the Feynman propagator of our approach does not obey the Hadamard-DeWitt expansion, a modified regularization scheme has to be applied to the vacuum expectation value of the stress-energy tensor. Our result differs from that of Dowker and Critchley and in particular does not show the standard trace anomaly for $m^2=0\mathrm{}$. For (the covering space of) anti-de Sitter space the self-adjointness condition eliminates the pathological effects of the violation of global hyperbolicity and amounts essentially to one of the "reflective boundary conditions" introduced by Avis et al. Quantization is straightforward and no vacuum instability is implied, but it is argued that the Feynman propagator cannot be of the Hadamard form despite the static character of space-time.