Massless Gupta-Bleuler vacuum on the (1+1)-dimensional de Sitter space-time

Abstract

We construct a causal, de Sitter, and conformally covariant massless free quantum field on the (1+1)-dimensional de Sitter space-time admitting a de Sitter invariant vacuum in an indefinite inner product space. The field is defined rigorously as an operator-valued distribution and is covariant in the usual strong sense: $V{}_{g}^{\ensuremath{-}1}\ensuremath{\varphi}(x)V{}_{g}=\ensuremath{\varphi}(g\ensuremath{\cdot}x)$ for any $g$ in the de Sitter group, where $V$ is a unitary representation of the de Sitter group on the space of states. We use the formalism of Gupta-Bleuler triplets which also allows for an explicit description of the gauge degree of freedom. As a consequence the model does not suffer from infrared divergences, contrary to what happened in previous treatments of this problem. The causality and the covariance of the theory are assured thanks to a suitable choice of the space of solutions of the classical field equation. We show that, although the field itself is not observable (it is gauge dependent), the stress tensor and the energy-momentum vector are. The energy operator ${P}_{0}$ is positive in all physical states, and vanishes in the vacuum. In addition, the field is conformally covariant and the model does not exhibit a conformal anomaly in the trace of the energy-momentum tensor.