Hyperspherical Rindler space, dimensional reduction, and de Sitter-space scalar field theory

Field theory in de Sitter space is related to flat-spacetime theory via dimensional reduction. de Sitter space can be embedded in Minkowski space of one higher dimension as a hyperboloid of one sheet. This paper examines the field theory imposed on de Sitter space by a free, massless, real scalar field in the embedding space. This corresponds to a collection of massive de Sitter-space scalars, of all masses above a positive lower bound: Half of the field normal modes in hyperspherical Rindler coordinates specialize to such fields on an embedded de Sitter space, while the other half vanish there. Dimensional reduction simply eliminates those noncontributing modes. The quantized field theory in hyperspherical Rindler space, hence, in de Sitter space, is treated here using both canonical operator expansions and the functional Schr\"odinger formalism. Transformations between field operators and state wave functionals in the Rindler--de Sitter theory and those in the Minkowski-space formulation are calculated explicitly. The Minkowski vacuum state corresponds to the Chernikov-Tagirov or Euclidean vacuum of the de Sitter-space fields. The Rindler--de Sitter `` particle'' content of this state, for various simple mode choices, can be time dependent, even nonmonotonic; it can also exhibit ``thermal'' features recalling those of rectangular Rindler-space theory or of Euclidean field theory in de Sitter space.