Group theoretical approach to quantum fields in de Sitter space II. The complementary and discrete series

TL;DR

This paper develops a group-theoretical framework for covariant quantum fields in de Sitter space, focusing on the complementary and discrete series, revealing new features and challenges in defining vacua and field properties.

Contribution

It introduces a novel algebraic approach to construct covariant quantum fields for light and massless cases in de Sitter space, including new vacuum definitions and regularization techniques.

Findings

Constructed covariant fields for complementary series using a non-trivial scalar product.

Proposed a generalized notion of in/out vacua compatible with de Sitter invariance.

Identified issues with covariant fields in the massless discrete series and proposed regularization methods.

Abstract

We use an algebraic approach based on representations of de Sitter group to construct covariant quantum fields in arbitrary dimensions. We study the complementary and the discrete series which correspond to light and massless fields and which lead new feature with respect to the massive principal series we previously studied (hep-th/0606119). When considering the complementary series, we make use of a non-trivial scalar product in order to get local expressions in the position representation. Based on these, we construct a family of covariant canonical fields parametrized by SU(1,1)/U(1). Each of these correspond to the dS invariant alpha-vacua. The behavior of the modes at asymptotic times brings another difficulty as it is incompatible with the usual definition of the in and out vacua. We propose a generalized notion of these vacua which reduces to the usual conformal vacuum in the conformally massless limit. When considering the massless discrete series we find that no covariant field obeys the canonical commutation relations. To further analyze this singular case, we consider the massless limit of the complementary scalar fields we previously found. We obtain canonical fields with a deformed representation by zero modes. The zero modes have a dS invariant vacuum with singular norm. We propose a regularization by a compactification of the scalar field and a dS invariant definition of the vertex operators. The resulting two-point functions are dS invariant and have a universal logarithmic infrared divergence.