Group theoretical approach to quantum fields in de Sitter space I. The principal series

TL;DR

The work develops a group-theoretical framework for quantum fields in de Sitter space by quantizing a massive scalar through unitary irreducible representations of SO0(1,n). A unique dS-invariant vacuum emerges, and the local field operator is built covariantly, with canonical commutation relations fixed up to a single normalization; the residual SU(1,1)/U(1) moduli reproduce the alpha-vacua when mapped to the standard QFT formalism via a squeezing operator. The BD vacuum is singled out by a Hadamard short-distance condition, and general alpha-vacua are obtained by SU(1,1)/U(1) rotations from this base. The framework extends to arbitrary dimensions, revealing dimension-dependent in/out Bogoliubov structure and, notably, no particle creation in odd dimensions. The approach offers a transparent, horizon-sensitive interpretation of vacuum selection and provides a route to deformations of de Sitter symmetry and their QFT consequences.

Abstract

Using unitary irreducible representations of the de Sitter group, we construct the Fock space of a massive free scalar field. In this approach, the vacuum is the unique dS invariant state. The quantum field is a posteriori defined by an operator subject to covariant transformations under the dS isometry group. This insures that it obeys canonical commutation relations, up to an overall factor which should not vanish as it fixes the value of hbar. However, contrary to what is obtained for the Poincare group, the covariance condition leaves an arbitrariness in the definition of the field. This arbitrariness allows to recover the amplitudes governing spontaneous pair creation processes, as well as the class of alpha vacua obtained in the usual field theoretical approach. The two approaches can be formally related by introducing a squeezing operator which acts on the state in the field theoretical description and on the operator in the present treatment. The choice of the different dS invariant schemes (different alpha vacua) is here posed in very simple terms: it is related to a first order differential equation which is singular on the horizon and whose general solution is therefore characterized by the amplitude on either side of the horizon. Our algebraic approach offers a new method to define quantum field theory on some deformations of dS space.