Localizability in de Sitter space for 2+1 dimensions

TL;DR

This paper extends the Newton-Wigner localization concept to 2+1 dimensional de Sitter space, analyzing the one-particle subspace, group representations, and time evolution, revealing trivial dynamics in the complementary series.

Contribution

It generalizes Newton-Wigner localization to three-dimensional de Sitter space and explicitly derives the time evolution of the localization operator.

Findings

One-particle subspace identified as an irreducible de Sitter group representation.

Explicit time evolution of the Newton-Wigner operator obtained.

No dynamics found in the complementary series.

Abstract

We extended the notion of Newton-Wigner localization, already constructed in the bi-dimensional de Sitter space, to the tri-dimensional case for both principal and complementary series. We identify the one-particle subspace, generated by the positive-energy modes solution of the Klein-Gordon equation, as an irreducible representation of the de Sitter group. The time-evolution of the Newton-Wigner operator was obtained explicitly, and for the complementary series the evolution is trivial, i.e., there are no dynamics. Also, we discussed heuristically the existing sign ambiguity when we do not require as a postulate that the Newton-Wigner functions must be proportional to their respective solutions in the representation of solutions of the Klein-Gordon equation.