[Paper Review] The de Sitter group and its representations: a window on the notion of de Sitterian elementary systems

Mohammad Enayati, Jean‐Pierre Gazeau+2 more|arXiv (Cornell University)|Jan 27, 2022

Noncommutative and Quantum Gravity Theories2 citations

TL;DR

This paper presents a rigorous group-theoretic framework for defining elementary quantum systems in de Sitter (dS) spacetime via unitary irreducible representations (UIR) of the dS group, establishing a quantum field theory (QFT) formulation analogous to Minkowski QFT but with a geometric Kubo-Martin-Schwinger (KMS) condition replacing the spectral condition. It introduces a consistent, invariant definition of mass in dS relativity through UIR invariants, reconciling 'massive' and 'massless' fields with their Minkowskian counterparts via group contraction.

ABSTRACT

We review the construction of ("free") elementary systems in de Sitter (dS) spacetime, in the Wigner sense, as associated with unitary irreducible representations (UIR's) of the dS (relativity) group. This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: "smooth" transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local ("tangent") Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. We review three decompositions of the dS group physically relevant for the description of dS spacetime and classical phase spaces of elementary systems living on it. We review the construction of (projective) dS UIR's issued from these group decompositions. (Projective) Hilbert spaces carrying the UIR's (in some restricted sense) identify quantum ("one-particle") states spaces of dS elementary systems. Adopting a well-established Fock procedure, based on the Wightman-G\"{a}rding axioms and on analyticity requirements in the complexified Riemannian manifold, we proceed with a consistent quantum field theory (QFT) formulation of elementary systems in dS spacetime. This dS QFT formulation closely parallels the corresponding Minkowskian one, while the usual spectral condition is replaced by a certain geometric Kubo-Martin-Schwinger (KMS) condition equivalent to a precise thermal manifestation of the associated "vacuum" states.

Motivation & Objective

• To establish a mathematically rigorous formulation of elementary quantum systems in de Sitter spacetime using unitary irreducible representations (UIR) of the dS group.
• To clarify the physical meaning of 'massive' and 'massless' fields in de Sitter relativity through invariant parameters in UIRs.
• To demonstrate a smooth transition from classical to quantum theory on dS spacetime via co-adjoint orbits and Hilbert space realizations.
• To show that the dS QFT formulation parallels Minkowskian QFT but replaces the spectral condition with a geometric KMS condition reflecting the Gibbons-Hawking temperature.
• To provide a consistent, unambiguous definition of mass in dS relativity that reduces correctly to the Minkowskian case under group contraction.

Proposed method

• Constructing UIRs of the dS group (SO₀(1,2) and SO₀(1,4)) via three key group decompositions: space-time-Lorentz, Cartan, and Iwasawa.
• Using co-adjoint orbits of the dS group as classical phase spaces for elementary systems, with orbits classified by Casimir invariants.
• Realizing UIRs globally on Hilbert spaces via principal, complementary, and discrete series representations, with unitarity conditions derived from analyticity and group structure.
• Applying the Wightman-Gårding axioms and analyticity in the complexified Riemannian manifold to construct a consistent quantum field theory on dS spacetime.
• Implementing group contraction procedures (dS → Poincaré → Galilean) to connect dS representations to Minkowskian ones and validate the mass definition.
• Deriving plane-wave solutions in tube domains and showing their role as generating functions for square-integrable eigenfunctions in dS QFT.

Experimental results

Research questions

• RQ1 How can elementary quantum systems in de Sitter spacetime be rigorously defined via unitary irreducible representations of the dS group?
• RQ2 What is the physical significance of the geometric Kubo-Martin-Schwinger (KMS) condition in dS quantum field theory, and how does it replace the spectral condition of Minkowskian QFT?
• RQ3 How is the concept of mass in de Sitter relativity defined invariantly, and how does it reduce to the Minkowskian mass under group contraction?
• RQ4 What role do the three group decompositions (space-time-Lorentz, Cartan, Iwasawa) play in the construction of classical and quantum phase spaces on dS spacetime?
• RQ5 How do the UIRs of the dS group (principal, complementary, discrete series) realize the quantum states of 'massive' and 'massless' fields, and what is their physical interpretation?

Key findings

• The dS quantum field theory formulation is fully consistent with Wightman-Gårding axioms and analyticity, yielding a QFT that parallels the Minkowskian one but with a geometric KMS condition replacing the spectral condition.
• The Gibbons-Hawking temperature emerges naturally as a thermal manifestation of the vacuum state in dS QFT, encoded in the KMS condition on the complexified manifold.
• The invariant parameters characterizing the UIRs—specifically the principal series and discrete series labels—provide a unique, unambiguous definition of mass in dS relativity, distinguishing 'massive' and 'massless' fields unambiguously.
• The group contraction of dS UIRs to Poincaré UIRs reproduces the standard Minkowskian mass and helicity quantum numbers, validating the dS mass definition as the correct relativistic generalization.
• The scalar and spinor UIRs in the principal series are realized on Hilbert spaces of square-integrable functions on the dS group, with explicit constructions of generating functions (dS plane waves) in tube domains.
• The dS UIRs in the complementary and discrete series are shown to be physically relevant for describing elementary systems, with the discrete series corresponding to finite-energy states and the complementary series to non-normalizable, 'fuzzy' states reflecting the intrinsic non-locality of dS spacetime.