# De Sitter Group

## Why is it interesting?

The simple existence of a non-vanishing cosmological constant in the universe means that Poincare is no longer the kinematic group of spacetime; this is a largely overlooked point.

To the extent that* ~$74 \simeq 100$ %, we can say that our universe is observed to be almost maximally symmetric and de Sitter.(*Dark energy amounts to ~74% of the universe […]). [..] Our universe might very well be described by $dS^4$ to a good approximation, as discussed in chapter VIII.2 and in the preceding section.

Einstein Gravity in a Nutshell - A. Zee

We will confront certain recent astronomical observations suggesting that, even in an empty universe, the event world may possess properties not reflected in the structure of Minkowski spacetime, at least on the cosmological scale. Remarkably, there is a viable alternative the deSitter spacetime, nearly 100 years old, that has precisely these properties and we will devote a little time to becoming acquainted with it.

The Geometry of Minkowski Spacetime - Naber

The present experimental value for the cosmological constant is tiny, but nonzero: $\Lambda \approx 1.19·10^{-52}$ $1/m^2$.

## Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Student

The de Sitter space is the curved space-time which has been most studied by quantum field theorists because it and the anti-de Sitter space are the unique maximally symmetric curved spacetimes Weinberg, 1972). Dirac Wave Equation in the de Sitter Universe E. A. Notte Cuello

## Researcher

As shown in Contractions of Representations of de Sitter Groups J. MICKELSSON* and J. NIEDERLE* (see also their references to Wigner and Inonu), it is fortunately the case that at least all the massive representations of the Poincaré group can be obtained by contracting representations of the deSitter group, meaning the approximation is also consistent on this level. What exactly the mass parameter itself means in deSitter space is a somewhat open question.

A longer overview over the Wigner classification for O(1,4)O(1,4), i.e. 4D deSitter space, and the physical meaning of the different possible representations is given in "Group theory and de Sitter QFT" by Boers.

Recall that in the flat spacetime case the particle states are labeled by the Poincar´e label (p, λ), whose values are closely related to the UIR label (m, s); m is used to define the mass-shell condition and s determines the range of λ. We identified m with the mass of the field quanta, and s with the spin. In a group theoretical sense, we associate these parameters with the eigenvalues of the Casimir operators (see section 3.5). Since the notion of mass is not as clear in de Sitter space as in Minkowski space in a field theoretical sense, we may ask ourselves if the Casimir operators of SO(4, 1) might bring resolvement to the issue. There is a very straightforward link between the Casimir operators and the wave equation, which we shall investigate by using the ambient space formalism in the next section. […]

The classification of SO(4, 1) UIRs reveals that there are three types, belonging either to the so-called principal, complementary or discrete series. In the zero curvature limit the principal series contracts to the massive Poincar´e UIRs, one particular complementary series UIR contracts to the massless scalar Poincar´e UIR, and a certain class of the discrete series contracts to the massless nonzero-spin Poincar´e UIRs (see section 4.3.2 for details). These UIRs contracting to the massless Poincar´e UIRs are uniquely extendable to UIRs of the conformal group SO(4, 2). All other SO(4, 1) UIRs have no (or no physically interesting) link to the Poincar´e group in the zero curvature limit. This last observation excludes a range of de Sitter fields from having a Minkowskian interpretation of their mass parameter in this sense.

For the representations of the De Sitter group, see http://thep.housing.rug.nl/sites/default/files/theses/Master%20thesis_Marco%20Boers.pdf and Gürsey, F.: Introduction to the De Sitter Group. In: Gürsey, F. (ed.) Group Theoretical Concepts and Methods in Elementary Particle Physics. Gordon and Breach, New York (1965

A great, very complete discussion can be found in http://onlinelibrary.wiley.com/doi/10.1002/prop.19770250112/epdf. Unfortunately, the paper is only available in German.

and "Positive Cosmological Constant and Quantum Theory" by Lev, Felix M.

"Gürsey (1963) presented the Casimir operators for the de Sitter group and concluded by showing that a particle in a de Sitter universe does not have a definite mass and spin, but definite eigenvalues of the two Casimir invariant operators of the group."Dirac Wave Equation in the de Sitter Universe E. A. Notte Cuello et. al.

Why was the DeSitter group ignored for such a long time?

The theories in which the kinematic invariance group is not the Poincare group but one of the de Sitter groups SO(1,4) or S0(2,3) have been considered by many physicists. From the group-theoretical and aesthetic points of view, the de Sitter invariance looks much more attractive than the Poincare invariance. However, recently the de Sitter invariance has been studied less intensively, since in the currently popular superstring theories the flatness of the spacetime is supposed from the beginning.

What is the connection to dark energy etc?

The expansion of the observed universe appears to be accelerating. A simple explanation of this phenomenon is provided by the non-vanishing of the cosmological constant in the Einstein equations. Arguments are commonly presented to the effect that this simple explanation is not viable or not sufficient, and therefore we are facing the "great mystery" of the "nature of a dark energy". We argue that these arguments are unconvincing, or ill-founded.Why all these prejudices against a constant? by Eugenio Bianchi, Carlo Rovelli

With such a formulation of symmetry, the fact that $Λ \neq 0$ means not that the space-time background is curved (since the notion of the space-time background is not physical) but that the symmetry algebra is the de Sitter algebra rather than the Poincare one. In particular, there is no need to involve dark energy or other fields for explaining this fact. As a consequence, instead of the cosmological constant problem we have a problem why nowadays Poincare symmetry is so good approximate symmetry. This is rather a problem of cosmology but not fundamental quantum physics.

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