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DeSitter Group

Why is it interesting?

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

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

The Poincare group is the contraction of the deSitter group in the limit $\Lambda \rightarrow 0$, analogous to how the Galilean group is the contraction of the Poincare group in the limit $c\rightarrow \infty$.

Thus a non-zero $\Lambda$ means that the exact spacetime symmetry group is the deSitter group and the Poincare group is only a good approximation because $\Lambda$ is so small.


Given the current picture of an increasingly expanding universe, it may very well be the case that future generations of students first learn about the de Sitter and the Newton-Hooke groups, while the Poincaré and the Galilei groups will be considered to be nothing but historical aberrations. Symmetries in Fundamental Physics by Kurt Sundermeyer

Since the renormalized value of the cosmological constant is experimentally quite small, $ρ ∼ 10^{-122}M_{Pl} \sim 10^{-3}$ eV (and positive – we live in de Sitter space), we can ignore it for terrestrial experiments. To account for a non- zero cosmological constant in quantum field theory requires field theory in curved space, a topic beyond the scope of this text.

Page 415 in Schwartz - Quantum Field Theory and the Standard Model

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

In addition:

"Apart from the current accelerating cosmic expansion, further motivation for the interest in de Sitter gravity comes from the inflationary era, during which one assumes that the universe was also described by a de Sitter phase." Aspects of Quantum Gravity in de Sitter Spaces Dietmar Klemm and Luciano Vanzo

In quantum field theory (in its present imperfect form) the Minkowski metric is the vacuum expectation value of the Riemannian metric. It seems unsafe to restrict the attention arbitrarily to the special case of vanishing cosmological constant, for this ease is unstable to deformations; a nonzero cosmological constant may, for example, appear spontaneously through renormalization. In that case the zeroth approximation (or vacuum expectation value) of the metric cannot be Minkowski, but must be de Sitter. Our previous analysis must therefore be modified by the substitution of the de Sitter group for the Poincare group from Massless particles, conformal group, and de Sitter Universe by E. Angelopoulos

General relativity can be formulated as a gauge theory of the deSitter group, see


The symmetry group that we use when we only consider the principle that all inertial frames of reference should be the same, called Galilean relativity, is the Galilean group.

If we add to this principle that there should be an invariant velocity $c$, we end up with special relativity and the corresponding symmetry group is the Poincare group.

If we then add a second principle that says there should be an invariant length scale $R$ (= an invariant energy scale $\Lambda$), the deSitter group is the group that we must use.

The deSitter group becomes the Poincare group in the contraction limit $R \rightarrow \infty$, where $R$ is the so-called deSitter radius. Oftentimes, people prefer to work with the cosmological constant $ \Lambda \propto \frac{1}{R^2}$ instead. Analogously, the Poincare group becomes the Galilean group in the $c \rightarrow \infty$ limit.


In this section things should be explained by analogy and with pictures and, if necessary, some formulas.


The motto in this section is: the higher the level of abstraction, the better.





advanced_tools/group_theory/desitter.1521886089.txt.gz · Last modified: 2018/03/24 10:08 (external edit)