DeSitter Group

Prerequisites: Poincare Group. Full path (9 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

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.

C.f. http://physics.stackexchange.com/questions/299621/the-desitter-group-vs-the-poincare-group-for-a-non-zero-cosmological-constant

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 http://journals.aps.org/prd/pdf/10.1103/PhysRevD.21.1466.

Intuitive

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 $R$ (equivalently an invariant energy scale $\hbar c/R$), 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.

The fact that the deSitter group contracts to the Poincare group in the $R\rightarrow \infty$ limit, means that the Poincare group is a good approximation as long as we consider systems with a length scale that is small compared to $R$. This is analogous to how the Galilean group is good enough as long as we are only dealing with velocities much smaller than the invariant velocity $c$.

Expressed differently: the deSitter group is only important for cosmological systems, which have a length scale comparable to $R$.

Alternatively, we can talk about the invariant energy scale $\hbar c\sqrt{\Lambda}$ set by the cosmological constant. The deSitter group contracts to the Poincare group in the $\Lambda \rightarrow 0$ limit. Thus the deSitter structure is not important, as long as we are dealing with energies much larger than $\hbar c\sqrt{\Lambda}$. In such systems such a small constant energy has no effect. The present day value for the cosmological constant is $\Lambda \approx 10^{-52} \mathrm{m^{-2}}$ and this means that present day effects of the deSitter group structure are tiny. This means, the Poincare group is a great approximate symmetry nowadays, because $\Lambda$ is almost zero.

However, the deSitter group could be very important in the early universe, too. For example, because it seems plausible that there was

Concrete

We use the metric signature $(-,+,+,+)$ and $c=1$. Four-dimensional de Sitter spacetime $dS_4$ is the hyperboloid $$\eta_{AB} X^A X^B = -(X^0)^2 + (X^1)^2 + (X^2)^2 + (X^3)^2 + (X^4)^2 = R^2$$ in a flat five-dimensional spacetime with coordinates $X^A$, $A = 0,\dots,4$, and metric $\eta_{AB} = \mathrm{diag}(-1,1,1,1,1)$. The constant $R$ is the de Sitter radius. The induced metric solves Einstein's equations in vacuum with cosmological constant $$\Lambda = \frac{3}{R^2} .$$ In the coordinates that cover half of the hyperboloid and describe an expanding universe, the line element reads $ds^2 = -dt^2 + e^{2t/R}\, d\vec x^{\,2}$, so the Hubble rate is constant, $H = 1/R = \sqrt{\Lambda/3}$.

Every linear transformation of the $X^A$ that preserves $\eta_{AB}$ maps the hyperboloid to itself. The de Sitter group is therefore $SO(4,1)$, the "Lorentz group" of the five-dimensional ambient space. It has $5\cdot 4/2 = 10$ generators $M_{AB} = -M_{BA}$, which act on functions of $X$ as $M_{AB} = i(X_A \partial_B - X_B\partial_A)$ with $X_A = \eta_{AB}X^B$. They obey $$[M_{AB}, M_{CD}] = i\left(\eta_{BC} M_{AD} - \eta_{AC} M_{BD} - \eta_{BD} M_{AC} + \eta_{AD} M_{BC}\right).$$

Contraction to the Poincaré algebra. Split the indices into $\mu = 0,\dots,3$ and the extra direction $4$. The six generators $M_{\mu\nu}$ form the Lorentz algebra $\mathfrak{so}(3,1)$. Near the point $X = (0,0,0,0,R)$ of the hyperboloid, the remaining four generators $M_{\mu 4}$ move points along the hyperboloid, so we define $$P_\mu = \frac{1}{R} M_{\mu 4} .$$ The commutation relations above give $$\begin{aligned} [M_{\mu\nu}, P_\rho] &= i\left(\eta_{\nu\rho} P_\mu - \eta_{\mu\rho} P_\nu\right), \\ [P_\mu, P_\nu] &= -\frac{i}{R^2}\, M_{\mu\nu} . \end{aligned}$$ The first line is the same as in the Poincaré algebra. The second line says that "translations" in different directions do not commute; their commutator is a Lorentz transformation suppressed by $1/R^2 = \Lambda/3$. In the limit $R\to\infty$ with $M_{\mu\nu}$ and $P_\mu$ held fixed, we recover the Poincaré algebra. This is an Inönü-Wigner contraction.

Casimir operator. The quadratic Casimir of $SO(4,1)$ splits as $$\frac{1}{2} M_{AB}M^{AB} = \frac{1}{2} M_{\mu\nu} M^{\mu\nu} + R^2 P_\mu P^\mu ,$$ so $\frac{1}{R^2}\cdot\frac{1}{2} M_{AB}M^{AB} \to P_\mu P^\mu = -m^2$ as $R\to\infty$. The de Sitter Casimir, divided by $R^2$, takes over the role of $P_\mu P^\mu = -m^2$. On scalar functions on $dS_4$ it acts as $\frac{1}{2}M_{AB}M^{AB} = -R^2\,\Box$, with $\Box$ the d'Alembertian of $dS_4$, so a scalar field obeying $(\Box - m^2)\phi = 0$ has Casimir eigenvalue $-m^2R^2$.

Anti-de Sitter. Flipping the sign of the extra direction, $\eta_{AB} = \mathrm{diag}(-1,1,1,1,-1)$ and $\eta_{AB}X^AX^B = -R^2$, gives anti-de Sitter spacetime with $\Lambda = -3/R^2$ and isometry group $SO(3,2)$. The same steps give $[P_\mu,P_\nu] = +\frac{i}{R^2}M_{\mu\nu}$, and the same contraction to the Poincaré algebra.

Abstract

As a manifold, $dS_4$ is the homogeneous space $$dS_4 \cong SO_0(4,1)/SO_0(3,1),$$ where $SO_0$ denotes the connected component of the identity and the stabiliser of a point is a copy of the Lorentz group. The full isometry group is $O(4,1)$, which has four connected components. Since $\dim SO(4,1) = 10 = n(n+1)/2$ for $n=4$, de Sitter space is maximally symmetric, with constant positive curvature. The universal (double) cover of $SO_0(4,1)$ is $\mathrm{Spin}(4,1) \cong Sp(1,1)$, the group of $2\times 2$ quaternionic matrices preserving a Hermitian form of signature $(1,1)$; this cover acts on spinor fields.

The Lie algebra $\mathfrak{so}(4,1)$ is a non-compact real form of the complex simple Lie algebra $\mathfrak{so}(5,\mathbb{C})$ of type $B_2$. It has rank two, so the centre of its universal enveloping algebra is generated by two Casimir operators: the quadratic one $\frac12 M_{AB}M^{AB}$ and a quartic one $W_A W^A$ built from $$W^A \propto \epsilon^{ABCDE} M_{BC} M_{DE} .$$ Under the contraction $P_\mu = M_{\mu 4}/R$, $R\to\infty$, the components $W^\mu/R$ become proportional to the Pauli-Lubanski vector, so $W_AW^A/R^2$ contracts to the Poincaré Casimir that measures spin. This is the precise sense in which a particle in de Sitter space is labelled by two Casimir eigenvalues rather than by a mass and a spin.

Since $SO_0(4,1)$ is non-compact and simple, all of its non-trivial unitary irreducible representations are infinite dimensional. They fall into principal, complementary and discrete series. Unlike Minkowski space, de Sitter space has no globally timelike Killing vector. There is therefore no generator of the group that could serve as a positive energy operator for all observers, and the spectrum condition used in the Wigner classification of Poincaré representations has no direct analogue. For scalar fields on $dS_4$ with $m^2 R^2 = \Delta(3-\Delta)$, the principal series corresponds to $m^2R^2 > 9/4$ (complex $\Delta$) and the complementary series to $0 < m^2R^2 < 9/4$ (real $\Delta$).

The sign of $[P_\mu,P_\nu]$ in the Concrete section distinguishes the two deformations of the Poincaré algebra that keep the Lorentz algebra and the Lorentz transformation property of $P_\mu$: $\mathfrak{so}(4,1)$ for $\Lambda>0$ and $\mathfrak{so}(3,2)$ for $\Lambda<0$. Viewing Minkowski space as $ISO(3,1)/SO(3,1)$ and replacing $ISO(3,1)$ by $SO(4,1)$ is also the starting point of Cartan-geometric formulations of gravity, in which the spacetime is modelled locally on de Sitter space instead of Minkowski space.

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Newton's Second Law
5. Vector Calculus
6. Newtonian Mechanics
7. Special Relativity
8. Lorentz Group
9. Poincare Group