Two-dimensional De Sitter Research Articles

Creation of spin-1/2 particles in de Sitter space–time

In this work we analyzed the particle creation process of spin-1/2 particles by considering a simple cosmological model, a two-dimensional de Sitter space–time. We choose two different charts, the static and global chart, on de Sitter space–time. We solved the Dirac equation for these two charts and then used Bogoliubov coefficients to relate vacuum states in the asymptotic regions of gravitational backgrounds. By using the Bogoliubov transformation technique we compute the density of particles created.

Holographic central charge for 2-dimensional de Sitter space

Recently, investigations have begun into a holographic duality for two-dimensional de Sitter space. In this paper, we evaluate the associated central charge, using a modified version of the canonical Hamiltonian method that was first advocated by Catelani {\it et al}. Our computation agrees with that of a prior work (Cadoni {\it et al}), but we argue that the method used here is, perhaps, aesthetically preferable on holographic grounds. We also confirm an agreement between the Cardy and thermodynamic entropy, thus providing further support for the conjectured two-dimensional de Sitter/conformal field theory correspondence.

AdS 2 supergravity and superconformal quantum mechanics

We investigate the asymptotic dynamics of topological anti-de Sitter supergravity in two dimensions. Starting from the formulation as a BF theory, it is shown that the AdS 2 boundary conditions imply that the asymptotic symmetries form a super-Virasoro algebra. Using the central charge of this algebra in Cardy’s formula, we exactly reproduce the thermodynamical entropy of AdS 2 black holes. Furthermore, we show that the dynamics of the dilaton and its superpartner reduces to that of superconformal transformations that leave invariant one chiral component of the stress tensor supercurrent of a two-dimensional conformal field theory. This dynamics is governed by a supersymmetric extension of the de Alfaro–Fubini–Furlan model of conformal quantum mechanics. Finally, two-dimensional de Sitter gravity is also considered, and the dS 2 entropy is computed by counting CFT states.

Two-dimensional dS/CFT correspondence

We investigate de Sitter/conformal field theory (dS/CFT) correspondence in two dimensions. We define the conserved mass of de Sitter spacetime and formulate the correspondence along the lines of anti-de Sitter/conformal field theory duality. Asymptotic symmetry group, mass, and central charge of de Sitter spacetime are equal to those of anti-de Sitter spacetime. The entropy of two-dimensional de Sitter spacetime is evaluated by applying Cardy formula. We calculate the boundary correlators induced by the propagation of the dilaton in two-dimensional de Sitter space. Although the dilaton is a tachyonic perturbation in the bulk, boundary conformal correlators have positive dimension.

DS/CFT correspondence in two dimensions

We discuss the quantization of a scalar particle moving in two-dimensional de Sitter space. We construct the conformal quantum mechanical model on the asymptotic boundary of de Sitter space in the infinite past. We obtain explicit expressions for the generators of the conformal group and calculate the eigenvalues of the Hamiltonian. We also show that two-point correlators are in agreement with the Green function one obtains from the wave equation in the bulk de Sitter space.

The Asymptotic Dynamics of two-dimensional (anti-)de Sitter Gravity

We show that the asymptotic dynamics of two-dimensional de Sitter or anti-de Sitter Jackiw-Teitelboim (JT) gravity is described by a generalized two-particle Calogero-Sutherland model. This correspondence is established by formulating the JT model of (A)dS gravity in two dimensions as a topological gauge theory, which reduces to a nonlinear 0+1-dimensional sigma model on the boundary of (A)dS space. The appearance of cyclic coordinates allows then a further reduction to the Calogero-Sutherland quantum mechanical model.

Symbolic calculus on the nilpotent orbits of SO 0(1,2)

The coadjoint conical orbits in so(1,2) ∗ ⋍ su(1,1) ∗ are the phase spaces of the zero mass particles on the two-dimensional (anti-)de Sitter space-time. It contains also, as an open dense subset, the phase space for massless particles on one-dimensional Minkowski space-time when one identifies the Poincaré group to a subgroup of the conformal group SO 0(2,2)⋍SO 0(1,2)×SO 0(1,2). On the other hand, the quantum representation associated to these systems is an indecomposable extension of the first term of the discrete series of representations of SO 0(1,2) ⋍ SU(1,1)/ Z 2. We present in this paper a symbol map linking this representation and this orbit. This calculus is invariant and behaves correctly in the classical limit. As a result we have obtained a conformally invariant symbolic calculus for massless particles on (anti-)de Sitter or Minkowski space-time in 1+1 dimension.

Quantization of the nilpotent orbits in so(1,2)* and massless particles on (anti-)de Sitter space–time

The nontrivial nilpotent orbits in so(1,2)*≂su(1,1)* are the phase spaces of the zero mass particles on the two-dimensional (anti-)de Sitter space–time. As is well known, the lack of global hyperbolicity (respectively, stationarity) for the anti-de Sitter (respectively, de Sitter) space–time implies that the canonical field quantization of its free massless field is not uniquely defined. One might nevertheless hope to get the one-particle quantum theory directly from an appropriate ‘‘first’’ quantization of the classical phase space. Unfortunately, geometric quantization (the orbit method) does not apply to the above orbits and a naive canonical quantization does not yield the correct result. To resolve these difficulties, we present a simple geometric construction that associates to them an indecomposable representation of SO0(1,2) on a positive semidefinite inner product space. It is shown that quotienting out its one-dimensional invariant subspace yields the first term of the holomorphic discrete series of representations of SO0(1,2)≂SU(1,1)/Z2. We interpret these results physically by showing that the above positive semidefinite inner product space is naturally isomorphic to a space of solutions of the conformally invariant zero-mass Klein–Gordon equation on the (anti)-de Sitter space–time, equipped with the usual Klein–Gordon inner product, obtained by integrating over a suitable spacelike hypersurface. As such, our version of geometric quantization selects in a natural way a quantization of the free massless particle. We show it is conformally [i.e., SO0(2,2)] invariant and behaves correctly in the classical limit.

Vacuum for a massless scalar field outside a collapsing body in de Sitter spacetime.

We present a general way to define regular vacuum states of a quantized massless scalar field in two-dimensional spacetimes with horizons. We discuss in more detail the cases of Schwarzschild-de Sitter spacetime and especially the exterior of a massive shell that collapses to form a black hole in a two-dimensional de Sitter spacetime. In the latter case a vacuum state is defined using modes of positive frequency with respect to the past cosmological horizon's affine parameter. In this vacuum static observers, long after the shell has collapsed, detect thermal fluxes coming from both the cosmological and the black-hole horizons, and characterized by the corresponding Hawking temperatures. The renormalized stress-energy tensor in this vacuum state is regular everywhere and has precisely the form one would expect from prior experience with de Sitter spacetime and with gravitational collapse spacetimes that have a vanishing cosmological constant.

Bubble formation in anisotropic cosmologies

We discuss the decay of the false vacuum in homogeneous cosmologies, both isotropic and anisotropic. If the false vacuum is isotropic, it is expected that the decay rate is dominated by spherically symmetric bubbles. However, if the universe during its past evolution was trapped in an anisotropic false vacuum one expects this anisotropy to be present in the decay products (bubbles) of that vacuum. In this paper we show the existence of two new, anisotropic types of bubble solution of Kantowski-Sachs type for a general double well potential. The first represents the decay of four-dimensional de Sitter space, and the second the decay of the product of the two-sphere and two-dimensional de Sitter space. We also derive qualitative features of the classical space-times emerging after tunneling.

Vacuum state of complex scalar fields in SO(2,1)-invariant backgrounds.

We consider complex scalar fields in two-dimensional de Sitter space minimally coupled to an SO(2,1)-invariant Maxwell potential and construct an ultraviolet-finite functional representation of the symmetry generators in the Schr\"odinger picture. We find that there is a unique vacuum state which is strictly SO(2,1) invariant.

Functional representation for the isometries of de Sitter space

We examine the Schrödinger picture for a spinless field theory in two-dimensional de Sitter space and construct an ultraviolet finite functional representation for the de Sitter Lie algebra. The one-parameter family of de Sitter vacua is found to be only phase-invariant, except for one value of the parameter where the state is truly invariant.

Trace anomalies in a two-dimensional de Sitter metric and black-body radiation

The author investigates the possibility of a Hawking-type effect due to the cosmological event horizon. He considers the reduced two-dimensional de Sitter metric. There appears to be radiation from the horizon towards the origin at temperature T defined by kT= square root Lambda '/2 pi where 3 Gamma ' is the cosmological constant. The observer dependence of this radiation is discussed.

Alternative vacuum states in static space-times with horizons

The paper develops several variants of the suggestion of Unruh (1976), to define vacuum initial conditions on the horizon through an analytic property of normal-mode solutions which expresses 'positive frequency' with respect to null translations on the horizon. It is verified that Unruh's condition corresponds to the absence of a flux of energy through an horizon surface, although there may be a flux parallel to the surface. A region with time-like isometries typically is bounded by four such surfaces, two of which may be the usual null infinites, I+or-. In general, Unruh's condition may be applied on two adjacent sides, forcing the appearance of a Hawking flux on the other two. In special cases, however, the opposite horizons can be in 'equilibrium', so that no radiation occurs. In particular, for two-dimensional de Sitter space the vacuum state thus obtained has a stress tensor proportional to the metric times the curvature scalar. If two horizons are not in equilibrium, then no state invariant under the isometries can yield a non-singular stress tensor.