We propose a novel prescription for computing the boundary stress tensor and charges of asymptotically de Sitter ͑dS͒ spacetimes from data at early or late time infinity. If there is a holographic dual to dS spaces, defined analogously to the AdS/conformal field theory correspondence, our methods compute the ͑Euclidean͒ stress tensor of the dual. We compute the masses of Schwarzschild-de Sitter black holes in four and five dimensions, and the masses and angular momenta of Kerr-de Sitter spaces in three dimensions. All these spaces are less massive than de Sitter space, a fact which we use to qualitatively and quantitatively relate de Sitter entropy to the degeneracy of possible dual field theories. Our results in general dimensions lead to a conjecture: Any asymptotically de Sitter spacetime with mass greater than de Sitter space has a cosmological singularity. Finally, if a dual to de Sitter space exists, the trace of our stress tensor computes the renormalized group ͑RG͒ equation of the dual field theory. Cosmological time evolution corresponds to RG evolution in the dual. The RG evolution of the c function is then related to changes in accessible degrees of freedom in an expanding universe.

We determine the energy eigenvalues and eigenfunctions of the harmonic oscillator where the coordinates and momenta are assumed to obey the modified commutation relations ͓x i ,p j ͔ϭiប͓(1ϩ␤ p 2 )␦ i j ϩ␤Јp i p j ͔. These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relations which appear in perturbative string theory. Our solutions illustrate how certain features of string theory may manifest themselves in simple quantum mechanical systems through the modification of the canonical commutation relations. We discuss whether such effects are observable in precision measurements on electrons trapped in strong magnetic fields.

We point out that for a large class of universes, holography implies that the most probable value for the cosmological constant is zero. In four space-time dimensions, the probability distribution takes the Baum-Hawking form, dP ϳ exp͑cM 2 p ͞L͒dL. The cosmological constant problem has a twofold meaning: It is a problem of fundamental physics, because the value of the cosmological constant L is tied to vacuum energy density. On the other hand, the cosmological constant tells us something about the large scale behavior of the universe, since a small cosmological constant implies that the observable universe is big and (nearly) flat. The problem is that there is an enormous discrepancy between the value of the vacuum energy density as predicted by quantum field theory of the standard-model degrees of freedom and the cosmologically observed value of L [2]. This discrepancy occurs already at very low-energy scales, of the order of eV, and clearly represents the most flagrant naturalness problem in today's physics.Thus, the cosmological constant relates the properties of the microscopic physics of the vacuum to the long-distance physics on cosmic scales. [This general philosophy was stressed in the wormhole approach to the cosmological constant problem (see [3] for the original references and [4,5] for a critique of this approach).] Therefore, the observable smallness of the cosmological constant should tell us something fundamental about the underlying microscopic theory of nature.In this note we study implications of holography [6,7], taken as a fundamental property of the microscopic theory of quantum gravity, for the cosmological constant problem. Assuming that the cosmological constant is a random variable, and that holographic entropy can be given a Boltzmannian interpretation, we point out that the most probable value of the cosmological constant in a holographic theory is zero, in ensembles of universes with finite-area holographic screens.It is not the aim of this paper to discuss the microscopic origin of any of these assumptions. In particular, the microscopic origin of holography and the randomness of the cosmological constant are clearly difficult problems whose solutions we do not claim to have understood. It is our goal to show that once these assumptions are satisfied, a simple and robust phenomenological argument implies a naturally small cosmological constant.

We discuss the issue of parity violation in quantum gravity. In particular, we study the coupling of fermionic degrees of freedom in the presence of torsion and the physical meaning of the Immirzi parameter from the viewpoint of effective field theory. We derive the low-energy effective Lagrangian which turns out to involve two parameters: one measuring the nonminimal coupling of fermions in the presence of torsion, the other being the Immirzi parameter. In the case of nonminimal coupling the effective Lagrangian contains an axial-vector interaction leading to parity violation. Alternatively, in the case of minimal coupling there is no parity violation and the effective Lagrangian contains only the usual axial-axial interaction. In this situation the real values of the Immirzi parameter are not at all constrained. On the other hand, purely imaginary values of the Immirzi parameter lead to violations of unitarity for the case of nonminimal coupling. Finally, the effective Lagrangian blows up for the positive and negative unit imaginary values of the Immirzi parameter.

We continue our investigation of the phenomenological implications of the ''deformed'' commutation relations ͓x i ,p j ͔ϭiប͓(1ϩ␤ p 2 )␦ i j ϩ␤Јp i p j ͔. These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relation which appears in perturbative string theory. In this paper, we consider the effects of the deformation on the classical orbits of particles in a central force potential. Comparison with observation places severe constraints on the value of the minimum length.