Raquel S. Garcia

Observables in causal set cosmology

For the theories known as classical sequential growth (CSG) models, it has been conjectured that, up to sets of measure zero, the “stem sets” generate the full σ-algebra of label-invariant measurable sets of causal sets. We prove this for a generic family of CSG models (the “generalized percolation models”). In consequence, we are able not only to identify the “observables” of these theories, but, more importantly, to provide them with an accessible physical interpretation. We suggest that the stem sets will play the same role of fundamental observable in the quantum analog of these theories, i.e. for quantum gravity.

Morse index and causal continuity. A criterion for topology change in quantum gravity

Studies in 1+1 dimensions suggest that causally discontinuous topology changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n-1 Morse points in topology changing spacetimes built from Morse functions. We establish a weaker form of this conjecture. Namely, if a Morse function f on a compact cobordism has critical points of index 1 or n-1, then all the Morse geometries associated with f are causally discontinuous, while if f has no critical points of index 1 or n-1, then there exist associated Morse geometries which are causally continuous.Studies in 1 + 1 dimensions suggest that causally discontinuous topology-changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n - 1 Morse points in topology-changing spacetimes built from Morse functions. We establish a weaker form of this conjecture. Namely, if a Morse function f on a compact cobordism has critical points of index 1 or n - 1, then all the Morse geometries associated with f are causally discontinuous, while if f has no critical points of index 1 or n - 1, then there exist associated Morse geometries which are causally continuous.

CAUSAL CONTINUITY IN DEGENERATE SPACETIMES

A change of spatial topology in a causal, compact spacetime cannot occur when the metric is globally Lorentzian. One can however construct a causal metric from a Riemannian metric and a Morse function on the background cobordism manifold, which is Lorentzian almost everywhere except that it is degenerate at each critical point of the function. We investigate causal structure in the neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is taken to be Cartesian flat in appropriate coordinates. For these geometries, we verify Borde and Sorkins conjecture that causal discontinuity occurs if and only if the Morse index is 1 or n-1.A change of spatial topology in a causal, compact spacetime cannot occur when the metric is globally Lorentzian. On any cobordism manifold, however, one can construct from a Morse function f and an auxiliary Riemannian metric hµ a causal metric gµ which is Lorentzian almost everywhere except that it degenerates to zero at each critical point of f. We investigate causal structure in the neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is taken to be Cartesian flat in appropriate coordinates. For these geometries, we verify the conjecture that causal discontinuity occurs if and only if the Morse index is 1 or n - 1.

A handlebody calculus for topology change

We consider certain interesting processes in quantum gravity which involve a change of spatial topology. We use Morse theory and the machinery of handlebodies to characterize topology changes as suggested by Sorkin. Our results support the view that the pair production of Kaluza-Klein monopoles and the nucleation of various higher-dimensional objects are allowed transitions with non-zero amplitude.We consider certain interesting processes in quantum gravity which involve a change of spatial topology. We use Morse theory and the machinery of handlebodies to characterise topology changes as suggested by Sorkin. Our results support the view that that the pair production of Kaluza-Klein monopoles and the nucleation of various higher dimensional objects are allowed transitions with non-zero amplitude.

The distribution of visual objects on the retina: connecting eye movements and cone distributions

Experimental data on the accuracy and frequency of saccades are incorporated into a model of the visual world and eye movements to determine the spatial distribution of visual objects on the retina. Visual scenes are represented as sequences of discrete small objects whose positions are initially uniformly distributed and then moved toward the center of the retina by eye movements. We then use this model to investigate whether the distribution of cones in the retina maximizes the information transferred about object position. Assuming for simplicity that a single cone is activated by the object, the rate of information transfer is maximized at the receptor stage if the probability that a target lies at a position on the retina is proportional to the local cone density. Although qualitatively it is easy to understand why the cone density is higher at the fovea, by linking the cone density with eye movements through information sampling theory, we provide an explanation for its quantitative variation across the retina. The human cone distribution and the object distribution in our model visual world are shown to have the same general form and are in close agreement between 5- and 30-deg eccentricity.

Existence of spinorial states in pure loop quantum gravity

We demonstrate the existence of spinorial states in a kinematical theory of canonical quantum gravity without matter. This should be regarded as evidence in support of the conjecture that bound states with particle properties appear in association with spatial regions of non-trivial topology. In asymptotically trivial general relativity the momentum constraint generates only a subgroup of the spatial diffeomorphisms. The remaining diffeomorphisms give rise to the mapping class group, which acts as a symmetry group on the phase space. This action induces a unitary representation on the loop state space of the Ashtekar formalism. Certain elements of the diffeomorphism group can be regarded as asymptotic rotations of space relative to its surroundings. We construct states that transform non-trivially under a 2-rotation: gravitational quantum states with fractional spin.

Understanding cone distributions from saccadic dynamics. Is information rate maximised

There is not yet a fully satisfactory and quantitative explanation as to why human cones have their particular distribution on the retina. If this distribution is to maximise the rate of the information transfer, it should be proportional to the probability distribution of locations of attended objects. We derive this probability distribution from experiment data on eye movements to test this hypothesis, which provides a quantitative link between saccadic dynamics and the cone distribution. c