Spatial Hypersurfaces in Causal Set Cosmology

Within the causal set approach to quantum gravity, a discrete analog of a spacelike region is a set of unrelated elements, or an antichain. In the continuum approximation of the theory, a moment-of-time hypersurface is well represented by an inextendible antichain. We construct a richer structure corresponding to a thickening of this antichain containing non-trivial geometric and topological information. We find that covariant observables can be associated with such thickened antichains and transitions between them, in classical stochastic growth models of causal sets. This construction highlights the difference between the covariant measure on causal set cosmology and the standard sum-over-histories approach: the measure is assigned to completed histories rather than to histories on a restricted spacetime region. The resulting re-phrasing of the sum-over-histories may be fruitful in other approaches to quantum gravity.Comment: Revtex, 12 pages, 2 figure

Quantum Dynamics without the Wave Function

When suitably generalized and interpreted, the path-integral offers an alternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one generalizes the path-integral-as-propagator to a {\it quantal measure} $\mu$ on the space $\Omega$ of all ``conceivable worlds'', and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such ``histories-based'' schemes new, and more ``realistic'' possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of $\mu$ in its sets of measure zero: such sets are to be ``precluded''. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of ``quantum logic'', would accommodate the contradictions by dualizing $\Omega$ to a space of ``co-events'' and effectively identifying reality with an element of this dual space.Comment: plainTeX, 24 pages, no figures. To appear in a special volume of {\it Journal of Physics A: Mathematical and General} entitled ``The Quantum Universe'' and dedicated to Giancarlo Ghirardi on the occasion of his 70th birthday. Most current version is available at http://www.physics.syr.edu/~sorkin/some.papers/ (or wherever my home-page may be

Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory

We present a computational tool that can be used to obtain the "spatial" homology groups of a causal set. Localisation in the causal set is seeded by an inextendible antichain, which is the analog of a spacelike hypersurface, and a one parameter family of nerve simplicial complexes is constructed by "thickening" this antichain. The associated homology groups can then be calculated using existing homology software, and their behaviour studied as a function of the thickening parameter. Earlier analytical work showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2d spacetimes our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2d and 3d simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is constant, or "stable" above the discreteness scale. Below this scale, the homology groups fluctuate rapidly as a function of the thickening parameter. This provides a necessary though not sufficient criterion to test for manifoldlikeness of a causal set.Comment: Latex, 46 pages, 43 .eps figures, v2 numerous changes to content and presentatio

Spacelike distance from discrete causal order

Any discrete approach to quantum gravity must provide some prescription as to how to deduce continuum properties from the discrete substructure. In the causal set approach it is straightforward to deduce timelike distances, but surprisingly difficult to extract spacelike distances, because of the unique combination of discreteness with local Lorentz invariance in that approach. We propose a number of methods to overcome this difficulty, one of which reproduces the spatial distance between two points in a finite region of Minkowski space. We provide numerical evidence that this definition can be used to define a `spatial nearest neighbor' relation on a causal set, and conjecture that this can be exploited to define the length of `continuous curves' in causal sets which are approximated by curved spacetime. This provides evidence in support of the ``Hauptvermutung'' of causal sets.Comment: 32 pages, 16 figures, revtex4; journal versio

Semiclassical Quantum Gravity: Obtaining Manifolds from Graphs

We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a unique manifold in an appropriate sense, and constructing the manifold when it exists. This problem arises in a variety of approaches to quantum gravity that assume a discrete structure at the fundamental level; the present work is motivated by the semiclassical sector of loop quantum gravity, so we will take the discrete structure to be a graph and the manifold to be a spatial slice in spacetime. We identify a class of graphs, those whose vertices have a fixed valence, for which such a construction can be specified. We define a procedure designed to produce a cell complex from a graph and show that, for graphs with which it can be carried out to completion, the resulting cell complex is in fact a PL-manifold. Graphs of our class for which the procedure cannot be completed either do not arise as edge graphs of manifold cell decompositions, or can be seen as cell decompositions of manifolds with structure at small scales (in terms of the cell spacing). We also comment briefly on how one can extend our procedure to more general graphs.Comment: 16 pages, 5 figure

Stable homology as an indicator of manifoldlikeness in causal set theory

We present a computational tool that can be used to obtain the \u27spatial\u27 homology groups of a causal set. Localization in the causal set is seeded by an inextendible antichain, which is the analogue of a spacelike hypersurface, and a one-parameter family of nerve simplicial complexes is constructed by \u27thickening\u27 this antichain. The associated homology groups can then be calculated using existing homology software, and their behaviour studied as a function of the thickening parameter. Earlier analytical work showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one-parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2D spacetimes, our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2D and 3D simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is constant, or \u27stable\u27, above the discreteness scale. Below this scale, the homology groups fluctuate rapidly as a function of the thickening parameter. This provides a necessary though not sufficient criterion to test for manifoldlikeness of a causal set