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

Codimension-2 surfaces and their Hilbert spaces: low-energy clues for holography from general covariance

We argue that the holographic principle may be hinted at already from low-energy considerations, assuming diffeomorphism invariance, quantum mechanics and Minkowski-like causality. We consider the states of finite spacelike hypersurfaces in a diffeomorphism-invariant QFT. A low-energy regularization is assumed. We note a natural dependence of the Hilbert space on a codimension-2 boundary surface. The Hilbert product is defined dynamically, in terms of transition amplitudes which are described by a path integral. We show that a canonical basis is incompatible with these assumptions, which opens the possibility for a smaller Hilbert-space dimension than canonically expected. We argue further that this dimension may decrease with surface area at constant volume, hinting at holographic area-proportionality. We draw comparisons with other approaches and setups, and propose an interpretation for the non-holographic space of graviton states at asymptotically-Minkowski null infinity.Comment: 13 pages, 9 eps figures. Added Section VI, improved presentation. Expanded and split the Introduction into two sections. Added Section VII. Added reference

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

A Bell Inequality Analog in Quantum Measure Theory

One obtains Bell's inequalities if one posits a hypothetical joint probability distribution, or {\it measure}, whose marginals yield the probabilities produced by the spin measurements in question. The existence of a joint measure is in turn equivalent to a certain causality condition known as ``screening off''. We show that if one assumes, more generally, a joint {\it quantal measure}, or ``decoherence functional'', one obtains instead an analogous inequality weaker by a factor of $\sqrt{2}$. The proof of this ``Tsirel'son inequality'' is geometrical and rests on the possibility of associating a Hilbert space to any strongly positive quantal measure. These results lead both to a {\it question}: ``Does a joint measure follow from some quantal analog of `screening off'?'', and to the {\it observation} that non-contextual hidden variables are viable in histories-based quantum mechanics, even if they are excluded classically.Comment: 38 pages, TeX. Several changes and added comments to bring out the meaning more clearly. Minor rewording and extra acknowledgements, now closer to published versio

Simulating causal collapse models

We present simulations of causal dynamical collapse models of field theories on a 1+1 null lattice. We use our simulations to compare and contrast two possible interpretations of the models, one in which the field values are real and the other in which the state vector is real. We suggest that a procedure of coarse graining and renormalising the fundamental field can overcome its noisiness and argue that this coarse grained renormalised field will show interesting structure if the state vector does on the coarse grained scale.Comment: 18 pages, 8 fugures, LaTeX, Reference added, discussion of probability distribution of labellings correcte

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

Time-Dependent Behavior of Linear Polarization in Unresolved Photospheres, With Applications for The Hanle Effect

Aims: This paper extends previous studies in modeling time varying linear polarization due to axisymmetric magnetic fields in rotating stars. We use the Hanle effect to predict variations in net line polarization, and use geometric arguments to generalize these results to linear polarization due to other mechanisms. Methods: Building on the work of Lopez Ariste et al., we use simple analytic models of rotating stars that are symmetric except for an axisymmetric magnetic field to predict the polarization lightcurve due to the Hanle effect. We highlight the effects for the variable line polarization as a function of viewing inclination and field axis obliquity. Finally, we use geometric arguments to generalize our results to linear polarization from the weak transverse Zeeman effect. Results: We derive analytic expressions to demonstrate that the variable polarization lightcurve for an oblique magnetic rotator is symmetric. This holds for any axisymmetric field distribution and arbitrary viewing inclination to the rotation axis. Conclusions: For the situation under consideration, the amplitude of the polarization variation is set by the Hanle effect, but the shape of the variation in polarization with phase depends largely on geometrical projection effects. Our work generalizes the applicability of results described in Lopez Ariste et al., inasmuch as the assumptions of a spherical star and an axisymmetric field are true, and provides a strategy for separating the effects of perspective from the Hanle effect itself for interpreting polarimetric lightcurves.Comment: 6 pages; 4 figures. Includes an extra figure found only in this preprint versio

Measurements of Sunyaev-Zel'dovich Effect Scaling Relations for Clusters of Galaxies

We present new measurements of the Sunyaev-Zel'dovich (SZ) effect from clusters of galaxies using the Sunyaev-Zel'dovich Infrared Experiment (SuZIE II). We combine these new measurements with previous cluster observations with the SuZIE instrument to form a sample of 15 clusters of galaxies. For this sample we calculate the central Comptonization, y, and the integrated SZ flux decrement, S, for each of our clusters. We find that the integrated SZ flux is a more robust observable derived from our measurements than the central Comptonization due to inadequacies in the spatial modelling of the intra-cluster gas with a standard Beta model. This is highlighted by comparing our central Comptonization results with values calculated from measurements using the BIMA and OVRO interferometers. On average, the SuZIE calculated central Comptonizations are approximately 60% higher in the cooling flow clusters than the interferometric values, compared to only approximately 12% higher in the non-cooling flow clusters. We believe this discrepancy to be in large part due to the spatial modelling of the intra-cluster gas. From our cluster sample we construct y-T and S-T scaling relations. The y-T scaling relation is inconsistent with what we would expect for self-similar clusters; however this result is questionable because of the large systematic uncertainty in the central Comptonization. The S-T scaling relation has a slope and redshift evolution consistent with what we expect for self-similar clusters with a characteristic density that scales with the mean density of the universe. We rule out zero redshift evolution of the S-T relation at 90% confidence.Comment: Accepted to Astrophysical Journal. 52 pages, 14 tables, 7 figures ;replaced to match ApJ accepted versio