Sumati Surya

On recovering continuum topology from a causal set

An important question that discrete approaches to quantum gravity must address is how continuum features of space-time can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the space-time continuum is a locally finite partial order. A new topology on causal sets using “thickened antichains” is constructed. This topology is then used to recover the homology of a globally hyperbolic space-time from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or “Hauptvermutung” of causal set theory.

Localized Branes and Black Holes

We address the delocalization of low dimensional D-branes and NS-branes when they are a part of a higher dimensional BPS black brane, and the homogeneity of the resulting horizon. We show that the effective delocalization of such branes is a classical effect that occurs when localized branes are brought together. Thus, the fact that the few known solutions with inhomogeneous horizons are highly singular need not indicate a singularity of generic D- and NS-brane states. Rather, these singular solutions are likely to be unphysical as they cannot be constructed from localized branes which are brought together from a finite separation.

Spatial hypersurfaces in causal set cosmology

Within the causal set approach to quantum gravity, a discrete analogue 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 sequential 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.

Evidence for the continuum in 2D causal set quantum gravity

We present evidence for a continuum phase in a theory of 2D causal set quantum gravity which contains a dimensionless non-locality parameter ∈ (0, 1]. We also find a phase transition between this continuum phase and a new crystalline phase which is characterized by a set of covariant observables. For a fixed size of the causal set, the transition temperature β−1c decreases monotonically with . The locus of the transition in the β2 versus plane asymptotically approaches to the infinite temperature axis, suggesting that the continuum phase survives the analytic continuation.

On extending the quantum measure

We point out that a quantum system with a strongly positive quantum measure or decoherence functional gives rise to a vector-valued measure whose domain is the algebra of events or physical questions. This gives an immediate handle on the question of the extension of the decoherence functional to the sigma algebra generated by this algebra of events. It is on the latter that the physical transition amplitudes directly give the decoherence functional. Since the full sigma algebra contains physically interesting questions, like the return question, extending the decoherence functional to these more general questions is important. We show that the decoherence functional, and hence the quantum measure, extends if and only if the associated vector measure does. We give two examples of quantum systems whose decoherence functionals do not extend: one is a unitary system with finitely many states, and the other is a quantum sequential growth model for causal sets. These examples fail to extend in the formal mathematical sense and we speculate on whether the conditions for extension are unphysically strong.

Discrete geometry of a small causal diamond

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance hCkiof k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for hCki using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the hCki. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

Causal set topology

The Causal Set Theory (CST) approach to quantum gravity is motivated by the observation that, associated with any causal spacetime (M,g) is a poset (M,@?), with the order relation @? corresponding to the spacetime causal relation. Spacetime in CST is assumed to have a fundamental atomicity or discreteness, and is replaced by a locally finite poset, the causal set. In order to obtain a well defined continuum approximation, the causal set must possess the requisite intrinsic topological and geometric properties that characterise a continuum spacetime in the large. The study of causal set topology is thus dictated by the nature of the continuum approximation. We review the status of causal set topology and present some new results relating poset and spacetime topologies. The hope is that in the process, some of the ideas and questions arising from CST will be made accessible to the larger community of computer scientists and mathematicians working on posets.

Echoes of asymptotic silence in causal set quantum gravity

We explore the idea of asymptotic silence in causal set theory and find that causal sets approximated by continuum spacetimes exhibit behaviour akin to asymptotic silence. We make use of an intrinsic definition of spatial distance between causal set elements in the discrete analogue of a spatial hypersurface. Using numerical simulations for causal sets approximated by D=2,3 and 4 dimensional Minkowski spacetime, we show that while the discrete distance rapidly converges to the continuum distance at a scale roughly an order of magnitude larger than the discreteness scale, it is significantly larger on small scales. This allows us to define an effective dimension which exhibits dimensional reduction in the ultraviolet, while monotonically increasing to the continuum dimension with increasing continuum distance. We interpret these findings as manifestations of asymptotic silence in causal set theory.

Boundary terms for causal sets

We propose a family of boundary terms for the action of a causal set with a spacelike boundary. We show that in the continuum limit one recovers the Gibbons-Hawking-York boundary term in the mean. We also calculate the continuum limit of the mean causal set action for an Alexandrov interval in flat spacetime. We find that it is equal to the volume of the codimension-2 intersection of the two light-cone boundaries of the interval.

Boundary Term Contribution to the Volume of a Small Causal Diamond

In his calculation of the spacetime volume of a small Alexandrov interval in 4 dimensions Myrheim introduced a term which he referred to as a surface integral [1]. The evaluation of this term has remained opaque and led subsequent authors to evaluate the volume using other techniques [2]. It is the purpose of this work to demystify this integral. We point out that it arises from the difference in the flat spacetime volumes of the curved and flat spacetime intervals. An explicit evaluation using first order degenerate perturbation theory shows that it adds a dimension independent factor to the flat spacetime volume as the lowest order correction. Our analysis admits a simple extension to a more general class of integrals over the same domain. Using a combination of techniques we also find that the next order correction to the volume vanishes.