arXiv subjects

Sumati Surya

Publications and source records attributed to Sumati Surya.

At least 37 records · Page 2Linked to original sources

arXiv2018

Scalar Field Green Functions on Causal Sets

We examine the validity and scope of Johnston's models for scalar field retarded Green functions on causal sets in 2 and 4 dimensions. As in the continuum, the massive Green function can be obtained from the massless one, and hence the key task in causal set theory is to first identify the massless Green function. We propose that the 2-d model provides a Green function for the massive scalar field on causal sets approximated by any topologically trivial 2 dimensional spacetime. We explicitly demonstrate that this is indeed the case in a Riemann normal neighbourhood. In 4-d the model can again be used to provide a Green function for the massive scalar field in a Riemann normal neighbourhood which we compare to Bunch and Parker's continuum Green function. We find that the same prescription can also be used for deSitter spacetime and the conformally flat patch of anti deSitter spacetime. Our analysis then allows us to suggest a generalisation of Johnston's model for the Green function for a causal set approximated by 3 dimensional flat spacetime.

gr-qc↗

arXiv2017

Finite Size Scaling in 2d Causal Set Quantum Gravity

We study the $N$-dependent behaviour of $\mathrm{2d}$ causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter $β$, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of $N$. Focussing on the $\mathrm{2d}$ causal set action $S$, we find that $β\langle S\rangle $ scales like $ N^ν$ where the scaling exponent $ν$ takes different values on either side of the phase transition. For $β> β_c$ we find that $ν=2$ which is consistent with our analytic predictions for a non-continuum phase in the large $β$ regime. For $β<β_c$ we find that $ν=0$, consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in $\mathrm{2d}$ causal set quantum gravity for $N \gtrsim 65$.

gr-qc↗

arXiv2017

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.

gr-qc↗

arXiv2017

Boundary and Corner Terms in the Action for General Relativity

We revisit the action principle for general relativity motivated by the path integral approach to quantum gravity. We consider a spacetime region whose boundary has piecewise $C^2$ components, each of which can be spacelike, timelike or null and consider metric variations in which only the pullback of the metric to the boundary is held fixed. Allowing all such metric variations we present a unified treatment of the spacelike, timelike and null boundary components using Cartan's tetrad formalism. Apart from its computational simplicity, this formalism gives us a simple way of identifying corner terms. We also discuss "creases" which occur when the boundary is the event horizon of a black hole. Our treatment is geometric and intrinsic and we present our results both in the computationally simpler tetrad formalism as well as the more familiar metric formalism. We recover known results from a simpler and more general point of view and find some new ones.

gr-qc↗

arXiv2015

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.

gr-qc↗

arXiv2015

The Hartle-Hawking wave function in 2d causal set quantum gravity

We define the Hartle-Hawking no-boundary wave function for causal set theory (CST) over the discrete analogs of spacelike hypersurfaces. Using Markov Chain Monte Carlo and numerical integration methods we analyse the wave function in non- perturbative 2d CST. We find that in the low temperature regime it is dominated by causal sets which have no continuum counterparts but possess physically interesting geometric properties. Not only do they exhibit a rapid spatial expansion with respect to the discrete proper time but also a high degree of spatial homogeneity. The latter is due to the extensive overlap of the causal pasts of the elements in the final discrete hypersurface and corresponds to high graph connectivity. Our results thus suggest new possibilities for the role of quantum gravity in the observable universe.

gr-qc↗

arXiv2015

Onset of the Asymptotic Regime for Finite Orders

We describe a Markov-Chain-Monte-Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one's choice. Implementing this algorithm for the uniform distribution, we explore the approach to the asymptotic regime in which almost every poset takes on the three-layer structure described by Kleitman and Rothschild (KR). By tracking the n-dependence of several order-invariants, among them the height of the poset, we observe an oscillatory behavior which is very unlike a monotonic approach to the KR regime. Only around n=40 or so does this "finite size dance" appear to give way to a gradual crossover to asymptopia which lasts until n=85, the largest n we have simulated.

math.CO↗

arXiv2014

Towards a Definition of Locality in a Manifoldlike Causal Set

It is a common misconception that spacetime discreteness necessarily implies a violation of local Lorentz invariance. In fact, in the causal set approach to quantum gravity, Lorentz invariance follows from the specific implementation of the discreteness hypothesis. However, this comes at the cost of locality. In particular, it is difficult to define a "local" region in a manifoldlike causal set, i.e., one that corresponds to an approximately flat spacetime region. Following up on suggestions from previous work, we bridge this lacuna by proposing a definition of locality based on the abundance of m-element order-intervals as a function of m in a causal set. We obtain analytic expressions for the expectation value of this function for an ensemble of causal set that faithfully embeds into an Alexandrov interval in d-dimensional Minkowski spacetime and use it to define local regions in a manifoldlike causal set. We use this to argue that evidence of local regions is a necessary condition for manifoldlikeness in a causal set. This in addition provides a new continuum dimension estimator. We perform extensive simulations which support our claims.

gr-qc↗

arXiv2012

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.

gr-qc↗

arXiv2012

The 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 of 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 the abundance 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 abundances of k-chains. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.

gr-qc↗

arXiv2011

Evidence for a Phase Transition in 2D Causal Set Quantum Gravity

We present evidence for a phase transition in a theory of 2D causal set quantum gravity which contains a dimensionless non-locality parameter $ε\in (0,1]$. The transition is between a continuum phase and a crystalline phase, characterised by a set of covariant observables. For a fixed size of the causal set the transition temperature $β_c^{-1}$ decreases monotonotically with $ε$. The line of phase transitions in the $β_c^2$ v/s $ε$ plane asymptotes to the infinite temperature axis, suggesting that the continuum phase survives the analytic continuation.

gr-qc↗

arXiv2011

Causal Topology in Future and Past Distinguishing Spacetimes

The causal structure of a strongly causal spacetime is particularly well endowed. Not only does it determine the conformal spacetime geometry when the spacetime dimension n >2, as shown by Malament and Hawking-King-McCarthy (MHKM), but also the manifold dimension. The MHKM result, however, applies more generally to spacetimes satisfying the weaker causality condition of future and past distinguishability (FPD), and it is an important question whether the causal structure of such spacetimes can determine the manifold dimension. In this work we show that the answer to this question is in the affirmative. We investigate the properties of future or past distinguishing spacetimes and show that their causal structures determine the manifold dimension. This gives a non-trivial generalisation of the MHKM theorem and suggests that there is a causal topology for FPD spacetimes which encodes manifold dimension and which is strictly finer than the Alexandrov topology. We show that such a causal topology does exist. We construct it using a convergence criterion based on sequences of "chain-intervals" which are the causal analogs of null geodesic segments. We show that when the region of strong causality violation satisfies a local achronality condition, this topology is equivalent to the manifold topology in an FPD spacetime.

gr-qc↗

arXiv2011

Directions in Causal Set Quantum Gravity

In the causal set approach to quantum gravity the spacetime continuum arises as an approximation to a fundamentally discrete substructure, the causal set, which is a locally finite partially ordered set. The causal set paradigm was elucidated in a classic paper by Bombelli, Lee, Meyer and Sorkin in 1987. While early kinematical results already showed promise, the program received a substantial impetus about a decade ago with the work of Rideout and Sorkin on a classical stochastic growth dynamics for causal sets. Considerable progress has been made ever since in our understanding of causal set theory while leaving undisturbed the basic paradigm. Recent highlights include a causal set expression for the Einstein-Hilbert action and the construction of a scalar field Feynman propagator on a fixed causal set. The aim of the present article is to give a broad overview of the results in causal set theory while pointing out directions for future investigations.

gr-qc↗

arXiv2010

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.

gr-qc↗

arXiv2009

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.

gr-qc↗

arXiv2008

Quantum Covers in Quantum Measure Theory

In standard measure theory the measure on the base set Omega is normalised to one, which encodes the statement that "Omega happens". Moreover, the rules imply that the measure of any subset A of Omega is strictly positive if and only if A cannot be covered by a collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation of a cover to quantum measure theory, the {\sl quantum cover}, which in addition to being a cover of A, satisfies the property that if every one of its elements has zero quantum measure, then so does A. We show that a large class of inextendible antichains in the associated powerset lattice provide quantum covers for Omega, for a quantum measure that derives from a strongly positive decoherence functional. Quantum covers, moreover, give us a new perspective on the Peres-Kochen-Specker theorem and its role in the anhomomorphic logic approach to quantum interpretation.

quant-ph↗

arXiv2008

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 continuum approximation thus sets the stage for the study of topology in CST. 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.

gr-qc↗

arXiv2008

A 2D model of Causal Set Quantum Gravity: The emergence of the continuum

Non-perturbative theories of quantum gravity inevitably include configurations that fail to resemble physically reasonable spacetimes at large scales. Often, these configurations are entropically dominant and pose an obstacle to obtaining the desired classical limit. We examine this "entropy problem" in a model of causal set quantum gravity corresponding to a discretisation of 2D spacetimes. Using results from the theory of partial orders we show that, in the large volume or continuum limit, its partition function is dominated by causal sets which approximate to a region of 2D Minkowski space. This model of causal set quantum gravity thus overcomes the entropy problem and predicts the emergence of a physically reasonable geometry.

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