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R. D. Sorkin

Publications and source records attributed to R. D. Sorkin.

11 recordsLinked to original sources
arXiv2004

A Classical Sequential Growth Dynamics for Causal Sets

Starting from certain causality conditions and a discrete form of general covariance, we derive a very general family of classically stochastic, sequential growth dynamics for causal sets. The resulting theories provide a relatively accessible ``half way house'' to full quantum gravity that possibly contains the latter's classical limit (general relativity). Because they can be expressed in terms of state models for an assembly of Ising spins living on the relations of the causal set, these theories also illustrate how non-gravitational matter can arise dynamically from the causal set without having to be built in at the fundamental level. Additionally, our results bring into focus some interpretive issues of importance for causal set dynamics, and for quantum gravity more generally.

gr-qc↗
arXiv2004

The Random Walk in Generalized Quantum Theory

One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a certain condition of ``strong positivity'', the most general Markovian, translationally invariant ``decoherence functional'' with nearest neighbor transitions.

gr-qc↗
arXiv2001

Spin and statistics in quantum gravity

We present a review of the spin and statistics of topological geons, particles in 3+1 quantum gravity. They can have half-odd-integral spin and fermionic statistics and since the underlying gravitational field is tensorial and bosonic, this is an example of ``emergent'' non-trivial spin and statistics as displayed by familiar non-gravitating objects such as skyrmions. We give the topological background and show that in a ``canonical'' quantization of gravity there is no spin-statistics correlation for topological geons. Allowing the topology of space to change, for example in a sum-over-histories approach, raises the possibility that a spin-statistics correlation can be recovered for geons. We review a conjectured set of rules powerful enough to give such a spin-statistics correlation for all topological geons. These would appear to rule out the possibility of parastatistics and may rule out spinorial and fermionic geons altogether.

gr-qc↗
arXiv2000

Evidence for a continuum limit in causal set dynamics

We find evidence for a continuum limit of a particular causal set dynamics which depends on only a single ``coupling constant'' $p$ and is easy to simulate on a computer. The model in question is a stochastic process that can also be interpreted as 1-dimensional directed percolation, or in terms of random graphs.

gr-qc↗
arXiv2000

On the "renormalization" transformations induced by cycles of expansion and contraction in causal set cosmology

We study the ``renormalization group action'' induced by cycles of cosmic expansion and contraction, within the context of a family of stochastic dynamical laws for causal sets derived earlier. We find a line of fixed points corresponding to the dynamics of transitive percolation, and we prove that there exist no other fixed points and no cycles of length two or more. We also identify an extensive ``basin of attraction'' of the fixed points but find that it does not exhaust the full parameter space. Nevertheless, we conjecture that every trajectory is drawn toward the fixed point set in a suitably weakened sense.

gr-qc↗
arXiv1999

A Causal Order for Spacetimes with $C^0$ Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves

We recast the tools of ``global causal analysis'' in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call $K^+$, and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime $\M$ which is free of causal cycles, one may define a causal curve simply as a compact connected subset of $\M$ which is linearly ordered by $K^+$. Our definitions all make sense for arbitrary $C^0$ metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general $C^0$ metric, the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive energy theorem, and may also prove useful in the study of topology change. We have tried to make our treatment self-contained by including proofs of all the facts we use which are not widely available in reference works on topology and differential geometry.

gr-qc↗
arXiv1999

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 Sorkin's conjecture that causal discontinuity occurs if and only if the Morse index is 1 or n-1.

gr-qc↗
arXiv1996

A Spin-Statistics Theorem for Certain Topological Geons

We review the mechanism in quantum gravity whereby topological geons, particles made from non-trivial spatial topology, are endowed with nontrivial spin and statistics. In a theory without topology change there is no obstruction to ``anomalous'' spin-statistics pairings for geons. However, in a sum-over-histories formulation including topology change, we show that non-chiral abelian geons do satisfy a spin-statistics correlation if they are described by a wave function which is given by a functional integral over metrics on a particular four-manifold. This manifold describes a topology changing process which creates a pair of geons from $R^3$.

gr-qc↗
arXiv1993

A Positive Mass Theorem Based on the Focusing and Retardation of Null Geodesics

A positive mass theorem for General Relativity Theory is proved. The proof is 4-dimensional in nature, and relies completely on arguments pertaining to causal structure, the basic idea being that positive energy-density focuses null geodesics, and correspondingly retards them, whereas a negative total mass would advance them. Because it is not concerned with what lies behind horizons, this new theorem applies in some situations not covered by previous positivity theorems. Also, because geodesic focusing is a global condition, the proof might allow a generalisation to semi-classical gravity, even though quantum violations of local energy conditions can occur there.

gr-qc↗
arXiv1992

Topological Spin-Statistics Theorem for Strings

Recently, a topological proof of the spin-statistics Theorem has been proposed for a system of point particles which does not require relativity or field theory, but assumes the existence of antiparticles. We extend this proof to a system of string loops in three space dimensions and show that by assuming the existence of antistring loops, one can prove a spin-statistics theorem for these string loops. According to this theorem, all unparametrized strings(such as flux tubes in superconductors and cosmic strings ) should be quantized as bosons. Also, as in the point particle case, we find that the theorem excludes nonabelian statistics.

hep-th↗