Morse Theory and the Topology of Configuration Space

The first and second homology groups are computed for configuration spaces of framed three-dimensional point particles with annihilation included, when up to two particles and an antiparticle are present

Topology Change and Causal Continuity

The result that, for a scalar quantum field propagating on a ``trousers'' topology in 1+1 dimensions, the crotch singularity is a source for an infinite burst of energy has been used to argue against the occurrence of topology change in quantum gravity. We draw attention to a conjecture due to Sorkin that it may be the particular type of topology change involved in the trousers transition that is problematic and that other topology changes may not cause the same difficulties. The conjecture links the singular behaviour to the existence of ``causal discontinuities'' in the spacetime and relies on a classification of topology changes using Morse theory. We investigate various topology changing transitions, including the pair production of black holes and of topological geons, in the light of these ideas.Comment: Latex, 28 pages, 10 figures, small changes in text (one figure removed), conclusions remain unchanged. Accepted for publication in Physical Review

Physical Logic

In R.D. Sorkin's framework for logic in physics a clear separation is made between the collection of unasserted propositions about the physical world and the affirmation or denial of these propositions by the physical world. The unasserted propositions form a Boolean algebra because they correspond to subsets of an underlying set of spacetime histories. Physical rules of inference, apply not to the propositions in themselves but to the affirmation and denial of these propositions by the actual world. This physical logic may or may not respect the propositions' underlying Boolean structure. We prove that this logic is Boolean if and only if the following three axioms hold: (i) The world is affirmed, (ii) Modus Ponens and (iii) If a proposition is denied then its negation, or complement, is affirmed. When a physical system is governed by a dynamical law in the form of a quantum measure with the rule that events of zero measure are denied, the axioms (i) - (iii) prove to be too rigid and need to be modified. One promising scheme for quantum mechanics as quantum measure theory corresponds to replacing axiom (iii) with axiom (iv) Nature is as fine grained as the dynamics allows.Comment: 14 pages, v2 published version with a change in the title and other minor change

A Causal Order for Spacetimes with C0C^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.Comment: Two small revisions to accomodate errors brought to our attention by R.S. Garcia. No change to chief results. 33 page

Scalar Field Theory on a Causal Set in Histories Form

We recast into histories-based form a quantum field theory defined earlier in operator language for a free scalar field on a background causal set. The resulting decoherence-functional resembles that of the continuum theory. The counterpart of the d'Alembertian operator is nonlocal and is a generalized inverse of the discrete retarded Green function. We comment on the significance of this and we also suggest how to include interactions.Comment: plainTeX, 25 pages, no figures. One paragraph added, other small changes. Most current version is available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/142.causet.dcf.pdf (or wherever my home-page may be, such as http://www.physics.syr.edu/~sorkin/some.papers/

A Lorentzian Gromov-Hausdoff notion of distance

This paper is the first of three in which I study the moduli space of isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I introduce a notion of Gromov-Hausdorff distance which makes this moduli space into a metric space. Further properties of this metric space are studied in the next papers. The importance of the work can be situated in fields such as cosmology, quantum gravity and - for the mathematicians - global Lorentzian geometry.Comment: 20 pages, 0 figures, submitted to Classical and quantum gravity, seriously improved presentatio

Stable non-uniform black strings below the critical dimension

The higher-dimensional vacuum Einstein equation admits translationally non-uniform black string solutions. It has been argued that infinitesimally non-uniform black strings should be unstable in 13 or fewer dimensions and otherwise stable. We construct numerically non-uniform black string solutions in 11, 12, 13, 14 and 15 dimensions. Their stability is investigated using local Penrose inequalities. Weakly non-uniform solutions behave as expected. However, in 12 and 13 dimensions, strongly non-uniform solutions appear to be stable and can have greater horizon area than a uniform string of the same mass. In 14 and 15 dimensions all non-uniform black strings appear to be stable.Comment: 26 pages, 11 figures. V2: reference added, matches published versio

Regulation of peripheral inflammation by spinal p38 MAP kinase in rats.

BackgroundSomatic afferent input to the spinal cord from a peripheral inflammatory site can modulate the peripheral response. However, the intracellular signaling mechanisms in the spinal cord that regulate this linkage have not been defined. Previous studies suggest spinal cord p38 mitogen-activated protein (MAP) kinase and cytokines participate in nociceptive behavior. We therefore determined whether these pathways also regulate peripheral inflammation in rat adjuvant arthritis, which is a model of rheumatoid arthritis.Methods and findingsSelective blockade of spinal cord p38 MAP kinase by administering the p38 inhibitor SB203580 via intrathecal (IT) catheters in rats with adjuvant arthritis markedly suppressed paw swelling, inhibited synovial inflammation, and decreased radiographic evidence of joint destruction. The same dose of SB203580 delivered systemically had no effect, indicating that the effect was mediated by local concentrations in the neural compartment. Evaluation of articular gene expression by quantitative real-time PCR showed that spinal p38 inhibition markedly decreased synovial interleukin-1 and -6 and matrix metalloproteinase (MMP3) gene expression. Activation of p38 required tumor necrosis factor alpha (TNFalpha) in the nervous system because IT etanercept (a TNF inhibitor) given during adjuvant arthritis blocked spinal p38 phosphorylation and reduced clinical signs of adjuvant arthritis.ConclusionsThese data suggest that peripheral inflammation is sensed by the central nervous system (CNS), which subsequently activates stress-induced kinases in the spinal cord via a TNFalpha-dependent mechanism. Intracellular p38 MAP kinase signaling processes this information and profoundly modulates somatic inflammatory responses. Characterization of this mechanism could have clinical and basic research implications by supporting development of new treatments for arthritis and clarifying how the CNS regulates peripheral immune responses

Quantum Reality Filters

An anhomomorphic logic \ascript ^* is the set of all possible realities for a quantum system. Our main goal is to find the "actual reality" \phi_a\in\ascript ^* for the system. Reality filters are employed to eliminate unwanted potential realities until only $\phi_a$ remains. In this paper, we consider three reality filters that are constructed by means of quantum integrals. A quantum measure $\mu$ can generate or actualize a \phi\in\ascript ^* if $\mu (A)$ is a quantum integral with respect to $\phi$ for a density function $f$ over events $A$. In this sense, $\mu$ is an "average" of the truth values of $\phi$ with weights given by $f$. We mainly discuss relations between these filters and their existence and uniqueness properties. For example, we show that a quadratic reality generated by a quantum measure is unique. In this case we obtain the unique actual quadratic reality.Comment: 25 page

Evidence for an entropy bound from fundamentally discrete gravity

The various entropy bounds that exist in the literature suggest that spacetime is fundamentally discrete, and hint at an underlying relationship between geometry and "information". The foundation of this relationship is yet to be uncovered, but should manifest itself in a theory of quantum gravity. We present a measure for the maximal entropy of spherically symmetric spacelike regions within the causal set approach to quantum gravity. In terms of the proposal, a bound for the entropy contained in this region can be derived from a counting of potential "degrees of freedom" associated to the Cauchy horizon of its future domain of dependence. For different spherically symmetric spacelike regions in Minkowski spacetime of arbitrary dimension, we show that this proposal leads, in the continuum approximation, to Susskind's well-known spherical entropy bound.Comment: 25 pages, 9 figures. Comment on Bekenstein bound added and smaller corrections. To be published in Class.Quant.Gra