Results 1  10
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26
Classical sequential growth dynamics for causal sets
 Physical Review D
, 2000
"... 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 ..."
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Cited by 46 (5 self)
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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 nongravitational 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. 1
Dual Formulation of Spin Network Evolution
, 1997
"... We illustrate the relationship between spin networks and their dual representation by labelled triangulations of space in 2+1 and 3+1 dimensions. We apply this to the recent proposal for causal evolution of spin networks. The result is labelled spatial triangulations evolving with transition amplitu ..."
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Cited by 26 (9 self)
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We illustrate the relationship between spin networks and their dual representation by labelled triangulations of space in 2+1 and 3+1 dimensions. We apply this to the recent proposal for causal evolution of spin networks. The result is labelled spatial triangulations evolving with transition amplitudes given by labelled spacetime simplices. The formalism is very similar to simplicial gravity, however, the triangulations represent combinatorics and not an approximation to the spatial manifold. The distinction between future and past nodes which can be ordered in causal sets also exists here. Spacelike and timelike slices can be defined and the foliation is allowed to vary. We clarify the choice of the two rules in the causal spin network evolution, and the assumption of trivalent spin networks for 2+1 spacetime dimensions and fourvalent for 3+1. As a direct application, the problem of the exponential growth of the causal model is remedied. The result is a clear and more rigid graphical...
Structural Issues in Quantum Gravity
, 1995
"... A discursive, nontechnical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of bo ..."
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Cited by 23 (1 self)
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A discursive, nontechnical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of both classical general relativity and standard quantum theory. This discussion is preceded by a short history of the last twentyfive years of research in quantum gravity, and concludes with speculations on what a future theory might look like.
Finitary Spacetime Sheaves of Quantum Causal Sets: Curving Quantum Causality, paper submitted to the International Journal of Theoretical Physics
, 2000
"... A locally finite, causal and quantal substitute for a locally Minkowskian principal fiber bundle P of modules of Cartan differential forms Ω over a bounded region X of a curved C ∞smooth differential manifold spacetime M with structure group G that of orthochronous Lorentz transformations L +: = SO ..."
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Cited by 19 (15 self)
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A locally finite, causal and quantal substitute for a locally Minkowskian principal fiber bundle P of modules of Cartan differential forms Ω over a bounded region X of a curved C ∞smooth differential manifold spacetime M with structure group G that of orthochronous Lorentz transformations L +: = SO(1,3) ↑, is presented. P is the structure on which classical Lorentzian gravity, regarded as a YangMills type of gauge theory of a sl(2, C)valued connection 1form A, is usually formulated. The mathematical structure employed to model this replacement of P is a principal finitary spacetime sheaf Pn of quantum causal sets Ωn with structure group Gn, which is a finitary version of the group G of local symmetries of General Relativity, and a finitary Lie algebra gnvalued connection 1form An on it, which is a section of its subsheaf Ω 1 n. An is physically interpreted as the dynamical field of a locally finite quantum causality, while its associated curvature Fn, as some sort of ‘finitary Lorentzian quantum gravity.
Quantum Measure Theory and its Interpretation
 Quantum Classical Correspondence: Proceedings of the 4 th Drexel Symposium on Quantum Nonintegrability, held Philadelphia, September 811
, 1994
"... We propose a realistic, spacetime interpretation of quantum theory in which reality constitutes a single history obeying a “law of motion ” that makes definite, but incomplete, predictions about its behavior. We associate a “quantum measure ” —S — to the set S of histories, and point out that —S — f ..."
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Cited by 17 (3 self)
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We propose a realistic, spacetime interpretation of quantum theory in which reality constitutes a single history obeying a “law of motion ” that makes definite, but incomplete, predictions about its behavior. We associate a “quantum measure ” —S — to the set S of histories, and point out that —S — fulfills a sum rule generalizing that of classical probability theory. We interpret —S— as a “propensity”, making this precise by stating a criterion for —S—=0 to imply “preclusion ” (meaning that the true history will not lie in S). The criterion involves triads of correlated events, and in application to electronelectron scattering, for example, it yields definite predictions about the electron trajectories themselves, independently of any measuring devices which might or might not be present. (In this way, we can give an objective account of measurements.) Two unfinished aspects of the interpretation involve
Algebraic description of spacetime foam
 Classical and Quantum Gravity
, 2001
"... A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its eve ..."
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Cited by 13 (10 self)
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A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its events by means of appropriate limit constructions. Physical Motivation In this paper we present an algebraic model of spacetime foam. The notion of spacetime foam has manifold and somewhat ambiguous meaning in the literature if only because the models vary. There is no unanimous agreement about what foam ‘really ’ pertains to mainly due to the fact that each of the mathematical models highlights different aspects of that concept. Here we use the term ‘foam ’ along the concrete but general lines originally introduced by Wheeler [42] who intended to refer to a spacetime with a dynamically variable, because quantally fluctuating, topology. The basic intuition is that at quantum scales even the topology of spacetime is subject to dynamics and interference. This conception of foam is in glaring contrast with general relativity, the classical theory of gravity, where spacetime is fixed to a topological manifold once and forever so that the sole dynamical variable is a higher level structure, namely, the spacetime geometry. It seems theoretically lame and rather ad hoc to regard the geometry of spacetime as being a dynamical variable that can in principle be measured (ie, an observable), while at the same time to think of its topology as a structure a priori fixed by the theoretician, an inert etherlike absolute background that is not liable to experimental investigation thus effectively an unobservable theoretical entity [9]. Especially in the quantum realm where everything seems
A new approach to quantising spacetime: I. Quantising on a general category
 Advances in Theoretical and Mathematical Physics
, 2003
"... A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for spacetime (or space). The starting point is the observation that entities of this type can be regarded as objects in a category whose arrows are structurepreserving maps. This motivates in ..."
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Cited by 11 (4 self)
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A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for spacetime (or space). The starting point is the observation that entities of this type can be regarded as objects in a category whose arrows are structurepreserving maps. This motivates investigating the general problem of quantising a system whose ‘configuration space ’ (or historytheory analogue) can be regarded as the set of objects Ob(Q) in a category Q. In this first of a series of papers, we study this question in general and develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold Q ≃ G/H where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of ‘arrow fields ’ on Q. Physically, this means that an arrow between two objects in the category is viewed as some sort of analogue of momentum. After finding the ‘category quantisation monoid’, we show how suitable representations can be constructed using a bundle of Hilbert spaces over Ob(Q). eprint archive:
Discrete Quantum Causal Dynamics
 International Journal of Theoretical Physics
, 2003
"... We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolut ..."
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Cited by 10 (5 self)
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We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences  such as covariance  follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1
Computational Complexity of Determining Which Statements about Causality Hold
 in Different SpaceTime Models”, Theoretical Computer Science
"... Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of spacetime. In this paper, we analyze the computational complexity of the corresponding deciding problems. In particular, we ..."
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Cited by 10 (7 self)
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Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of spacetime. In this paper, we analyze the computational complexity of the corresponding deciding problems. In particular, we show that: • for Minkowski spacetime, the deciding problem is as difficult as the Tarski’s problem of deciding elementary geometry, while • for a natural model of primordial spacetime, the corresponding deciding problem is of the lowest possible complexity. 1
NonCommutative Topology for Curved Quantum Causality
 International Journal of Theoretical Physics
"... A quantum causal topology is presented. This is modeled after a noncommutative scheme type of theory for the curved finitary spacetime sheaves of the nonabelian incidence Rota algebras that represent ‘gravitational quantum causal sets’. The finitary spacetime primitive algebra scheme structures fo ..."
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Cited by 8 (8 self)
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A quantum causal topology is presented. This is modeled after a noncommutative scheme type of theory for the curved finitary spacetime sheaves of the nonabelian incidence Rota algebras that represent ‘gravitational quantum causal sets’. The finitary spacetime primitive algebra scheme structures for quantum causal sets proposed here are interpreted as the kinematics of a curved and reticular local quantum causality. Dynamics for quantum causal sets is then represented by appropriate scheme morphisms, thus it has a purely categorical description that is manifestly ‘gaugeindependent’. Hence, a schematic version of the Principle of General Covariance of General Relativity is formulated for the dynamically variable quantum causal sets. We compare our noncommutative schemetheoretic curved quantum causal topology with some recent C ∗quantale models for nonabelian generalizations of classical commutative topological spaces or locales, as well as with some relevant recent results obtained from applying sheaf and topostheoretic ideas to quantum logic proper. Motivated by the latter, we organize our finitary spacetime primitive algebra schemes of curved quantum causal sets into a toposlike structure, coined ‘quantum topos’, and argue that it is a sound model of a structure that Selesnick has anticipated to underlie Finkelstein’s reticular and curved quantum causal net. At the end we conjecture that the fundamental quantum timeasymmetry that Penrose has expected to be the main characteristic of the elusive ‘true quantum gravity ’ is possibly of a kinematical or structural rather than of a dynamical character, and we also discuss the possibility of a unified description of quantum logic and quantum gravity in quantum topostheoretic terms.