Results 1  10
of
12
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 ..."
Abstract

Cited by 13 (10 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 8 (8 self)
 Add to MetaCart
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.
I.: Finitary Čechde Rham cohomology: much ado without C ∞  smoothness
 International Journal of Theoretical Physics
, 2001
"... Cordially dedicated to Jim Lambek, teacher, colleague and friend. The present paper comes to continue [40] and study the curved finitary spacetime sheaves of incidence algebras presented therein from a Čech cohomological perspective. In particular, we entertain the possibility of constructing a non ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
Cordially dedicated to Jim Lambek, teacher, colleague and friend. The present paper comes to continue [40] and study the curved finitary spacetime sheaves of incidence algebras presented therein from a Čech cohomological perspective. In particular, we entertain the possibility of constructing a nontrivial de Rham complex on these finite dimensional algebra sheaves along the lines of the first author’s axiomatic approach to differential geometry via the theory of vector and algebra sheaves [35, 36]. The upshot of this study is that important ‘classical ’ differential geometric constructions and results usually thought of as being intimately associated with C∞smooth manifolds carry through, virtually unaltered, to the finitaryalgebraic regime with the help of some quite universal, because abstract, ideas taken mainly from sheafcohomology as developed in [35, 36]. At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets [47, 40], we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity. 1 The general question motivating our quest
FinitaryAlgebraic ‘Resolution ’ of the Inner Schwarzschild Singularity
, 2004
"... A ‘resolution ’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a pointparticle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
A ‘resolution ’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a pointparticle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and, in extenso, Calculusfree purely algebraic (:sheaftheoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed via Sorkin’s finitary (:locally finite) poset substitutes of continuous manifolds in their Gel’fanddual picture in terms of discrete differential incidence algebras and the finitary spacetime sheaves thereof. It is shown that the Einstein equations hold not only at the finitary poset level of ‘discrete events’—as it were, when only finitely many ‘degrees of freedom ’ of the gravitational field are involved, so that no infinity or uncontrollable divergence of the latter arises at all in our inherently finitisticalgebraic scenario, let alone that the law of gravity—still modelled in ADG by a differential equation proper—breaks down in any (differential geometric) sense in the vicinity of the locus of the pointmass as it is currently maintained in the usual manifold based analysis of spacetime singularities in General Relativity (GR), but also that
C∞Smooth Singularities
 Chimeras of the Spacetime Manifold, in preparation
, 2001
"... Abstract. We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the socalled “singularities”, that emanated from looking at the theme, in terms of ADG (: abstract differential geometry). Thus, according to the latter perspective, we can involve “singularit ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract. We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the socalled “singularities”, that emanated from looking at the theme, in terms of ADG (: abstract differential geometry). Thus, according to the latter perspective, we can involve “singularities ” in our arguments, while still employing fundamental differentialgeometric notions, as connections, curvature, metric and the like, retaining also the form of standard important relations of the classical theory (e.g. Einstein and/or YangMills equations, in vacuum), even within that generalized context of ADG. To wind up, we can extend (in point of fact, calculate) over singularities classical differentialgeometric relations/equations, without altering their forms and/or changing the standard arguments; the change concerns thus only the way, we employ the usual differential geometry of smooth manifolds, so that the base “space ” acquires now a quite secondary rôle, not contributing, at all (!), to the differentialgeometric technique, we apply, the latter being thus, by definition, directly referred to the objects involved, that “live on the space”, not being, of course, i p s o f a c t o “singular”!
On geometric topological algebras
 J. Math. Anal. Appl
"... Abstract. Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the “structure sheaf algebras ” in many—in point of fact, in all—the situations of a geometrical character that occur, thus far, in several mathematical disciplines, as f ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Abstract. Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the “structure sheaf algebras ” in many—in point of fact, in all—the situations of a geometrical character that occur, thus far, in several mathematical disciplines, as for instance, differential and/or algebraic geometry, complex (geometric) analysis etc. It is proved that at the basis of this type of algebras lies the sheaftheoretic notion of (functional) localization, which, in the particular case of a given topological algebra, refers to the respective “Gel’fand transform algebra ” over the spectrum of the initial algebra. As a result, one further considers the socalled “geometric topological algebras”, having special cohomological properties, in terms of their “Gel’fand sheaves”, being also of a particular significance for (abstract) differentialgeometric applications; yet, the same class of algebras is still “closed”, with respect to appropriate inductive limits, a fact which thus considerably broadens the sort of the topological algebras involved, hence, as we shall see, their
Finitary, Causal and Quantal Vacuum Einstein Gravity
, 2002
"... We continue recent work [73, 74] and formulate the gravitational vacuum Einstein equations over a locally finite spacetime by using the basic axiomatics, techniques, ideas and working philosophy of Abstract Differential Geometry. The main kinematical structure involved, originally introduced and exp ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
We continue recent work [73, 74] and formulate the gravitational vacuum Einstein equations over a locally finite spacetime by using the basic axiomatics, techniques, ideas and working philosophy of Abstract Differential Geometry. The main kinematical structure involved, originally introduced and explored in [73], is a curved principal finitary spacetime sheaf of incidence algebras, which have been interpreted as quantum causal sets, together with a nontrivial locally finite spinLoretzian connection on it which lays the structural foundation for the formulation of a covariant dynamics of quantum causality in terms of sheaf morphisms. Our scheme is innately algebraic and it supports a categorical version of the principle of general covariance that is manifestly independent of a background C ∞smooth spacetime manifold M. Thus, we entertain the possibility of developing a ‘fully covariant’ path integraltype of quantum dynamical scenario for these connections that avoids ab initio various problems that such a dynamics encounters in other current quantization schemes for gravity—either canonical (Hamiltonian), or covariant (Lagrangian)—involving an external, base differential spacetime manifold, namely, the choice of a diffeomorphisminvariant
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity, submitted to the
 International Journal of Theoretical Physics
"... ..."
Sheafifying Consistent Histories
, 2001
"... Isham’s topostheoretic perspective on the logic of the consistenthistories theory [34] is extended in two ways. First, the presheaves of consistent sets of history propositions in their corresponding topos originally proposed in [34] are endowed with a Vietoristype of topology and subsequently the ..."
Abstract
 Add to MetaCart
Isham’s topostheoretic perspective on the logic of the consistenthistories theory [34] is extended in two ways. First, the presheaves of consistent sets of history propositions in their corresponding topos originally proposed in [34] are endowed with a Vietoristype of topology and subsequently they are sheafified with respect to it. The category resulting from this sheafification procedure is the topos of sheaves of sets varying continuously over the Vietoristopologized base poset category of Boolean subalgebras of the universal orthoalgebra UP of quantum history propositions. The second extension of the topos in [34] consists in endowing the stalks of the aforementioned sheaves, which were originally inhabited by structureless sets, with further algebraic structure