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The tomographic histories approach is presented. As an inverse problem, we recover in an operational way the effective topology of the extended configuration space of a system. This means that from a series of experiments we get a set of points corresponding to events. The difference between effective and actual topology is drawn. We deduce the topology of the extended configuration space of a non-relativistic system, using certain concepts from the consistent histories approach to Quantum Mechanics, such as the notion of a record. A few remarks about the case of a relativistic system, preparing the ground for a forthcoming paper sequel to this, are made in the end.

The tomographic histories approach is presented. As an inverse problem, we recover in an operational way the effective topology of the extended configuration space of a system. This means that from a series of experiments we get a set of points corresponding to events. The difference between effective and actual topology is drawn. We deduce the topology of the extended configuration space of a non-relativistic system, using certain concepts from the consistent histories approach to Quantum Mechanics, such as the notion of a record. A few remarks about the case of a relativistic system, preparing the ground for a forthcoming paper sequel to this, are made in the end.

A synopsis-cum-update of work in the past half-decade or so on applying the algebraico-categorical concepts, technology and general philosophy of Abstract Differential Geometry (ADG) to various issues in current classical and quantum gravity research is presented. The exposition is mainly discursive, with conceptual, interpretational and philosophical matters emphasized throughout, while their formal technical-mathematical underpinnings have been left to the original papers. The general position is assumed that Quantum Gravity is in need of a new mathematical, novel physical concepts and principles introducing, framework in which old and current problems can be reformulated, readdressed and potentially retackled afresh. It is suggested that ADG can qualify as such a theoretical framework. © 2006 Springer Science+Business Media, Inc.

The following is a brief talk that opened and attempted to set the atmosphere for the first ‘Glafka–2004: Iconoclastic Approaches to Quantum Gravity’ international theoretical physics conference. It aimed to capture the general spirit of the meeting, as well as to inspire and unite its participants under the following envisioned ‘cause’: to bring together and scrutinize certain important current quantum gravity research approaches in a fresh, unconventional, almost unorthodox, way.

A ‘resolution’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a point-particle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and, in extenso, Differential Calculus-free purely algebraic (:sheaf-theoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed. As in previous works [Mallios, A. and Raptis, I. (2001). Finitary spacetime sheaves of quantum causal sets: Curving quantum causality. International Journal of Theoretical Physics, 40, 1885 [gr-qc/0102097]; Mallios, A. and Raptis, I. (2002). Finitary Cech-de Rham cohomology. International Journal of Theoretical Physics, 41, 1857 [gr-qc/0110033]; Mallios, A. and Raptis, I. (2003). Finitary, causal and quantal vacuum Einstein gravity. International Journal of Theoretical Physics42, 1479 [gr-qc/0209048]], which this paper continues, the starting point for the present application of ADG is Sorkin's finitary (:locally finite) poset (:partially ordered set) substitutes of continuous manifolds in their Gel'fand-dual 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,’ but also at a suitable ‘classical spacetime continuum limit’ of the said finitary sheaves and the associated differential triads that they define ADG-theoretically. The upshot of this is two-fold: On the one hand, the field equations are seen to hold when only finitely many events or ‘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 finitistic-algebraic scenario. On the other hand, the law of gravity—still modelled in ADG by a differential equation proper—does not break down in any (differential geometric) sense in the vicinity of the locus of the point-mass as it is traditionally maintained in the usual manifold-based analysis of spacetime singularities in General Relativity (GR). At the end, some brief remarks are made on the potential import of ADG-theoretic ideas in developing a genuinely background-independent Quantum Gravity (QG). A brief comparison between the ‘resolution’ proposed here and a recent resolution of the inner Schwarzschild singularity by Loop QG means concludes the paper.

We claim that both multipartiteness and localization of subsystems of compound quantum systems are of an essentially relative nature crucially depending on the set of operationalistically available states. In a more general setting, to capture the relativity and variability of our structures with respect to the observation means, sheaves of algebras may need be introduced. We provide the general formalism based on algebras which exhibits the relativity of multipartiteness and localization.

We continue recent work (Mallios and Raptis, International Journal of Theoretical Physics40, 1885, 2001; in press) and formulate the gravitational vacuum Einstein equations over a locally finite space-time by using the basic axiomatics, techniques, ideas, and working philosophy of Abstract Differential Geometry. The main kinematical structure involved, originally introduced and explored in (Mallios and Raptis, International Journal of Theoretical Physics40, 1885, 2001), is a curved principal finitary space-time sheaf of incidence algebras, which have been interpreted as quantum causal sets, together with a nontrivial locally finite spin-Loretzian 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 \(\mathcal{C}^\infty\)-smooth space-time manifold M. Thus, we entertain the possibility of developing a “fully covariant” path integral-type 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 space-time manifold, namely, the choice of a diffeomorphism-invariant measure on the moduli space of gauge-equivalent (self-dual) gravitational spin-Lorentzian connections and the (Hilbert space) inner product that could in principle be constructed relative to that measure in the quantum theory—the so-called “inner product problem,” as well as the “problem of time” that also involves the Diff(M) “structure group” of the classical \(\mathcal{C}^\infty\)-smooth space-time continuum of general relativity. Hence, by using the inherently algebraico—sheaf—theoretic and calculus-free ideas of Abstract Differential Geometry, we are able to draw preliminary, albeit suggestive, connections between certain nonperturbative (canonical or covariant) approaches to quantum general relativity (e.g., Ashtekar's new variables and the loop formalism that has been developed along with them) and Sorkin et al.'s causal set program. As it were, we “noncommutatively algebraize,” “differential geometrize” and, as a result, dynamically vary causal sets. At the end, we anticipate various consequences that such a scenario for a locally finite, causal and quantal vacuum Einstein gravity might have for the obstinate (from the viewpoint of the smooth continuum) problem of \(\mathcal{C}^\infty\)-smooth space-time singularities.

The present paper continues (Mallios & Raptis, International Journal of Theoretical Physics, 2001, 40, 1885) and studies the curved finitary spacetime sheaves of incidence algebras presented therein from a Cech 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 (Mallios, Geometry of Vector Sheaves: An Axiomtic Approach to Differential Geometry, Vols. 1–2, Kluwer, Dordrecht, 1998a; Mathematica Japonica (International Plaza), 1998b, 48, 93). 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 finitary-algebraic regime with the help of some quite universal, because abstract, ideas taken mainly from sheaf-cohomology as developed in Mallios (1998a,b). At the end of the paper, and due to the fact that the incidence algebras involved have been interpreted as quantum causal sets (Raptis, International Journal of Theoretical Physics, 2000, 39, 1233; Mallios & Raptis, 2001), we discuss how these ideas may be used in certain aspects of current research on discrete Lorentzian quantum gravity.

An algebraic quantization procedure for discretized spacetime models is suggestedbased on the duality between finitary substitutes and their incidence algebras.The provided limiting procedure that yields conventional manifold characteristicsof spacetime structures is interpreted in the quantum algebraic framework as acorrespondence principle.

The notion of finitary spacetime sheaves is introduced based on locally finiteapproximations of the continuous topology of a bounded region of a spacetimemanifold. Finitary spacetime sheaves are seen to be sound mathematical modelsof approximations of continuous spacetime observables.

A scheme for an algebraic quantization of the causal sets of Sorkin et al. ispresented. The suggested scenario is along the lines of a similar algebraizationand quantum interpretation of finitary topological spaces due to Zapatrin and thisauthor. To be able to apply the latter procedure to causal sets Sorkin's 'semanticswitch' from 'partially ordered sets as finitary topological spaces' to 'partiallyordered sets as locally finite causal sets' is employed. The result is the definition of'quantum causal sets'. Such a procedure and its resulting definition are physicallyjustified by a property of quantum causal sets that meets Finkelstein's requirementfor 'quantum causality' to be an immediate, as well as an algebraically represented,relation between events for discrete locality's sake. The quantum causal setsintroduced here are shown to have this property by direct use of a result fromthe algebraization of finitary topological spaces due to Breslav, Parfionov, andZapatrin.