Ioannis Raptis

Algebraic description of spacetime foam

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.

Finitary, Causal, and Quantal Vacuum Einstein Gravity

AbstractWe 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

Finitary Čech-de Rham Cohomology

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Spacetime Topology from the Tomographic Histories Approach: Part II

-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

Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold

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Quantum Space-Time as a Quantum Causal Set

-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., Ashtekars 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