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Kontsevich’s formality theorem and the consequent star-product formula rely on the construction of an L ∞-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley–Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich’s proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich’s star-product is described. [ABSTRACT FROM AUTHOR]

It is shown that the algebra of diffeomorphism-invariant charges of the Nambu–Goto string cannot be quantized in the framework of canonical quantization. The argument is shown to be independent of the dimension of the underlying Minkowski space. [ABSTRACT FROM AUTHOR]

If a state is passive for uniformly accelerated observers in n-dimensional (n>=2) anti–de Sitter (Ads) space–time (i.e., cannot be used by them to operate a perpetuum mobile), they will (a) register a universal value of the Unruh temperature, (b) discover a PCT symmetry, and (c) find that observables in complementary wedge-shaped regions necessarily commute with each other in this state. The stability properties of such a passive state induce a “geodesic causal structure” on AdS and concommitant locality relations. It is shown that observables in these complementary wedge-shaped regions fulfill strong additional independence conditions. In two-dimensional AdS these even suffice to enable the derivation of a nontrivial, local, covariant net indexed by bounded space–time regions. All these results are model-independent and hold in any theory which is compatible with a weak notion of space–time localization. Examples are provided of models satisfying the hypotheses of these theorems. [ABSTRACT FROM AUTHOR]

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 $\backslash cal\{C\}^\{\backslash infty\}$-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 $\backslash cal\{C\}^\{\backslash infty\}$-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 $\backslash cal\{C\}^\{\backslash infty\}$-smooth space-time singularities. [ABSTRACT FROM AUTHOR]