arXiv subjects
Publications and source records attributed to Anastasios Mallios.
Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the classical differential geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to classical symplectic geometry. The full investigation will be presented elsewhere.
In the current debate referring to the construction of a tenable background independent theory of Quantum Gravity we introduce the notion of topos-theoretic relativization of physical representability and demonstrate its relevance concerning the merging of General Relativity and Quantum Theory. For this purpose we show explicitly that the dynamical mechanism of physical fields can be constructed by purely algebraic means, in terms of connection inducing functors and their associated curvatures, independently of any background substratum. The application of this mechanism in General Relativity is constrained by the absolute representability of the theory in the field of real numbers. The relativization of physical representability inside operationally selected topoi of sheaves forces an appropriate interpretation of the mechanism of connection functors in terms of a generalized differential geometric dynamics of the corresponding fields in the regime of these topoi. In particular, the relativization inside the topos of sheaves over commutative algebraic contexts makes possible the formulation of quantum gravitational dynamics by suitably adapting the functorial mechanism of connections inside that topos.
The smooth gravitational singularities of the differential spacetime manifold based General Relativity (GR) are viewed from the perspective of the background manifold independent and, in extenso, Calculus-free Abstract Differential Geometry (ADG). In particular, the inner Schwarzschild singularity is being `resolved' ADG-theoretically in two different ways. A plethora of important mathematical, physical and philosophical issues in current classical and quantum gravity research are addressed and tackled.
The paper concerns the fictitious entanglement of the so-called ``singularities'' in problems, pertaining to quantum gravity, due, in point of fact, to the way we try to employ, in that context, differential geometry, the latter being associated, in effect, by far, classically (:smooth manifolds), on the basis of an erroneous correspondence between what we may call/understand, as ``physical space'' and the ``cartesian-newtonian'' one.
The ``geometry'', in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG), along with resulting, thereby, potential physical consequences, in what, in particular, concerns physical ``gauge theories'', when the latter are viewed as being, anyway, of a ``geometrical character''. Yet, ``physical geometry'', in connection with physical laws and the associated with them, within the context of ADG, ``differential'' equations (whence, no background spacetime manifold is needed thereat), are also under discussion.
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 sheaf-theoretic 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 so-called ``geometric topological algebras'', having special cohomological properties, in terms of their ``Gel'fand sheaves'', being also of a particular significance for (abstract) differential-geometric 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 potential applications as well.
We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the so-called ``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 differential-geometric notions such as connections, curvature, metric and the like, retaining also the form of standard important relations of the classical theory (e.g. Einstein and/or Yang-Mills equations, in vacuum), even within that generalized context of ADG. To wind up, we can extend (in point of fact, {calculate) over singularities classical differential-geometric 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 quite a secondary role, not contributing at all (!) to the differential-geometric technique/mechanism that we apply. Thus, the latter by definition refers directly to the objects being involved--the objects that ``live on that space'', which by themselves are not, of course, ipso facto ``singular''!
We continue recent work 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 whole construction is `fully covariant', `inherently quantum' (both expressions are analytically explained in the paper) and genuinely smooth background spacetime independent.
Applying the classical Serre-Swan theorem, as this is extended to topological (non-normed) algebras, one attains a classification of elementary particles via their spin-structure. In this context, our argument is virtually based on a ``correspondence principle'' of S. A. Selesnick, formulated herewith in a sheaf-theoretic language, presisely speaking, in terms of vector sheaves. This then leads directly to second quantization, as well as, to other applications of geometric (pre)quantization theory.