Finitary, Causal and Quantal Vacuum Einstein Gravity Anastasios Mallios∗and Ioannis Raptis† Abstract We continue recent work [74, 75] 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 [74], is a curved principal finitary spacetime sheaf of incidence algebras, which have been interpreted as quantum causal sets, together with a non-trivial locally finite spin-Loretzian connection on it which lays the structural founda- tion 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 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 exter- nal, base differential spacetime 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 arXiv:gr-qc/0209048v6 19 Oct 2002 well as the ‘problem of time’ that also involves the Diff(M) ‘structure group’ of the classi- cal C∞-smooth spacetime 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 non-perturbative (canonical or covariant) approaches to quantum general relativity (eg, Ashtekar’s new vari- ables 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 conse- quences 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 C∞- smooth spacetime singularities, thus we prepare the ground for a forthcoming paper [76]. PACS numbers: 04.60.-m, 04.20.Gz, 04.20.-q Key words: quantum gravity, causal sets, differential incidence algebras of locally finite partially ordered sets, abstract differential geometry, sheaf theory, sheaf cohomology, category theory ∗Algebra and Geometry Section, Department of Mathematics, University of Athens, Panepistimioupolis Zo- grafou 157 84, Athens, Greece; e-mail: [email protected] †Theoretical Physics Group, Blackett Laboratory, Imperial College of Science, Technology and Medicine, Prince Consort Road, South Kensington, London SW7 2BZ, UK; e-mail: [email protected] 1 2 Anastasios Mallios and Ioannis Raptis “...the theory that space is continuous is wrong, because we get...infinities [viz. ‘singulari- ties’] and other similar difficulties ...[while] the simple ideas of geometry, extended down to infinitely small, are wrong...” [39] .................................. “...at the Planck-length scale, classical differential geometry is simply incompatible with quantum theory...[so that] one will not be able to use differential geometry in the true quantum-gravity theory...” [52] 1 Prologue cum Physical Motivation In the last century, the path that we have followed to unite quantum mechanics with general relativity into a coherent, both technically and conceptually, quantum theory of gravity has been a long and arduous one, full of unexpected twists and turns, surprising detours, branchings and loops—even disheartening setbacks and impasses, as well as hopes, disappointments or even dis- illusionments at times. Certainly though, the whole enterprize has been supported and nurtured by impressive technical ingenuity, and creative imagination coming from physicists and mathe- maticians alike. All in all, it has been a trip of adventure, discovery and intellectual reward for all who have been privileged to be involved in this formidable quest. Arguably then, the attempt to arrive at a conceptually sound and ‘calculationally’ finite quantum gravity must be regarded and hailed as one of the most challenging and inspired endeavors in theoretical physics research that must be carried over and be zestfully continued in the new millenium. Admittedly, however, a cogent theoretical scenario for quantum gravity has proved to be stubbornly elusive not least because there is no unanimous agreement about what ought to qualify as the ‘proper’ approach to a quantum theoresis of spacetime and gravity. Generally speaking, most of the approaches fall into the following three categories:1 1. ‘General relativity conservative’: The general aim of the approaches falling into this category is to quantize classical gravity somehow. Thus, the mathematical theory on which general relativity—in fact, any field theory whether classical or quantum—is based, namely, the differential geometry of C∞-manifolds (ie, the usual differential calculus on manifolds), is essentially retained2 and it is used to treat the gravitational field quantum field-theoretically. Both the non-perturbative canonical and covariant (ie, path integral or ‘action-weighed sum-over-histories’) approaches to ‘quantum general relativity’, topological quantum field theories, as well as, to a large extent, higher dimensional (or extended objects’) theories like (super)string and membrane schemes arguably belong to this category. 1These categories should by no means be regarded as being mutually exclusive or exhaustive, and they certainly reflect only these authors’ subjective criteria and personal perspective on the general characteristics of various approaches to quantum gravity. This coarse classification will be useful for the informal description of our finitary and causal approach to Lorentzian vacuum quantum gravity to be discussed shortly. 2That is, in general relativity spacetime is modelled after a C∞-smooth manifold. Purely mathematically speaking, approaches in this category could also be called ‘C∞-smoothness or differential manifold conservative’. Finitary, Causal and Quantal Vacuum Einstein Gravity 3 2. ‘Quantum mechanics conservative’: The general spirit here is to start from general quantum principles such as algebraic operationality, noncommutativity and finitism (‘discreteness’) about the structure of spacetime and its dynamics, and then try to derive somehow general relativistic attributes, as it were, from within the quantum framework. Such approaches assume up-front that quantum theory is primary and fundamental, while the classical ge- ometrical smooth spacetime continuum and its dynamics secondary and derivative (emer- gent) from the deeper quantum dynamical realm. For instance, Connes’ noncommutative geometry [24, 57] and, perhaps more notably, Finkelstein’s quantum relativity [45, 106]3 may be classified here. 3. ‘Independent’: Approaches in this category assume neither quantum mechanics nor general relativity as a fundamental, ‘fixed’ background theory relative to which the other must be modified to suit. Rather, they start independently from principles that are neither quantum mechanical nor general relativistic per se, and proceed to construct a theory and a suitable mathematical formalism to accompany it that later may be interpreted as a coherent amalgamation (or perhaps even extension) of both. It is inevitable with such ‘iconoclastic’ schemes that in the end both general relativity and quantum mechanics may appear to be modified to some extent. One could assign to this category Penrose’s combinatorial spin-networks [83, 101] and its current relativistic spin-foam descendants [13, 15, 84], Regge’s homological spacetime triangulations and simplicial gravity [94], as well as Sorkin et al.’s causal sets [18, 115, 116, 95, 117]. It goes without saying that this is no place for us to review in any detail the approaches mentioned above.4 Rather, we wish to continue a finitary, causal and quantal sheaf-theoretic approach to spacetime and vacuum Lorentzian gravity that we have already started to develop in [74, 75]. This approach, as we will argue subsequently, combines characteristics from all three categories above and, in particular, the mathematical backbone which supports it, Abstract Differential Geometry (ADG) [67, 68, 70, 71, 73], was originally conceived in order to evade the C∞-smooth spacetime manifold M (and consequently its diffeomorphism group Diff(M)) underlying (and creating numerous problems for) the various approaches in 1. For, it must be emphasized up- front, ADG is an axiomatic formulation of differential geometry which does not use any C∞-notion from the usual differential calculus—the classical differential geometry of smooth manifolds. To summarize briefly what we have already accomplished in this direction,5 in [74] we com- 3In fact, Finkelstein maintains that “all is quantum. Anything that appears to be classical has not yet been resolved into its quantum elements” (David Finkelstein in private communication). 4For reviews of and different perspectives on