Toroidal Quantum Gravity I. Foundations

Abstract We develop a background-independent quantum spacetime picture (of relational origin) in which the basic kinematical objects are noncommutative (nc) tori equipped with spectral triples, which correspond to "quantum worldsheets', that classically are seen as infinitesimal spacetime regions with no internal structure unlike our case. Each micro process carries metric data via its Dirac operator and Weyl factor, so areas and distances are defined spectrally--without presupposing a manifold. Transitions between micro processes are controlled by the oriented couplings between sheets determined by cross-noncommutation and a SO(1,3)^+_{mathbb{Z}}-valued discrete holonomy. The construction is inspired by considerations on the canonical quantization of GR, as well as borrowing from other approaches, like LQG, causal sets, etc. In contrast to standard canonical quantization approaches, we take full advantage of modern formulations of the *covariant* Poisson bracket of GR. This allows us to incorporate temporal and causal notions in a natural way, something which is elusive in the former. The result concerning the discretization of time durations, among others, is thus a characteristic landmark of our approach. Furthermore, in our scheme we obtain a remarkable new interpretation of the transition time durations: they correspond to a purely nc part of the metric which is analogous (although, with some key differences) to the purely nc metric interpretation given to the Higgs field in the almost commutative standard model (the `tensor-product the algebra with the one from a two-points discrete space' paradigm), phenomenon which has no analogue in classical continuum metric geometry. In particular, no extraneous elements like extra dimensions, new particle types, or background geometries are assumed nor needed. The theory calls for a rich and diverse set of mathematical tools, thus offering an interesting playground to investigate their interactions. Physical applications will be covered in a second companion paper.