Spectral Quadruples

A set of data supposed to give possible axioms for spacetimes. It is hoped that such a proposal can serve to become a testing ground on the way to a general formulation. At the moment, the axioms are known to be sufficient for cases with a sufficient number of symmetries, in particular for 1+1 de Sitter spacetime. A major shortcoming of applications of noncommutative geometry to high energy physics (including gravity) so far has been the Euclidean formulation of such models. E.g., the fermion doubling in the noncomutative description of the standard model can be traced to the use of a metric with Euclidean signature and concepts like causality are needed for the formulation of realistic quantum field theories. Another important problem, closely connected with the one above is the absence of an action principle. One may formulate an " action " through a Lagrange density integrated over the entire (Euclidean) space [1]. For the moment that space can only be compact and boundary values cannot be incorporated. The solutions of the equations of motion (respectively the extramal points of the action) are then naturally non-unique. For a complete action principle, one should be able to give the values of the fields at on two arbitrary spacelike hypersurfaces, i.e. the initial and final field configuration, as boundary conditions under which the extremum of the action is to be found. This goal is at the moment still quite far away. Globally hyperbolic spacetimes of dimension d + 1 (physically, the Lorentz signi-ture is relevant only) may always be written as Σ × R, where Σ is a d-dimensional manifold. This corresponds to a foliation of the spacetime along the time axis R. The spacelike hypersurface Σ t is at each time homeomorphic to Σ. It is then at hand to deal with both of the above sketched problems by working with a (d + 1)-splitting of spacetime and to describe the spacelike hypersurfaces Σ t by spectral triples. The time coordinate t is then understood as a parameter and thus a whole family of spectral triples is obtained. However, attempts to transfor these ideas into an axiomatics for " noncommutative causal (Lorentzian) spin manifolds " have up to now had little success. One reason for this is certainly the technical difficulty of translating particular toy models into an algebraic language. As in the Euclidean case, also here the additional symmetries of …