Related papers: Emergence of Time from a Twisted Spectral Triple i…
This article demonstrates how the transition from a (Riemannian) twisted spectral triple to a pseudo-Riemannian spectral triple arises within an almost-commutative spectral triple. This opens a new perspective on the Lorentzian signature…
We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple…
With the bare essentials of noncommutative geometry (defined by a spectral triple), we first describe how it naturally gives rise to gauge theories. Then, we quickly review the notion of twisting (in particular, minimally) noncommutative…
We study a noncommutative analogue of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime…
The subject of this PhD thesis is noncommutative geometry - more specifically spectral triples - and how it can be generalized to semi-Riemannian manifolds generally, and Lorentzian manifolds in particular. The first half of this thesis…
In 1983, Hartle and Hawking proposed the no-boundary proposal, suggesting that the universe has no beginning in the sense of a spacetime singularity or boundary. Nevertheless, there is an origin of time. Mathematically, this involves…
Building upon previous works characterizing GRW space-times using concircular and torse-forming vectors, this paper investigates a Lorentzian manifold equipped with a concircularly semi-symmetric metric connection. We demonstrate that such…
In this short communication, we examine the relevance of the signature of the space-time metric in the construction of the product of a pseudo-Riemannian spectral triple with a finite triple describing the internal geometry. We obtain…
We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and…
In the early eighties Hartle and Hawking put forth that signature-type change may be conceptually interesting, paving the way to the so-called 'no boundary' proposal for the initial conditions for the universe. Such singularity-free…
We introduce the notion of a semi-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of semi-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in…
The theory of noncommutative geometry provides an interesting mathematical background for developing new physical models. In particular, it allows one to describe the classical Standard Model coupled to Euclidean gravity. However,…
An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and…
We show that the structure of an almost-commutative spectral triple emerges in a semi-classical limit from a geometric construction on a configuration space of gauge connections. The geometric construction resembles that of a spectral…
We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying, in the process, both concrete and abstract definitions of real commutative and…
The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures --such as triad and area operators-- exhibit a non-commutativity. At first…
A three-dimensional quasi-Fuchsian Lorentzian manifold $M$ is a globally hyperbolic spacetime diffeomorphic to $\Sigma\times (-1,1)$ for a closed orientable surface $\Sigma$ of genus $\geq 2$. It is the quotient $M=\Gamma\backslash…
This is a review of recent results regarding the application of Connes' noncommutative geometry to the Standard Model, and beyond. By twisting (in the sense of Connes-Moscovici) the spectral triple of the Standard Model, one does not only…
In the Hartle-Hawking ``no boundary'' approach to quantum cosmology, a real tunneling geometry is a configuration that represents a transition from a compact Riemannian spacetime to a Lorentzian universe. I complete an earlier proof that in…
In this document we present a twistor correspondence for half-flat almost-Grassmannian structures on real and complex manifolds. We provide foundational results regarding local theory in the complex setting and a global correspondence when…