We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a...

In this work we examine generalized Connes-Lott models, with C ⊕ C as finite algebra, over the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, wh...

We describe a higher-order spectral analysis-based approach for detecting people by recognizing human motion such as walking or running. The periodic attribute of human motion lends itself to efficient spectral inspection. In the proposed method, the stride length is determined in every frame as the image sequence evolves. The bispectrum which is the Fourier transform of the triple correlation ...

A result of Lewis on the extreme properties of the inner product of two vectors in a Cartan subspace of a semisimple Lie algebra is extended. The framework used is an Eaton triple which has a reduced triple. Applications are made for determining the minimizers and maximizers of the distance function considered by Chu and Driessel with spectral constraint.

We review the motivation, construction and physical interpretation of a semi-finite spectral triple obtained through a rearrangement of central elements of loop quantum gravity. The triple is based on a countable set of oriented graphs and the algebra consists of generalized holonomy loops in this set. The Dirac type operator resembles a global functional derivation operator and the interaction...

Let (A,H, D) be a spectral triple, namely: A is a C∗-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H, D) with additional properties which guaranty that the Con...

Through the example of the quantum symplectic 4-sphere, we discuss how the notion of twisted spectral triple fits into the framework of quantum homogeneous spaces.

Given a spectral triple of compact type with a real structure in the sense of [Da̧browski L., J. Geom. Phys. 56 (2006), 86–107] (which is a modification of Connes’ original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometrie...