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A New Spectral Triple over a Space of Connections

A New Spectral Triple over a Space of Connections,10.1007/s00220-009-0758-8,Communications in Mathematical Physics,Johannes Aastrup,Jesper Møller Grim

A New Spectral Triple over a Space of Connections   (Citations: 10)
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A new construction of a semifinite spectral triple on an algebra of holonomy loops is presented. The construction is canonically associated to quantum gravity and is an alternative version of the spectral triple presented in [1].
Journal: Communications in Mathematical Physics - COMMUN MATH PHYS , vol. 290, no. 1, pp. 389-398, 2009
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    • ...Its existence - as a mathematical entity - was established in [4,5]...
    • ...In [5] it was clear that the construction would work for a large class of such systems of graphs and no...
    • ...In this section we outline the construction of the semi-finite spectral triple first presented in [3,4] and further developed in [5]...
    • ...However, for reasons explained in [5] it turns out that D1 and D � should of the form...
    • ...The change of variables in (8) is the key step to construct D� n . However, there will be many different partitions of the line segment which simplify the structure maps and lead to different Dirac type operators, see Fig. 4. This ambiguity was also commented on in [5]...

    Johannes Aastrupet al. On SemiClassical States of Quantum Gravity and Noncommutative Geometry

    • ...As a fully geometrical theory (the action depends only on the eigenvalues of the Dirac operator and so it is pure geometry and diffeomorphism invariant) its quantisation would involve quantum gravity in some sense [1]...
    • ...Aastrup, Grimstrup and Nest [1],[2] already established a link between Connes’s noncommutative geometry and loop quantum gravity...
    • ...Instead of ~ always taking a certain value, it is viewed as a continuous parameter taking values in an interval of the real line [0,1] and the classical limit is obtained as it ‘goes to zero’...

    R. A. DAWE MARTINS. CATEGORIFIED NONCOMMUTATIVE MANIFOLDS

    • ...As a fully geometrical theory (the action depends only on the eigenvalues of the Dirac operator and so it is pure geometry and diffeomorphism invariant) its quantisation would involve quantum gravity in some sense [1]...
    • ...Aastrup, Grimstrup and Nest [1],[2] already established a link between Connes’s noncommutative geometry and loop quantum gravity...
    • ...Instead of ~ always taking a certain value, it is viewed as a continuous parameter taking values in an interval of the real line [0,1] and the classical limit is obtained as it ‘goes to zero’...

    R. A. Dawe Martins. Some constructions in Category theory and Noncommutative geometry

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