Moduli of quantum Riemannian geometries on ⩽4 points

Majid, Shahn; Raineri, E.

doi:10.1063/1.1804231

Cited by 2 publications

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References 15 publications

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“…We have seen in Proposition 3.2 that following the edges of each colour (they will form non-intersecting loops) provides a basis {ω a }. These generalise the bases {e a } on a Cayley graph and it is shown in [12] that such a colouring has the interpretation of a vielbein of 'permutation type'. More of the geometry of such colourings is a direction for study elsewhere.…”

Section: 2mentioning

confidence: 99%

Noncommutative Riemannian geometry on graphs

Majid

2013

Journal of Geometry and Physics

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Abstract. We show that arising out of noncmmutatve geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a 'finite manifold structure' on the set of vertices. We equip any graph with a canonical 'Euclidean metric' and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.

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Section: 2mentioning

confidence: 99%

Noncommutative Riemannian geometry on graphs

Majid

2013

Journal of Geometry and Physics

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“…However, for quantum gravity we need to work with the entire moduli of geometries. This has been investigated in detail for small finite sets in [MR2] and we discuss the results now.…”

Section: Quantum Gravity On Finite Setsmentioning

confidence: 99%

“…Such an approach has been carried out in the U (1) not gravitational case in [M8]; the integrals over α, while still divergent, are now ordinary not usual functional integrals and the theory is renormalisable. The gravitational case has not been carried so far but the moduli of α and its curvature have been investigated in the simplest n = 2 cases [MR2]. For example, consider the square with type (ii) colouring.…”

Section: Quantum Gravity On Finite Setsmentioning

confidence: 99%

“…The full picture when we include varying λ a has additional terms involving their derivatives and a regularity condition crosscoupling the a, b systems (otherwise there is a kind of factorisation), see [MR2] for details but with some different notations. Note that the model in the type (ii) case is very different from the simplicial interpretation in the type (i) models, and indeed we don't appear to have a Gauss-Bonnet theorem in the naive sense of x R(x) = 0 for a torus.…”

Section: Quantum Gravity On Finite Setsmentioning

confidence: 99%

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Algebraic Approach to Quantum Gravity III: Non-Commmutative Riemannian Geometry

Majid

Quantum Gravity

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Abstract. This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that arises naturally as the classical limit; a theory with nonsymmetric metric and a skew version of metric compatibilty. Meanwhile, in quantum gravity a key ingredient of our approach is the proposal that the differential structure of spacetime is something that itself must be summed over or 'quantised' as a physical degree of freedom. We illustrate such a scheme for quantum gravity on small finite sets. IntroductionWhy is quantum gravity so hard? Surely it is because of its nonrenormalisable nature leading to UV divergences that cannot be tamed. However, UV divergences in quantum field theory arise from the assumption that the classical configurations being summed over are defined on a continuum. This is an assumption that is not based on observation but on mathematical constructs that were invented in conjunction with the classical geometry visible in the 19th century. A priori it would therefore be more reasonable to expect classical continuum geometry to play a role only in a macroscopic limit and not as a fundamental ingredient. While this problem might not be too bad for matter fields in a fixed background, the logical nonsensicality of putting the notion of a classical manifold that is supposed to emerge from quatum gravity as the starting point for quantum gravity inside the functional path integral, is more severe and this is perhaps what makes gravity special. The idea of course is to have the quantum theory centred on classical solutions but also to take into account nearby classical configurations with some weight. However, taking that as the actual definition is wishful and rather putting 'the cart before the horse'. Note that string theory also assumes a continuum for the strings to move in, so does not address the fundamental problem either.2000 Mathematics Subject Classification. 83C65, 58B32, 20C05,58B20, 83C27.

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Moduli of quantum Riemannian geometries on ⩽4 points

Cited by 2 publications

References 15 publications

Noncommutative Riemannian geometry on graphs

Noncommutative Riemannian geometry on graphs

Algebraic Approach to Quantum Gravity III: Non-Commmutative Riemannian Geometry

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