- #1

tom.stoer

Science Advisor

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Except for the work of Torsten I am not aware of any paper which discusses these topics.

Some ideas:

1) all theories for QG I am aware of do either use manifolds and smootheness (string theory, geometrodynamics, shape dynamics, ...) or are constructed from them (LQG, CDT, ...)

2) in some theories the number of dimensions is fixed, whereas in other theories they become dynamical (spectral dimension)

3) we know that piecewise-wise linear and smooth manifolds are not necessarily the same

4) we know that for dim > 3 the classification of manifolds is not decidable, that there is no algorithm to distinguish between arbitrary manifolds, and that manifolds cannot be "listed"

5) we know that in dim = 4 there are in general uncountably many smoothness structures for a given (non-compact) manifold

5') however if we restrict the domain to manifolds which allow for a global foliation M

6) due to (4) we are not able to define a "sum over all manifolds" and due to (5) we are not able to define a "sum over all smoothness structures for a given manifold"

7) in all approaches I am aware of a kind of gauge fixing is required; therefore we need to know the diffeomorphism group / mapping class group and the fibre bundle topology (Gribov ambiguities) for local Lorentz invariance (for the connections and n-Beins)

Is there any paper discussing these topics and their relevance in the contex of QG?

Some ideas:

1) all theories for QG I am aware of do either use manifolds and smootheness (string theory, geometrodynamics, shape dynamics, ...) or are constructed from them (LQG, CDT, ...)

2) in some theories the number of dimensions is fixed, whereas in other theories they become dynamical (spectral dimension)

3) we know that piecewise-wise linear and smooth manifolds are not necessarily the same

4) we know that for dim > 3 the classification of manifolds is not decidable, that there is no algorithm to distinguish between arbitrary manifolds, and that manifolds cannot be "listed"

5) we know that in dim = 4 there are in general uncountably many smoothness structures for a given (non-compact) manifold

5') however if we restrict the domain to manifolds which allow for a global foliation M

^{4}= R * M^{3}then (3-5) become trivial6) due to (4) we are not able to define a "sum over all manifolds" and due to (5) we are not able to define a "sum over all smoothness structures for a given manifold"

7) in all approaches I am aware of a kind of gauge fixing is required; therefore we need to know the diffeomorphism group / mapping class group and the fibre bundle topology (Gribov ambiguities) for local Lorentz invariance (for the connections and n-Beins)

Is there any paper discussing these topics and their relevance in the contex of QG?

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