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Epilogue II.C. Background Independence and Problem of Time at Deeper Levels of Structure

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The Problem of Time

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 190))

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Abstract

Humanity’s modelling of nature has hitherto almost always been at most dynamical at the metric and differentiable levels of structure. The current epilogue entertains Riemann’s curiosity for other possibilities through to Isham’s identification that Background Independence arguments in Quantum Gravity have little reason to stop at the metric and differentiable levels. We thus consider Background Independence also at the level of topological manifolds, topological spaces, metric spaces and sets. In particular, we show that the current book’s Background Independence and Problem of Time study additionally very largely extends to further levels of structure. We consider this firstly by replacing an upper level of structure with a deeper one, such as passing from differential geometry to purely topological manifolds. Secondly, by passing from an upper level of structure being dynamical to a tower of levels of structure being dynamical. Examples of the latter include passing from differential geometry to differential topology or from conventional canonical general relativity on a fixed spatial topological manifold to its counterpart admitting topology change. We end by posing many new research projects in this fascinating and mostly hitherto untapped field of study, and support this epilogue with the freely available online Appendices S and T on technicalities concerning the deeper levels of structure.

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Notes

  1. 1.

    This is left imprecise due to, for instance, stratified differentiable manifolds retaining differentiability in some neighbourhoods.

  2. 2.

    See [55] for limitations on tower versions of such formulations of Temporal Relationalism.

  3. 3.

    Antichains are subsets of a poset such that the elements within each of which bear no ordering relations. Maximal antichains are the largest possible ones, in some ways analogous to global slices or Cauchy surfaces in Geometrodynamics.

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  1. DAMTP, Centre for Mathematical Sciences, Cambridge, UK

    Edward Anderson

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Anderson, E. (2017). Epilogue II.C. Background Independence and Problem of Time at Deeper Levels of Structure. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_38

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