Abstract
We develop here a very general ‘G-Act G-All’ implementation of Configurational Relationalism using group and fibre bundle mathematics. This refers to attaining group invariance by following group action by an operation using all of the group. It is useful in the study of configuration spaces, of which the freely available online Appendices G, H, I and N provide a wide range of foundational examples. Applications include both whole-universe modelling—of interest in foundational and theoretical physics—and Kendall-type shape statistics (which works for subsystems and is useful throughout the natural sciences).
‘G-Act G-All’ gives some indication of how much generality is required to remain inside one of the Problem of Time facet ‘gates’ during subsequent attempts to pass through further gates.
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Notes
- 1.
See Appendix F.4 for the definitions of \(\mathfrak{g}\)-fibre bundles and section.
- 2.
From here on, terming these ‘Metric Scale and Shape RPM’ and ‘Metric Shape RPM’ respectively helps get this point across while distinguishing them from yet further RPMs outlined below.
- 3.
\(\mathbb{S}^{3}\) is in many senses the simplest closed model for space. Its closest rival is the 3-torus \(\mathbb{T}^{3}\), which, due to arising from the simplest topological identification of \(\mathbb{R}^{3}\), is locally flat. However, this has less (global) Killing vectors, by 6 forming \(\mathit{SO}(4)\) to 3 forming \(\times_{i = 1}^{3} U(1)\).
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Anderson, E. (2017). Configuration Spaces and Their Configurational Relationalism. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_14
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