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5.1 Symmetries and backgrounds

It is impossible to introduce symmetries in a completely background-independent manner. The mathematical action of a symmetry group is defined by a mapping between abstract points, which do not exist in a diffeomorphism-invariant setting (if one, for instance, considers only equivalence classes up to arbitrary diffeomorphisms).

More precisely, while the full theory has as background only a differentiable or analytic manifold Σ, a symmetric model has as background a symmetric manifold (Σ, S) consisting of a differentiable or analytic manifold Σ together with an action of a symmetry group S : Σ → Σ. How strong the additional structure is depends on the symmetry used. The strongest symmetry in gravitational models is realized with spatial isotropy, which implies a unique spatial metric up to a scale factor. The background is thus equivalent to a conformal space.

All constructions in a given model must take its symmetry into account since otherwise its particular dynamics, for instance, could not be captured. The structure of models thus depends on the different types of background realized for different symmetry groups. This can not only lead to simplifications but also to conceptual differences, and it is always instructive to keep a complete view on different models as well as the full theory. Since the loop formalism is general enough to encompass all relevant models, there are many ways to compare and relate different systems. It is thus possible to observe characteristic features of (metric) background independence even in cases where more structure is available.


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