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We use the vacuum Bianchi I model as an example to investigate the concept of physical evolution in Loop Quantum Cosmology (LQC) in the absence of the massless scalar field which has been used so far in the literature as an internal time.
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Similar ideas have been employed to discuss the relation between different gauges and define gauge invariants in perturbation theory. See D. Brizuela, J.M. Martin-Garcia, and G.A. Mena Marugan, Phys. Rev. D 76
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The invariance can be recovered by allowing an extra dependence of the μ ¯ \bar{\mu} -parameters of the holonomies on the (ratios) of the coordinate lengths of the elementary cell edges, but at the cost of losing an interpretation of the relation with the full theory of Loop Quantum Gravity
Cited in the paper.
On the contrary, the prescription adopted in Ref. [ 10 ] leads to the relation μ ¯ i μ ¯ j | p k | = 2 Δ \bar{\mu}_{i}\bar{\mu}_{j}|p_{k}|=2\Delta (such that ϵ i j k ≠ 0 \epsilon_{ijk}\neq 0 )
Cited in the paper.
In what follows, terms in the WDW quantization are underlined to distinguish them from their analogs in the loop quantization
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