We extend former results for coherent states on the circle in the literature in two ways. On the one hand, we show that expectation values of fractional powers of momentum operators can be computed exactly analytically by means of Kummer's confluent hypergeometric functions. Earlier, these expectation values have only been obtained by using suitable estimates. On the other hand, we consider the Zak transformation not only to map harmonic oscillator coherent states to coherent states on the circle as it has been discussed before, but we also use the properties of the Zak transformation to derive a relation between matrix elements with respect to coherent states in L 2 (Ê) and L 2 (S 1 ). This provides an alternative way for computing semiclassical matrix elements for coherent states on the circle. In certain aspects, this method simplifies the semiclassical computations in particular if one is only interested in the classical limit, that is the zeroth order term in the semiclassical expansion.
The relational formalism based on geometrical clocks and Dirac observables in linearized canonical cosmological perturbation theory is used to introduce an efficient method to find evolution equations for gauge invariant variables. Our method generalizes an existing technique by Pons, Salisbury and Sundermeyer [1, 2] to relate the evolution of gauge invariant observables with the one of gauge variant quantities, and is applied as a demonstration for the longitudinal and spatially flat gauges. Gauge invariant evolution equations for the Bardeen potential and the Mukhanov-Sasaki variable are derived in the extended ADM phase space. Our method establishes a full agreement at the dynamical level between the canonical and conventional cosmological perturbation theory at the linear order using Dirac observables. * kristina.giesel@gravity.fau.de † psingh@lsu.edu ‡ david.winnekens@fau.de 1 A path integral formulation of cosmological perturbation theory in reduced phase space was presented in [8].primary constraints. Given the relation in (A5), we obtainF ρλ J (y n+4 , y n+5 , y)}C ρ (y n+4 )Π λ (y n+5 ) + d 3 y n+4 d 3 y n+5 {G I (x),C ρ (y n+4 )} F ρλ J (y n+4 , y n+5 , y)Π λ (y n+5 )
Using a new procedure based on Kummer’s Confluent Hypergeometric Functions, we investigate the question of singularity avoidance in loop quantum gravity (LQG) in the context of U(1)3 complexifier coherent states and compare obtained results with already existing ones. Our analysis focuses on the dynamical operators, denoted by ˆqi0I0(r), whose products are the analogue of the inverse scale factor in LQG and also play a pivotal role for other dynamical operators such as matter Hamiltonians or the Hamiltonian constraint. For graphs of cubic topology and linear powers in ˆqi0I0(r), we obtain the correct classical limit and demonstrate how higher order corrections can be computed with this method. This extends already existing techniques in the way how the involved fractional powers are handled. We also extend already existing formalisms to graphs with higher-valent vertices. For generic graphs and products of ˆqi0I0(r), using estimates becomes inevitable and we investigate upper bounds for these semiclassical expectation values. Compared to existing results, our method allows to keep fractional powers involved in ˆqi0I0(r) throughout the computations, which have been estimated by integer powers elsewhere. Similar to former results, we find a non-zero upper bound for the inverse scale factor at the initial singularity. Additionally, our findings provide some insights into properties and related implications of the results that arise when using estimates and can be used to look for improved estimates.
I k B. Towards an improved estimate C. On new estimates D. Comparison with the Brunnemann and Thiemann estimates V. Conclusion and outlook A. The Poisson (re-)summation formula B. The 9 integrations C. Estimates D. Detailed derivation of the semiclassical continuum limit for graphs of cubic topology E. Comparison of the KCHF procedure and the one used by Sahlmann and Thiemann applied to the U(1) case References
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