}, author = {Nekovar, Stefan and Sahlmann, Hanno}, faupublication = {yes}, peerreviewed = {automatic}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Gaussian} {Measures} and {Representations} of the {Holonomy}-{Flux} {Algebra}}, year = {2014} } @misc{faucris.111423664, author = {Zilker, Thomas and Thiemann, Thomas}, faupublication = {yes}, peerreviewed = {automatic}, title = {{Quantum} {Simplicity} {Constraints} and {Area} {Spectrum}}, year = {2010} } @masterthesis{faucris.119253244, author = {Zilker, Thomas and Giesel, Kristina}, faupublication = {yes}, peerreviewed = {automatic}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Manifestly} {Gauge} {Invariant} {Cosmological} {Perturbation} {Theory}}, year = {2013} } @misc{faucris.120355224, author = {Sahlmann, Hanno and Beier, Udo}, faupublication = {yes}, peerreviewed = {automatic}, title = {{Defects} as a model of spacetime foam - {Spinor} and vector structures on the defect}, year = {2015} } @masterthesis{faucris.108049744, abstract = {The dynamical laws of an evolving system determine how the system will look at later times, given some initial state the system started from. In quantum cosmology, one ap- plies this idea to the earliest moments of the universe, which requires one to have some idea about the intial conditions at the beginning of time. One proposal describing these initial conditions in the context of canonical quantum gravity is due to Hartle and Hawk- ing. Their proposal entails a preferred initial state for the universe which is based on a Euclidean path integral over all compact positive definite four-geometries which have a given three-geometry as a boundary. The wave function constructed this way satisfies the constraint equations of general relativity in ADM variables in a formal sense.

},
author = {Frembs, Markus and Sahlmann, Hanno},
faupublication = {yes},
peerreviewed = {automatic},
title = {{The} holonomy-flux algebra in low dimensions},
year = {2013}
}
@article{faucris.115364084,
abstract = {Quantum geometry, i.e., the quantum theory of intrinsic and extrinsic spatial geometry, is a cornerstone of loop quantum gravity. Recently, there have been many new ideas in this field, and I will review some of them. In particular, after a brief description of the main structures and results of quantum geometry, I review a new description of the quantized geometry in terms of polyhedra, new results on the volume operator, and a way to incorporate a classical background metric into the quantum description. Finally I describe a new type of exponentiated flux operator, and its application to Chern-Simons theory and black holes. © Published under licence by IOP Publishing Ltd.},
author = {Sahlmann, Hanno},
doi = {10.1088/1742-6596/360/1/012007},
faupublication = {no},
journal = {Journal of Physics : Conference Series},
peerreviewed = {No},
title = {{New} insights in quantum geometry},
volume = {360},
year = {2012}
}
@masterthesis{faucris.200383145,
abstract = {The objective of this Master’s thesis is to consider the well-known framework of Weyl algebras and

quasifree states in order to find out if it is possible to apply the general ideas, coming from algebraic

quantum theory, to the theory of loop quantum gravity.

Starting from the U(1) toy-model of the canonical commutation relation of the holonomy-flux

algebra, underlying loop quantum gravity, we construct a Weyl C^{*}-algebra generated by so-called

Weyl elements that arise from combining holonomies and exponentiated electric fluxes, which are

the canonically conjugated variables of the theory. Quasifree states are a certain notion of Gaussian

states, directly emerging from Weyl algebras. Because it seems to be impossible to establish such

states on the algebra we found, we develop a different notion states that is only Gaussian in one of

the variables and hence is referred to as almost-quasifree states. For such a state, which is Gaussian

in the fluxes, we find a representation on a Hilbert space that combines the Hilbert space of loop

quantum gravity with the Fock space of a scalar field.

For the canonical commutation relation of the actual theory, which involves SU(2) Yang-Mills

holonomies and the corresponding fluxes, we try to generalize our results. It is possible to define

Weyl-like elements for holonomies along a single path and a set of exponentiated fluxes. We work

toward a notion of elements that take care of more distinct edges or even graphs. It is, however,

not clear if these also generate a C*-algebra. Without an underlying Weyl algebra we successfully

generalize the almost-quasifree representation, found for the toy-model, and analyze its properties

by re-deriving the area operator of loop quantum gravity in this new representation.

String theory is one of the candidates for a theory that not only describes the nature of gravity at microscopic scales but also unites all fundamental forces into one common framework. This is possible by the simple assumption that all matter is given by small one-dimensional objects – strings – that may be open or closed.

Yet, an analysis of the theory suggests that in the quantum formulation the diffeomorphism in- variance of the theory seems to be broken in the sense that the constraints that generate the diffeomorphisms are only implemented weakly. Since the constraint algebra is the 1+1 dimen- sional version of the Dirac hypersurface deformation algebra of general relativity, the idea is that the physical subspace that corresponds to the kernel of the constraints are the diffeomorphism invariant states.

We therefore analyze the role of the hypersurface deformation algebra in the standard quantization of the bosonic string in further detail. It turns out that the spatial diffeomorphism constraint is anomaly-free, however, one also obtains that neither the physical states of the theory lie in the kernel of these constraints nor do they create physical states.

We analyze the action of the constraints in terms of the oscillations on the strings with an eye towards implementing spatial diffeos strongly. It turns out that the action is so complicated that we were unable to nd solutions.

Therefore, in a second part of the work we focus on a construction of a manifest diffeomorphism invariant quantization of the string in the spirit of previous work by Thiemann, i.e. the states are independent of the chosen parametrization. With the help of the Gelfand-Naimark-Segal construc- tion we obtain a Hilbert space and discuss a concrete representation that excites distinct points on the string. In the end we obtain a Fock space for a subalgebra of the mentioned representation and start a discussion of the diffeomorphism constraint and the mass within our framework.

The bosonic string also is a model for the hypersurface deformation algebra in loop quantum gravity and the issues of anomaly freeness. Thus our work might shed light on associated questions

},
author = {Wolz, Florian and Sahlmann, Hanno},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{On} spatially diffeomorphism invariant quantizations of the bosonic string},
year = {2015}
}
@article{faucris.201147872,
abstract = {We study the asymptotic behaviour of the vertex amplitude for the EPRL spin foam model extended to include timelike tetrahedra. We analyze both, tetrahedra of signature --- (standard EPRL), as well as of signature +-- (Hnybida-Conrady extension), in a unified fashion. However, we assume all faces to be of signature --. We find that the critical points of the extended model are described again by 4-simplices and the phase of the amplitude is equal to the Regge action. Interestingly, in addition to the Lorentzian and Euclidean sectors there appear also split signature 4-simplices.},
author = {Sahlmann, Hanno and Kaminski, Wojciech and Kisielowski, Marcin},
doi = {10.1088/1361-6382/aac6a4},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
keywords = {quantum gravity;spin foam;EPRL model;stationary phase analysis;large spin limit},
peerreviewed = {Yes},
title = {{Asymptotic} analysis of the {EPRL} model with timelike tetrahedra},
volume = {35},
year = {2018}
}
@article{faucris.120514944,
abstract = {A simple model is constructed which allows to compute modified dispersion relations with effects from loop quantum gravity. Different quantization choices can be realized and their effects on the order of corrections studied explicitly. A comparison with more involved semiclassical techniques shows that there is agreement even at a quantitative level. Furthermore, by contrasting Hamiltonian and Lagrangian descriptions we show that possible Lorentz symmetry violations may be blurred as an artifact of the approximation scheme. Whether this is the case in a purely Hamiltonian analysis can be resolved by an improvement in the effective semiclassical analysis. © 2005 The American Physical Society.},
author = {Sahlmann, Hanno and Bojowald, Martin and Morales-Tecotl, Hugo},
doi = {10.1103/PhysRevD.71.084012},
faupublication = {no},
journal = {Physical Review D - Particles, Fields, Gravitation and Cosmology},
pages = {1-7},
peerreviewed = {unknown},
title = {{Loop} quantum gravity phenomenology and the issue of {Lorentz} invariance},
volume = {71},
year = {2005}
}
@masterthesis{faucris.118093844,
abstract = {This thesis is devoted to the study of the quantum theory of charged black holes in the context of loop quantum gravity, extending the model of the quantum black hole as considered so far in the literature. We therefore consider Maxwell theory coupled to gravity de ned on a spacetime manifold with internal boundary described by an isolated horizon, construct the Hamiltonian formulation of the classical system, quantize the corresponding symplectic phase space and nally go over to the computation of the black hole entropy. We consider the spherically symmetric case in the U(1) framework as well as the distorted case following the SU(2) approach. The resulting picture depends signi cantly on the choices made for the quantization and the de nition of the state counting. We show that there is a choice such that the Bekenstein-Hawking relation holds. At the end, we use the theory in order to address the question whether there is a correspondence between the highly damped quasinormal modes and the area spectra of quantum charged black holes in the framework of loop quantum gravity. },
author = {Eder, Konstantin and Sahlmann, Hanno},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Quantum} theory of charged black hole horizons},
year = {2017}
}
@article{faucris.115355504,
abstract = {As a toy model for the implementation of the diffeomorphism constraint, the interpretation of the resulting states and the treatment of ordering ambiguities in loop quantum gravity, we consider the Hilbert space of spatially diffeomorphism-invariant states for a scalar field. We give a very explicit formula for the scalar product on this space and discuss its structure. Then we turn to the quantization of a certain class of diffeomorphism-invariant quantities on that space and discuss in detail the ordering issues involved. On a technical level these issues bear some similarity to those encountered in full loop quantum gravity. © 2007 IOP Publishing Ltd.},
author = {Sahlmann, Hanno},
doi = {10.1088/0264-9381/24/18/003},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {4601-4615},
peerreviewed = {Yes},
title = {{Exploring} the diffeomorphism-invariant {Hilbert} space of a scalar field},
volume = {24},
year = {2007}
}
@article{faucris.201148142,
abstract = {We describe the quantum theory of isolated horizons with electromagnetic or non-Abelian gauge charges in a setting in which both the gauge and gravitational field are quantized. We consider the distorted case, and its spherically symmetric limit. We show that the gravitational horizon d.o.f. give rise to the Bekenstein-Hawking relation, with lower-order terms giving some corrections for small black holes. We also demonstrate that one can include matter d.o.f. in the state counting. We show that one can expect (potentially divergent) contributions proportional to the area, as well as logarithmic corrections proportional to the horizon charge. This is qualitatively similar to results on matter contributions obtained with other methods in the literature.},
author = {Sahlmann, Hanno and Eder, Konstantin},
doi = {10.1103/PhysRevD.97.086016},
faupublication = {yes},
journal = {Physical Review D},
peerreviewed = {Yes},
title = {{Quantum} theory of charged isolated horizons},
volume = {97},
year = {2018}
}
@incollection{faucris.116261244,
abstract = {We introduce a notion of a weak Poisson structure on a manifold *M* modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra A⊆C∞(M)" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">A⊆C∞(M) which has to satisfy a non-degeneracy condition (the differentials of elements of A" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">A separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields. Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras. As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form *κ* on a Lie algebra g" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">g and a *κ*-skew-symmetric derivation *D* a weak affine Poisson structure on g" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">g itself. This leads naturally to a concept of a Hamiltonian *G*-action on a weak Poisson manifold with a g" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">g

-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.

}, author = {Neeb, Karl-Hermann and Thiemann, Thomas and Sahlmann, Hanno}, booktitle = {Springer Proceedings in Mathematics & Statistics}, editor = {V. Dobrev}, faupublication = {yes}, isbn = {978-4-431-55284-0}, pages = {105-136}, peerreviewed = {unknown}, publisher = {Springer Japan}, title = {{Weak} {Poisson} structures on infinite dimensional manifolds and hamiltonian actions}, url = {https://arxiv.org/abs/1402.6818}, volume = {111}, year = {2015} } @article{faucris.115349784, abstract = {We report on a new approach to the calculation of Chern-Simons theory expectation values, using the mathematical underpinnings of loop quantum gravity, as well as the Duflo map, a quantization map for functions on Liealgebras. These new developments can be used in the quantum theory for certain types of black hole horizons, and they may offer new insights for loop quantum gravity, Chern-Simons theory and the theory of quantum groups. © 2012 American Physical Society.}, author = {Sahlmann, Hanno and Thiemann, Thomas}, doi = {10.1103/PhysRevLett.108.111303}, faupublication = {yes}, journal = {Physical Review Letters}, peerreviewed = {Yes}, title = {{Chern}-simons expectation values and quantum horizons from loop quantum gravity and the duflo map}, volume = {108}, year = {2012} } @article{faucris.115336364, abstract = {Generalized connections and their calculus have been developed in the context of quantum gravity. Here we apply them to abelian Chern-Simons theory. We derive the expectation values of holonomies in U(1) Chern-Simons theory using Stokes' theorem, flux operators and generalized connections. A framing of the holonomy loops arises in our construction, and we show how, by choosing natural framings, the resulting expectation values nevertheless define a functional over gauge invariant cylindrical functions.The abelian theory considered in the present article is the test case for our method. It can also be applied to the non-abelian theory. Results will be reported in a companion article. © 2011 Elsevier B.V.}, author = {Sahlmann, Hanno and Thiemann, Thomas}, doi = {10.1016/j.geomphys.2011.10.012}, faupublication = {yes}, journal = {Journal of Geometry and Physics}, keywords = {Abelian Chern-Simons theory; Generalized connections; Loop quantum gravity}, pages = {204-212}, peerreviewed = {Yes}, title = {{Abelian} {Chern}-{Simons} theory, {Stokes}' theorem, and generalized connections}, volume = {62}, year = {2012} } @misc{faucris.201058655, author = {Wichert, Josef and Sahlmann, Hanno}, faupublication = {yes}, peerreviewed = {automatic}, title = {{What} does the {Penrose} operator measure in loop quantum gravity?}, year = {2016} } @article{faucris.108871884, abstract = {In the setting of vector-valued quantum fields obeying a linear wave-equation in a globally hyperbolic, stationary spacetime, it is shown that the two-point functions of passive quantum states (mixtures of ground- or KMS-states) fulfill the microlocal spectrum condition (which in the case of the canonically quantized scalar field is equivalent to saying that the two-pnt function is of Hadamard form). The fields can be of bosonic or fermionic character. We also give an abstract version of this result by showing that passive states of a topological *-dynamical system have an asymptotic pair correlation spectrum of a specific type.}, author = {Sahlmann, Hanno and Verch, Rainer}, faupublication = {no}, journal = {Communications in Mathematical Physics}, pages = {705-731}, peerreviewed = {Yes}, title = {{Passivity} and microlocal spectrum condition}, url = {https://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=0034343139&origin=inward}, volume = {214}, year = {2000} } @article{faucris.109996304, abstract = {The Hartle-Hawking state is a proposal for a preferred initial state for quantum gravity, based on a path integral over all compact Euclidean four-geometries which have a given three-geometry as a boundary. The wave function constructed this way satisfies the (Lorentzian) Hamiltonian constraint of general relativity in ADM variables in a formal sense. In this article, we address the question of whether this construction is dependent on the canonical variables used. We give a precise derivation of the properties of the Hartle-Hawking state in terms of formal manipulations of the path integral expressions. Then we mimic the construction in terms of Ashtekar-Barbero variables, and observe that the resulting wave function does not satisfy the Lorentzian Hamiltonian constraint even in a formal sense. We also investigate this issue for the relativistic particle, with a similar conclusion. We finally suggest a modification of the proposal that does satisfy the constraint at least in a formal sense and start to consider its implications in quantum cosmology. We find that for certain variables, and in the saddle point approximation, the state is very similar to the Ashtekar-Lewandowski state of loop quantum gravity. In the process, we have calculated the on-shell action for several cosmological models in connection variables.}, author = {Dhandhukiya, Satya and Sahlmann, Hanno}, doi = {10.1103/PhysRevD.95.084047}, faupublication = {yes}, journal = {Physical Review D}, peerreviewed = {unknown}, title = {{Towards} {Hartle}-{Hawking} states for connection variables}, volume = {95}, year = {2017} } @article{faucris.107360264, abstract = {The no-boundary wavefunction of quantum gravity usually assigns only very small probability to long periods of inflation. This was a reason to doubt about the no-boundary wavefunction to explain the observational universe. We study the no-boundary proposal in the context of multi-field inflation to see whether the number of fields changes the situation. For a simple model, we find that indeed the no-boundary wavefunction can give higher probability for sufficient inflation, but the number of fields involved N has to be very high, e.g., N ≃ m. © 2013 IOP Publishing Ltd.}, author = {Hwang, Dong-il and Kim, Soo A and Lee, Bum-Hoon and Sahlmann, Hanno and Yeom, Dong-han}, doi = {10.1088/0264-9381/30/16/165016}, faupublication = {yes}, journal = {Classical and Quantum Gravity}, peerreviewed = {Yes}, title = {{No}-boundary measure and preference for large e-foldings in multi-field inflation}, volume = {30}, year = {2013} } @article{faucris.120782464, abstract = {In this article, we investigate the assumption of equipartition of energy in arguments for the entropic nature of gravity. It has already been pointed out by other authors that equipartition is not valid for low temperatures. Here we additionally point out that it is similarly not valid for systems with bounded energy. Many explanations for black hole entropy suggest that the microscopic systems responsible have a finite dimensional state space, and thus finite maximum energy. Assuming this to be the case leads to drastic corrections to Newton's law for high gravitational fields, and, in particular, to a singularity in acceleration at finite radius away from a point mass. This is suggestive of the physics at the Schwarzschild radius. We show, however, that the location of the singularity scales differently. © 2011 American Physical Society.}, author = {Sahlmann, Hanno}, doi = {10.1103/PhysRevD.84.104010}, faupublication = {no}, journal = {Physical Review D - Particles, Fields, Gravitation and Cosmology}, peerreviewed = {Yes}, title = {{Energy} equipartition and minimal radius in entropic gravity}, volume = {84}, year = {2011} } @article{faucris.123560184, abstract = {The Duflo map is a valuable tool for operator ordering in contexts in which Kirillov-Kostant brackets and their quantizations play a role. A priori, the Duflo map is only defined on the subspace of the symmetric algebra over a Lie algebra consisting of elements invariant under the adjoint action. Here we discuss extensions to the whole symmetric algebra, as well as their application to the calculation of Chern-Simons theory expectation values.}, author = {Sahlmann, Hanno and Zilker, Thomas}, doi = {10.1016/j.geomphys.2017.07.022}, faupublication = {yes}, journal = {Journal of Geometry and Physics}, pages = {297 - 308}, peerreviewed = {Yes}, title = {{Extensions} of the {Duflo} map and {Chern}-{Simons} expectation values}, volume = {121}, year = {2017} } @article{faucris.123504524, abstract = {In the framework of loop quantum gravity, we define a new Hilbert space of states which are solutions of a large number of components of the diffeomorphism constraint. On this Hilbert space, using the methods of Thiemann, we obtain a family of gravitational scalar constraints. They preserve the Hilbert space for every choice of lapse function. Thus adjointness and commutator properties of the constraint can be investigated in a straightforward manner. We show how the space of solutions of the symmetrized constraint can be defined by spectral decomposition, and the Hilbert space of physical states by subsequently fully implementing the diffeomorphism constraint. The relationship of the solutions to those resulting from a proposal for a symmetric constraint operator by Thiemann remains to be elucidated.}, author = {Sahlmann, Hanno and Lewandowski, Jerzy}, doi = {10.1103/PhysRevD.91.044022}, faupublication = {yes}, journal = {Physical Review D - Particles, Fields, Gravitation and Cosmology}, peerreviewed = {unknown}, title = {{Symmetric} scalar constraint for loop quantum gravity}, volume = {91}, year = {2015} } @article{faucris.123191244, abstract = {Two-dimensional hard-particle systems are rather easy to simulate but surprisingly difficult to treat by theory. Despite their importance from both theoretical and experimental points of view, theoretical approaches are usually qualitative or at best semi-quantitative. Here, we present a density functional theory based on the ideas of fundamental measure theory for two-dimensional hard-disk mixtures, which allows for the first time an accurate description of the structure of the dense fluid and the equation of state for the solid phase within the framework of density functional theory. The properties of the solid phase are obtained by freely minimizing the functional. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3687921]}, author = {Roth, Roland and Mecke, Klaus and Oettel, Martin}, doi = {10.1063/1.3687921}, faupublication = {yes}, journal = {Journal of Chemical Physics}, keywords = {density functional theory;equations of state}, peerreviewed = {Yes}, title = {{Communication}: {Fundamental} measure theory for hard disks: {Fluid} and solid}, volume = {136}, year = {2012} } @article{faucris.115373104, abstract = {In a remarkable numerical analysis of the spectrum of states for a spherically symmetric black hole in loop quantum gravity, Corichi, Diaz-Polo and Fernandez-Borja found that the entropy of the black hole horizon increases in what resembles discrete steps as a function of area. In the present article we reformulate the combinatorial problem of counting horizon states in terms of paths through a certain space. This formulation sheds some light on the origins of this steplike behavior of the entropy. In particular, using a few extra assumptions we arrive at a formula that reproduces the observed step length to a few tenths of a percent accuracy. However, in our reformulation the periodicity ultimately arises as a property of some complicated process, the properties of which, in turn, depend on the properties of the area spectrum in loop quantum gravity in a rather opaque way. Thus, in some sense, a deep explanation of the observed periodicity is still lacking. © 2007 The American Physical Society.}, author = {Sahlmann, Hanno}, doi = {10.1103/PhysRevD.76.104050}, faupublication = {no}, journal = {Physical Review D - Particles, Fields, Gravitation and Cosmology}, peerreviewed = {Yes}, title = {{Toward} explaining black hole entropy quantization in loop quantum gravity}, volume = {76}, year = {2007} } @article{faucris.107356964, abstract = {In loop quantum gravity, states of the gravitational field turn out to be excitations over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also important to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.}, author = {Sahlmann, Hanno and et al.}, author_hint = {Koslowski T., Sahlmann H.}, doi = {10.3842/SIGMA.2012.026}, faupublication = {no}, journal = {Symmetry Integrability and Geometry-Methods and Applications}, keywords = {Geometric condensate; Loop quantum gravity; Representations}, peerreviewed = {Yes}, support_note = {Author relations incomplete. You may find additional data in field 'author_hint'}, title = {{Loop} quantum gravity vacuum with nondegenerate geometry}, volume = {8}, year = {2012} } @article{faucris.109458404, abstract = {We present an introduction to the canonical quantization of gravity performed in loop quantum gravity, based on lectures held at the 3rd quantum geometry and quantum gravity school in Zakopane in 2011. A special feature of this introduction is the inclusion of new proposals for coupling matter to gravity that can be used to deparametrize the theory, thus making its dynamics more tractable. The classical and quantum aspects of these new proposals are explained alongside the standard quantization of vacuum general relativity in loop quantum gravity.}, author = {Giesel, Kristina and Sahlmann, Hanno}, faupublication = {yes}, journal = {PoS - Proceedings of Science}, pages = {55}, peerreviewed = {Yes}, title = {{From} {Classical} {To} {Quantum} {Gravity}: {Introduction} to {Loop} {Quantum} {Gravity}}, volume = {C11-02-28}, year = {2011} } @masterthesis{faucris.201058905, author = {Wichert, Josef and Sahlmann, Hanno}, faupublication = {yes}, peerreviewed = {automatic}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{The} ideas of {Kaluza} and {Klein} in the context of loop quantum gravity}, year = {2018} } @masterthesis{faucris.200464729, abstract = {

The inflationary epoch at the beginning of the universe is commonly described within the frame- work of (linear) cosmological perturbation theory. The corresponding equation of motion for the gauge-invariant perturbations is the Mukhanov-Sasaki equation, which resembles a time-dependent harmonic oscillator. At first we will consider a mechanical analogue of the Mukhanov-Sasaki equa- tion and use the known Lewis-Riesenfeld invariant and the extended phase space formalism in- troduced in previous works in order to analyse the system. These techniques allow to construct an extended canonical transformation that maps an explicitly time-dependent Hamiltonian into a time-independent one. The generators of this symplectic map can in turn be canonically quan- tised on the original part of the phase space, which is the constraint hypersurface of the extended theory, connecting to recent publications. Our further analysis leads us to a closed form of the time-evolution operator for the single-mode Mukhanov-Sasaki Hamiltonian, that is to the associ- ated Dyson series. We will analyse the characteristic properties of this time-evolution operator and discuss whether it can be extended to the full Fock space. Finally we give an outlook towards possible applications of these techniques to inflationary quantum cosmology.