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@article{faucris.110388344,
abstract = {In her recent work, Dittrich generalized Rovelli's idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here We use this framework and propose how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to Compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables, so generated and interesting representations of the latter will be those for which a suitable preferred Subgroup is realized unitarily. We sketch how Such a programme might look for general relativity. We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for general relativity such that they are spatially diffeomorphism invariant. This has the important Consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed master constraint programme.},
author = {Thiemann, Thomas},
doi = {10.1088/0264-9381/23/4/006},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1163-1180},
peerreviewed = {Yes},
title = {{Reduced} phase space quantization and {Dirac} observables},
volume = {23},
year = {2006}
}
@article{faucris.123226004,
abstract = {The framework developed here is the classical cornerstone on which the semiclassical analysis in a new series of papers called `gauge theory coherent states' is based.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {3293-3338},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics ({QSD}): {VII}. {Symplectic} structures and continuum lattice formulations of gauge field theories},
volume = {18},
year = {2001}
}
@article{faucris.110416504,
abstract = {Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. In this paper we review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantization and, more specifically, with the master constraint quantization for first-class constraints. For first-class constraints with nontrivial structure functions the equivalence can only be established by passing to Abelian(ized) constraints which is always possible locally in phase space. Generically, the correct configuration space path integral measure deviates from the exponential of the Lagrangian action. The corrections are especially severe if the theory suffers from second-class secondary constraints. In a companion paper we compute these corrections for the Holst and Plebanski formulations of GR on which current spin foam models are based.},
author = {Han, Muxin and Thiemann, Thomas},
doi = {10.1088/0264-9381/27/22/225019},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{On} the relation between operator constraint, master constraint, reduced phase space and path integral quantization},
volume = {27},
year = {2010}
}
@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.110401104,
abstract = {We investigate a certain distributional extension of the group of spatial diffeomorphisms in loop quantum gravity. This extension, which is given by the automorphisms Aut(P) of the path groupoid P, was proposed by Velhinho and is inspired by category theory. These automorphisms have much larger orbits than piecewise analytic diffeomorphisms. In particular, we will show that graphs with the same combinatorics but different generalized knotting classes can be mapped into each other. We describe the automorphism-invariant Hilbert space and comment on how a combinatorial formulation of LQG might arise.},
author = {Bahr, Benjamin and Thiemann, Thomas},
doi = {10.1088/0264-9381/26/23/235022},
faupublication = {no},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Automorphisms} in loop quantum gravity},
volume = {26},
year = {2009}
}
@article{faucris.110405724,
abstract = {The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.},
author = {Thiemann, Thomas and Winkler, Oliver},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2561-2636},
peerreviewed = {Yes},
title = {{Gauge} field theory coherent states ({GCS}): {II}: {Peakedness} properties},
volume = {18},
year = {2001}
}
@article{faucris.110414744,
abstract = {We employ the techniques introduced in the companion papers (Bodendorfer et al 2011 arXiv: 1105.3703 [gr-qc]; arXiv: 1105.3704 [gr-qc]; arXiv: 1105.3705 [gr-qc]) to derive a connection formulation of Lorentzian general relativity coupled to Dirac fermions in dimensions D + 1 >= 3 with a compact gauge group. The technique that accomplishes that is similar to the one that has been introduced in 3+1 dimensions already. First one performs a canonical analysis of Lorentzian general relativity using the time gauge and then introduces an extension of the phase space analogous to the one employed in [1] to obtain a connection theory with SO(D + 1) as the internal gauge group subject to additional constraints. The success of this method rests heavily on the strong similarity of the Lorentzian and Euclidean Clifford algebras. A quantization of the Hamiltonian constraint is provided.},
author = {Bodendorfer, Norbert and Thiemann, Thomas and Thurn, Andreas},
doi = {10.1088/0264-9381/30/4/045004},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{New} variables for classical and quantum gravity in all dimensions: {IV}. {Matter} coupling},
volume = {30},
year = {2013}
}
@article{faucris.123221384,
abstract = {They turn out, as expected, to be non-local and form naturally a set of countable cardinality.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
month = {Jan},
pages = {59-88},
peerreviewed = {Yes},
title = {{COMPLETE} {QUANTIZATION} {OF} {A} {DIFFEOMORPHISM} {INVARIANT} {FIELD}-{THEORY}},
volume = {12},
year = {1995}
}
@article{faucris.123480104,
abstract = {We introduce a new top down approach to canonical quantum gravity, called algebraic quantum gravity (AQG). The quantum kinematics of AQG is determined by an abstract *-algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single master constraint operator. While AQG is inspired by loop quantum gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure. A natural Hilbert space representation acquires the structure of an infinite tensor product (ITP) whose separable strong equivalence class Hilbert subspaces (sectors) are left invariant by the quantum dynamics. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states. Given such data, the corresponding coherent state defines a sector in the ITP which can be identified with a usual QFT on the given manifold and background. Thus, AQG contains QFT on all curved spacetimes at once, possibly has something to say about topology change and provides the contact with the familiar low energy physics. In particular, in two companion papers we develop semiclassical perturbation theory for AQG and LQG and thereby show that the theory admits a semiclassical limit whose infinitesimal gauge symmetry agrees with that of general relativity. In AQG everything is computable with sufficient precision and no UV divergences arise due to the background independence of the fundamental combinatorial structure. Hence, in contrast to lattice gauge theory on a background metric, no continuum limit has to be taken. There simply is no lattice regulator that must be sent to zero.},
author = {Giesel, Kristina and Thiemann, Thomas},
doi = {10.1088/0264-9381/24/10/003},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2465-2497},
peerreviewed = {Yes},
title = {{Algebraic} quantum gravity ({AQG}): {I}. {Conceptual} setup},
volume = {24},
year = {2007}
}
@article{faucris.110416284,
abstract = {In this paper, we discuss several approaches to solve the quadratic and linear simplicity constraints in the context of the canonical formulations of higher dimensional general relativity and supergravity developed in our companion papers. Since the canonical quadratic simplicity constraint operators have been shown to be anomalous in any dimension D >= 3 in Class. Quantum Grav. 30 045003, non-standard methods have to be employed to avoid inconsistencies in the quantum theory. We show that one can choose a subset of quadratic simplicity constraint operators which are non-anomalous among themselves and allow for a natural unitary map of the spin networks in the kernel of these simplicity constraint operators to the SU(2)-based Ashtekar-Lewandowski Hilbert space in D = 3. The linear constraint operators on the other hand are non-anomalous by themselves; however, their solution space is shown to differ in D = 3 from the expected Ashtekar-Lewandowski Hilbert space. We comment on possible strategies to make a connection to the quadratic theory. Also, we comment on the relation of our proposals to the existing work in the spin foam literature and how these works could be used in the canonical theory. We emphasize that many ideas developed in this paper are certainly incomplete and should be considered as suggestions for possible starting points for more satisfactory treatments in the future.},
author = {Bodendorfer, Norbert and Thiemann, Thomas and Thurn, Andreas},
doi = {10.1088/0264-9381/30/4/045005},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{On} the implementation of the canonical quantum simplicity constraint},
volume = {30},
year = {2013}
}
@article{faucris.110404404,
abstract = {The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly infinite lattice.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2025-2064},
peerreviewed = {Yes},
title = {{Gauge} field theory coherent states ({GCS}): {I}. {General} properties},
volume = {18},
year = {2001}
}
@article{faucris.123570084,
abstract = {In this paper, we provide the techniques and proofs for the results presented in our companion paper concerning the consistency check on volume and triad operator quantization in loop quantum gravity.},
author = {Giesel, Kristina and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/18/012},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {5693-5771},
peerreviewed = {Yes},
title = {{Consistency} check on volume and triad operator quantization in loop quantum gravity: {II}},
volume = {23},
year = {2006}
}
@phdthesis{faucris.121044704,
author = {Thurn, Andreas and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Higher} {Dimensional} and {Supersymmetric} {Extensions} of {Loop} {Quantum} {Gravity}},
year = {2013}
}
@article{faucris.110426404,
abstract = {The loop transform in quantum gauge field theory can be recognized as the Fourier transform (or characteristic functional) of a measure on the space of generalized connections module gauge transformations. Since this space is a compact Hausdorff space, conversely, we know from the Riesz-Markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular Borel measure on the moduli space. In the present article we show how one can compute the finite joint distributions of a given characteristic functional, that is, we derive the inverse loop transform. (C) 1998 American Institute of Physics. [S002-2488(98)00302-8].},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {1236-1248},
peerreviewed = {Yes},
title = {{The} inverse loop transform},
volume = {39},
year = {1998}
}
@article{faucris.123917244,
abstract = {We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which an SU(2) connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities, and does not require any renormalization. We show that the length operator admits self-adjoint extensions and compute part of its spectrum which, like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly quantized in units of the Planck length. The length operator contains full and direct information about all the components of the metric tensor which facilitates the construction of so-called weave states which approximate a given classical three-geometry. (C) 1998 American Institute of Physics.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {3372-3392},
peerreviewed = {Yes},
title = {{A} length operator for canonical quantum gravity},
volume = {39},
year = {1998}
}
@article{faucris.110389444,
abstract = {We analyze the stability under time evolution of complexitier coherent states (CCS) in one-dimensional mechanical systems. A system of coherent states is called stable if it evolves into another coherent state. It turns out that a system can only possess stable CCS if the classical evolution of thc variable z = e-14(i xc)q pound for a given complexifier C depends only on z itself and not on its complex conjugate. This condition is very restrictive in general so that only a few systems exist that obey this condition. However, it is possible to access a wider class of models that in principle may allow for stable coherent states associated with certain regions in the phase space by introducing action-angle coordinates.},
author = {Zipfel, Antonia and Thiemann, Thomas},
doi = {10.1103/PhysRevD.93.084030},
faupublication = {yes},
journal = {Physical Review D},
peerreviewed = {unknown},
title = {{Stable} coherent states},
volume = {93},
year = {2016}
}
@article{faucris.110382404,
abstract = {The present first paper aims at clarifying the classical structures that underlies this formalism, namely projective limits of symplectic manifolds [27, subsection 2.1]. In particular, this allows us to discuss accurately the issues hindering an easy implementation of the dynamics in this context, and to formulate a strategy for overcoming them [27, subsection 4.1]. (C) 2016 Elsevier B.V. All rights reserved.},
author = {Lanery, Suzanne and Thiemann, Thomas},
doi = {10.1016/j.geomphys.2016.10.010},
faupublication = {yes},
journal = {Journal of Geometry and Physics},
keywords = {Field theory;Projective limits;Symplectic geometry;Algebras of observables;Constrained Hamiltonian systems},
month = {Jan},
pages = {6-39},
peerreviewed = {Yes},
title = {{Projective} limits of state spaces {I}. {Classical} formalism},
volume = {111},
year = {2017}
}
@phdthesis{faucris.111430484,
author = {Bodendorfer, Norbert and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Loop} {Quantization} of {Supergravity} {Theories}},
year = {2013}
}
@article{faucris.110370964,
abstract = {This article, as the first of three, aims at establishing the (time-dependent) Born-Oppenheimer approximation, in the sense of space adiabatic perturbation theory, for quantum systems constructed by techniques of the loop quantum gravity framework, especially the canonical formulation of the latter. The analysis presented here fits into a rather general framework and offers a solution to the problem of applying the usual Born-Oppenheimer ansatz for molecular (or structurally analogous) systems to more general quantum systems (e.g., spin-orbit models) by means of space adiabatic perturbation theory. The proposed solution is applied to a simple, finite dimensional model of interacting spin systems, which serves as a non-trivial, minimal model of the aforesaid problem. Furthermore, it is explained how the content of this article and its companion affect the possible extraction of quantum field theory on curved spacetime from loop quantum gravity (including matter fields). Published by AIP Publishing.},
author = {Stottmeister, Alexander and Thiemann, Thomas},
doi = {10.1063/1.4954228},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
peerreviewed = {Yes},
title = {{Coherent} states, quantum gravity, and the {Born}-{Oppenheimer} approximation. {I}. {General} considerations},
volume = {57},
year = {2016}
}
@article{faucris.110372284,
abstract = {In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the proposed Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity. Published by AIP},
author = {Stottmeister, Alexander and Thiemann, Thomas},
doi = {10.1063/1.4960823},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
peerreviewed = {Yes},
title = {{Coherent} states, quantum gravity, and the {Born}-{Oppenheimer} approximation. {III}.: {Applications} to loop quantum gravity},
volume = {57},
year = {2016}
}
@article{faucris.123835184,
abstract = {Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosmological experimental data provided e. g. by cosmic microwave anisotropy probes. A key issue in that theory is to extract the gauge-invariant degrees of freedom which allow unambiguous comparison between theory and experiment. When one goes beyond first (linear) order, the task of writing the Einstein equations expanded to nth order in terms of quantities that are gauge-invariant up to terms of higher orders becomes highly non-trivial and cumbersome. This fact has prevented progress for instance on the issue of the stability of linear perturbation theory and is a subject of current debate in the literature. In this series of papers we circumvent these difficulties by passing to a manifestly gauge-invariant framework. In other words, we only perturb gauge-invariant, i.e. measurable quantities, rather than gauge variant ones. Thus, gauge invariance is preserved non-perturbatively while we construct the perturbation theory for the equations of motion for the gauge-invariant observables to all orders. In this first paper we develop the general framework which is based on a seminal paper due to Brown and Kuchar as well as the relational formalism due to Rovelli. In the second, companion, paper we apply our general theory to FRW cosmologies and derive the deviations from the standard treatment in linear order. As it turns out, these deviations are negligible in the late universe, thus our theory is in agreement with the standard treatment. However, the real strength of our formalism is that it admits a straightforward and unambiguous, gauge-invariant generalization to higher orders. This will also allow us to settle the stability issue in a future publication.},
author = {Giesel, Kristina and Hofmann, Stefan and Thiemann, Thomas and Winkler, Oliver},
doi = {10.1088/0264-9381/27/5/055005},
faupublication = {no},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Manifestly} gauge-invariant general relativistic perturbation theory: {I}. {Foundations}},
volume = {27},
year = {2010}
}
@article{faucris.115353084,
abstract = {We consider a novel derivation of the expectation values of holonomies in Chern-Simons theory, based on Stokes' Theorem and the functional properties of the Chern-Simons action. It involves replacing the connection by certain functional derivatives under the path integral. It turns out that ordering choices have to be made in the process, and we demonstrate that, quite surprisingly, the Duflo isomorphism gives the right ordering, at least in the simple cases that we consider. In this way, we determine the expectation values of unknotted, but possibly linked, holonomy loops for SU(2) and SU(3), and sketch how the method may be applied to more complicated cases. Our manipulations of the path integral are formal but well motivated by a rigorous calculus of integration on spaces of generalized connections which has been developed in the context of loop quantum gravity. © 2011 Elsevier B.V.},
author = {Sahlmann, Hanno and Thiemann, Thomas},
doi = {10.1016/j.geomphys.2011.02.013},
faupublication = {yes},
journal = {Journal of Geometry and Physics},
keywords = {Chern-Simons theory; Duflo map; Loop quantum gravity},
pages = {1104-1121},
peerreviewed = {Yes},
title = {{Chern}-{Simons} theory, {Stokes}' theorem, and the {Duflo} map},
volume = {61},
year = {2011}
}
@inproceedings{faucris.108293944,
author = {Thiemann, Thomas},
faupublication = {no},
month = {Jan},
pages = {585-587},
peerreviewed = {unknown},
title = {{CANONICAL} {QUANTIZATION} {OF} {A} {MINISUPERSPACE} {MODEL} {FOR} {GRAVITY} {USING} {SELF}-{DUAL} {VARIABLES}},
year = {1994}
}
@article{faucris.122529484,
abstract = {Canonical quantization of constrained systems with first-class constraints via Dirac's operator constraint method proceeds by the theory of Rigged Hilbert spaces, sometimes also called refined algebraic quantization. This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the master constraint method has been proposed which replaces the individual constraints by a weighted sum of absolute squares of the constraints. Now the direct integral decomposition (DID) methods, which are closely related to Rigged Hilbert spaces, become available and have been successfully tested in various situations. It is relatively straightforward to relate the rigging inner product to the path integral that one obtains via reduced phase space methods. However, for the master constraint, this is not at all obvious. In this paper we find sufficient conditions under which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at least locally in phase space. Then the master constraint DID for those Abelian constraints can be directly related to the rigging map and therefore has a path integral formulation. (C) 2010 American Institute of Physics. [doi:10.1063/1.3486359]},
author = {Han, Muxin and Thiemann, Thomas},
doi = {10.1063/1.3486359},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
keywords = {Dirac equation;Hilbert spaces;integral equations;Lie algebras;master equation;quantisation (quantum theory)},
peerreviewed = {Yes},
title = {{On} the relation between rigging inner product and master constraint direct integral decomposition},
volume = {51},
year = {2010}
}
@article{faucris.109461924,
abstract = {In the previous paper (Giesel and Thiemann 2006 Conceptual setup Preprint gr-qc/0607099) a new combinatorial and thus purely algebraical approach to quantum gravity, called algebraic quantum gravity (AQG), was introduced. In the framework of AQG, existing semiclassical tools can be applied to operators that encode the dynamics of AQG such as the master constraint operator. In this paper, we will analyse the semiclassical limit of the (extended) algebraic master constraint operator and show that it reproduces the correct infinitesimal generators of general relativity. Therefore, the question of whether general relativity is included in the semiclassical sector of the theory, which is still an open problem in LQG, can be significantly improved in the framework of AQG. For the calculations, we will substitute SU(2) with U(1)(3). That this substitution is justified will be demonstrated in the third paper ( Giesel and Thiemann 2006 Semiclassical perturbation theory Preprint gr-qc/0607101) of this series.},
author = {Giesel, Kristina and Thiemann, Thomas},
doi = {10.1088/0264-9381/24/10/004},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2499-2564},
peerreviewed = {Yes},
title = {{Algebraic} quantum gravity ({AQG}): {II}. {Semiclassical} analysis},
volume = {24},
year = {2007}
}
@article{faucris.109131704,
abstract = {We perform a canonical, reduced phase space quantization of general relativity by loop quantum gravity (LQG) methods. The explicit construction of the reduced phase space is made possible by the combination of (a) the Brown-Kuchar mechanism in the presence of pressure-free dust fields which allows to deparametrize the theory and (b) Rovelli's relational formalism in the extended version developed by Dittrich to construct the algebra of gauge-invariant observables. Since the resulting algebra of observables is very simple, one can quantize it using the methods of LQG. Basically, the kinematical Hilbert space of non-reduced LQG now becomes a physical Hilbert space and the kinematical results of LQG such as discreteness of spectra of geometrical operators now have physical meaning. The constraints have disappeared; however, the dynamics of the observables is driven by a physical Hamiltonian which is related to the Hamiltonian of the standard model (without dust) and which we quantize in this paper.},
author = {Giesel, Kristina and Thiemann, Thomas},
doi = {10.1088/0264-9381/27/17/175009},
faupublication = {no},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Algebraic} quantum gravity ({AQG}): {IV}. {Reduced} phase space quantization of loop quantum gravity},
volume = {27},
year = {2010}
}
@article{faucris.110381744,
abstract = {In a seminal paper, Kaminski et al for the first time extended the definition of spin foam models to arbitrary boundary graphs. This is a prerequisite in order to make contact to the canonical formulation of loop quantum gravity whose Hilbert space contains all these graphs. This makes it finally possible to investigate the question whether any of the presently considered spin foam models yields a rigging map for any of the presently defined Hamiltonian constraint operators. We postulate a rigging map by summing over all abstract spin foams with arbitrary but given boundary graphs. The states induced on the boundary of these spin foams can then be identified with elements in the gauge invariant Hilbert space H-0 of the canonical theory. Of course, such a sum over all spin foams is potentially divergent and requires a regularization. Such a regularization can be obtained by introducing specific cut-offs and a weight for every single foam. Such a weight could be for example derived from a generalized formal group field theory allowing for arbitrary interaction terms. Since such a derivation is, however, technical involved we forgo to present a strict derivation and assume that there exist a weight satisfying certain natural axioms, most importantly a gluing property. These axioms are motivated by the requirement that spin foam amplitudes should define a rigging map ( physical inner product) induced by the Hamiltonian constraint. In the analysis of the resulting object we are able to identify an elementary spin foam transfer matrix that allows to generate any finite foam as a finite power of the transfer matrix. It transpires that the sum over spin foams, as written, does not define a projector on the physical Hilbert space. This statement is independent of the concrete spin foam model and Hamiltonian constraint. However, the transfer matrix potentially contains the necessary ingredient in order to construct a proper rigging map in terms of a modified transfer matrix.},
author = {Thiemann, Thomas and Zipfel, Antonia},
doi = {10.1088/0264-9381/31/12/125008},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
keywords = {loop quantum gravity;spin foam;Hamiltonian constraint;rigging map},
peerreviewed = {Yes},
title = {{Linking} covariant and canonical {LQG} {II}: spin foam projector},
volume = {31},
year = {2014}
}
@misc{faucris.124241084,
author = {Bärenz, Manuel and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
title = {{Cartan} {Geometries} and {Spin} {Network} {Quantisation}},
year = {2011}
}
@article{faucris.118400964,
abstract = {The technique introduced here can also be used to produce a couple of completely well-defined regulated operators including but not exhausting (i) the Euclidean Wheeler-DeWitt operator, (ii) the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, (iii) length operators, (iv) Hamiltonian operators of the matter sector and (v) the generators of the asymptotic Poincare group including the quantum ADM energy.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Physics Letters B},
keywords = {Lorentzian;four-dimensional quantum gravity;Wheeler-DeWitt constraint;generator of the Wick rotation;euclidean Hamiltonian},
pages = {257-264},
peerreviewed = {Yes},
title = {{Anomaly}-free formulation of non-perturbative, four-dimensional {Lorentzian} quantum gravity},
volume = {380},
year = {1996}
}
@phdthesis{faucris.121572704,
author = {Stottmeister, Alexander and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{On} the {Embedding} of {Quantum} {Field} {Theory} on {Curved} {Spacetimes} into {Loop} {Quantum} {Gravity}},
year = {2015}
}
@article{faucris.110425964,
abstract = {Spin foam models are an attempt at a covariant or path integral formulation of canonical loop quantum gravity. The construction of such models usually relies on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional 'Regge-like' constraints arising from simplicial triangulations and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a 'cubulation' of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: firstly, the structure of its amplitudes differs from the standard spin foam models. Secondly, the tetrad n-point functions admit a 'Wick-like' structure. Thirdly, the restriction to simple representations does not automatically occur-unless one makes use of the time gauge, just as in the classical theory.},
author = {Baratin, Aristide and Flori, Cecilia and Thiemann, Thomas},
doi = {10.1088/1367-2630/14/10/103054},
faupublication = {yes},
journal = {New Journal of Physics},
peerreviewed = {Yes},
title = {{The} {Holst} spin foam model via cubulations},
volume = {14},
year = {2012}
}
@article{faucris.123219624,
abstract = {We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four space-time dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state-independent factor. (C) 1998 American Institute of Physics.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {3347-3371},
peerreviewed = {Yes},
title = {{Closed} formula for the matrix elements of the volume operator in canonical quantum gravity},
volume = {39},
year = {1998}
}
@article{faucris.123228864,
abstract = {We combine (i) background-independent loop quantum gravity (LQG) quantization techniques, (ii) the mathematically rigorous framework of algebraic quantum field theory (AQFT) and (iii) the theory of integrable systems resulting in the invariant Pohlmeyer charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space. While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge, new, non-trivial solution to the representation problem. This solution exists (1) for any target space dimension, (2) for Minkowski signature of the target space, (3) without tachyons, (4) manifestly ghost free (no negative norm states), (5) without fixing a worldsheet or target space gauge, (6) without (Virasoro) anomalies (zero central charge), (7) while preserving manifest target space Poincare invariance and (8) without picking up UV divergences. The existence of this stable solution is, on one hand, exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string. On the other hand, if such solutions are found, then this would prove that neither a critical dimension (D = 10, 11, 26) nor supersymmetry is a prediction of string theory. Rather, these would be features of the particular Fock representation of current string theory and hence would not be generic. The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper, we treat the more complicated case of curved target spaces.},
author = {Thiemann, Thomas},
doi = {10.1088/0264-9381/23/6/007},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1923-1970},
peerreviewed = {Yes},
title = {{The} {LQG} string - loop quantum gravity quantization of string theory: {I}. {Flat} target space},
volume = {23},
year = {2006}
}
@masterthesis{faucris.122308384,
author = {Thurn, Andreas and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Constraint} {Analysis} of the {D}+1 dimensional {Palatini} action},
year = {2009}
}
@article{faucris.110418924,
abstract = {In the first paper of this series, an extension of the Ashtekar-Lewandowski state space of loop quantum gravity was set up with the help of a projective formalism introduced by Kijowski. The motivation for this work was to achieve a more balanced treatment of the position and momentum variables (also known as holonomies and fluxes). While this is the first step toward the construction of states semi-classical with respect to a full set of observables, one uncovers a deeper issue, which we analyse in the present article in the case of real-valued holonomies. Specifically, we show that, in this case, there does not exist any state on the holonomy-flux algebra in which the variances of the holonomy and flux observables would all be finite, let alone small. It is important to note that this obstruction cannot be bypassed by further enlarging the quantum state space, for it arises from the structure of the algebra itself. Away out would be to suitably restrict the algebra of observables: we take the first step in this direction in a companion paper. Published by AIP Publishing.},
author = {Lanery, Suzanne and Thiemann, Thomas},
doi = {10.1063/1.4983133},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
peerreviewed = {Yes},
title = {{Projective} loop quantum gravity. {II}. {Searching} for semi-classical states},
volume = {58},
year = {2017}
}
@article{faucris.120553004,
abstract = {Loop quantum cosmology (LQC), mainly due to Bojowald, is not the cosmological sector of loop quantum gravity (LQG). Rather, LQC consists of a truncation of the phase space of classical general relativity to spatially homogeneous situations which is then quantized by the methods of LQG. Thus, LQC is a quantum-mechanical toy model (finite number of degrees of freedom) for LQG (a genuine QFT with an infinite number of degrees of freedom) which provides important consistency checks. However, it is a non-trivial question whether the predictions of LQC are robust after switching on the inhomogeneous fluctuations present in full LQG. Two of the most spectacular findings of LQC are that: (1) the inverse scale factor is bounded from above on zero-volume eigenstates which hints at the avoidance of the local Curvature singularity and (2) the quantum Einstein equations are non-singular which hints at the avoidance of the global initial singularity. This rests on (1) a key technique developed for LQG and (2) the fact that there are no inhomogeneous excitations. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is not bounded from above on zero-volume eigenstates. Thus, in full LQG, if curvature singularity avoidance is realized, then not in this simple way. In fact, it turns out that the boundedness of the inverse scale factor is neither necessary nor sufficient for the curvature singularity avoidance and that non-singular evolution equations are neither necessary nor sufficient for initial singularity avoidance because none of these criteria are formulated in terms of observable quantities. After outlining what would be required, we present the results of a calculation for LQG which could be a first indication that our criteria at least for curvature singularity avoidance are satisfied in LQG.},
author = {Thiemann, Thomas and Brunnemann, Johannes},
doi = {10.1088/0264-9381/223/5/001},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1395-1427},
peerreviewed = {Yes},
title = {{On} (cosmological) singularity avoidance in loop quantum gravity},
volume = {23},
year = {2006}
}
@article{faucris.109990364,
abstract = {A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang-Mills continuum theories with gauge groups SU(N) and U(1) and spacetime topologies R-1 x R-1 and R-1 x S-1. (For the U(1) theory, we also consider the S-1 x S-1 topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder-Schrader axioms for gauge theories is proposed, These axioms are satisfied in the present model. (C) 1997 American Institute of Physics.},
author = {Ashtekar, Abhay and Lewandowski, Jerzy and Marolf, Donald and Mourao, José Manuel and Thiemann, Thomas},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {5453-5482},
peerreviewed = {Yes},
title = {{SU}({N}) quantum {Yang}-{Mills} theory in two dimensions: {A} complete solution},
volume = {38},
year = {1997}
}
@article{faucris.123831664,
abstract = {In the past, the possibility to employ (scalar) material reference systems in order to describe classical and quantum gravity directly in terms of gauge invariant (Dirac) observables has been emphasized frequently. This idea has been picked up more recently in loop quantum gravity with the aim to perform a reduced phase space quantization of the theory, thus possibly avoiding problems with the (Dirac) operator constraint quantization method for a constrained system. In this work, we review the models that have been studied on the classical and/or the quantum level and parametrize the space of theories considered so far. We then describe the quantum theory of a model that, to the best of our knowledge, has only been considered classically so far. This model could arguably be called the optimal one in this class of models considered as it displays the simplest possible true Hamiltonian, while at the same time reducing all constraints of general relativity.},
author = {Giesel, Kristina and Thiemann, Thomas},
doi = {10.1088/0264-9381/32/13/135015},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
keywords = {scalar material reference systems;loop quantum gravity;Dirac observables},
peerreviewed = {Yes},
title = {{Scalar} material reference systems and loop quantum gravity},
volume = {32},
year = {2015}
}
@phdthesis{faucris.123917024,
author = {Lanéry, Suzanne and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Projective} {State} {Spaces} for {Theories} of {Connections}},
year = {2015}
}
@article{faucris.118419444,
abstract = {QGR therefore is, by definition, not a unified theory of all interactions in the standard sense, since such a theory would require a new symmetry principle. However, it unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Lecture Notes in Physics},
month = {Jan},
pages = {41-135},
peerreviewed = {unknown},
title = {{Lectures} on {Loop} {Quantum} {Gravity}},
volume = {631},
year = {2003}
}
@article{faucris.122538724,
abstract = {Recently, the master constraint programme for loop quantum gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Wheeler-DeWitt constraint equations in terms of a single master equation. While the proposal has some promising abstract features, it was until now barely tested in known models. In this series of five papers we fill this gap, thereby adding confidence to the proposal. We consider a wide range of models with increasingly more complicated constraint algebras, beginning with a finite-dimensional, Abelian algebra of constraint operators which are linear in the momenta and ending with an infinite-dimensional, non-Abelian algebra of constraint operators which closes with structure functions only and which are not even polynomial in the momenta. In all these models, we apply the master constraint programme successfully; however, the full flexibility of the method must be exploited in order to complete our task. This shows that the master constraint programme has a wide range of applicability but that there are many, physically interesting Subtleties that must be taken care of in doing so. In particular, as we will see, that we can possibly construct a master constraint operator for a nonlinear, that is, interacting quantum field theory underlines the strength of the background-independent formulation of LQG. In this first paper, we prepare the analysis of our test models by outlining the general framework of the master constraint programme. The models themselves will be studied in the remaining four papers. As a side result, we develop the direct integral decomposition (DID) programme for solving quantum constraints as an alternative to refined algebraic quantization (RAQ).},
author = {Dittrich, Bianca and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/4/001},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1025-1065},
peerreviewed = {Yes},
title = {{Testing} the master constraint programme for loop quantum gravity: {I}. {General} framework},
volume = {23},
year = {2006}
}
@masterthesis{faucris.123268464,
author = {Bodendorfer, Norbert and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Canonical} {Analysis} of {Gravity} {Theories} without the {Time} {Gauge}},
year = {2009}
}
@article{faucris.123223584,
abstract = {In this paper, we generalize the treatment of isolated horizons in loop quantum gravity, resulting in a Chern-Simons theory on the boundary in the four-dimensional case, to non-distorted isolated horizons in 2(n + 1)-dimensional spacetimes. The key idea is to generalize the four-dimensional isolated horizon boundary condition by using the Euler topological density E-(2n) of a spatial slice of the black hole horizon as a measure of distortion. The resulting symplectic structure on the horizon coincides with the one of higher-dimensional SO(2(n + 1))-Chern-Simons theory in terms of a Peldan-type hybrid connection Gamma(0) and resembles closely the usual treatment in (3 + 1) dimensions. We comment briefly on a possible quantization of the horizon theory. Here, some subtleties arise since higher-dimensional non-Abelian Chern-Simons theory has local degrees of freedom. However, when replacing the natural generalization to higher dimensions of the usual boundary condition by an equally natural stronger one, it is conceivable that the problems originating from the local degrees of freedom are avoided, thus possibly resulting in a finite entropy.},
author = {Bodendorfer, Norbert and Thiemann, Thomas and Thurn, Andreas},
doi = {10.1088/0264-9381/31/5/055002},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
keywords = {loop quantum gravity;higher dimensions;black holes;Chern-Simons theory},
peerreviewed = {Yes},
title = {{New} variables for classical and quantum gravity in all dimensions: {V}. {Isolated} horizon boundary degrees of freedom},
volume = {31},
year = {2014}
}
@misc{faucris.111921524,
author = {Liegener, Klaus and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
title = {{Hamiltonian} {Constraint} in {Loop} {Quantum} {Gravity}},
year = {2012}
}
@article{faucris.123228644,
abstract = {This is the fifth and final paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. Here we consider interacting quantum field theories, specifically we consider the non-Abelian Gauss constraints of Einstein-Yang-Mills theory and 2+1 gravity. Interestingly, while Yang-Mills theory in 4D is not yet rigorously defined as ail ordinary (Wightman) quantum field theory oil Minkowski space, in background-independent quantum field theories such as loop quantum gravity (LQG) this might become possible by working in a new, background-independent representation. While for the Gauss Constraint the master constraint can be solved explicitly, for the 2+1 theory we are only able to rigorously define the master constraint operator. We show that the, by other methods known, physical Hilbert is contained in the kernel of the master constraint, however, to systematically derive it by Only using spectral methods is as complicated as for 3+1 gravity and we therefore leave the complete analysis for 3+1 gravity.},
author = {Dittrich, Bianca and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/4/005},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1143-1162},
peerreviewed = {Yes},
title = {{Testing} the master constraint programme for loop quantum gravity: {V}. {Interacting} field theories},
volume = {23},
year = {2006}
}
@article{faucris.123480324,
abstract = {The construction of Dirac observables, that is, gauge-invariant objects, in general relativity is technically more complicated than in other gauge theories such as the standard model due to its more complicated gauge group which is closely related to the group of spacetime diffeomorphisms. However, the explicit and usually cumbersome expression of Dirac observables in terms of gauge noninvariant quantities is irrelevant if their Poisson algebra is sufficiently simple. Precisely that can be achieved by employing the relational formalism and a specific type of matter proposed originally by Brown and Kuchar, namely pressureless dust fields. Moreover one is able to derive a compact expression for a physical Hamiltonian that drives their physical time evolution. The resulting gauge-invariant Hamiltonian system is obtained by Higgs-ing the dust scalar fields and has an infinite number of conserved charges which force the Goldstone bosons to decouple from the evolution. In previous publications we have shown that explicitly for cosmological perturbations. In this paper we analyse the spherically symmetric sector of the theory and it turns out that the solutions are in one-to-one correspondence with the class of Lemaitre-Tolman-Bondi metrics. Therefore, the theory is capable of properly describing the whole class of gravitational experiments that rely on the assumption of spherical symmetry.},
author = {Giesel, Kristina and Tambornino, Johannes and Thiemann, Thomas},
doi = {10.1088/0264-9381/27/10/105013},
faupublication = {no},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{LTB} spacetimes in terms of {Dirac} observables},
volume = {27},
year = {2010}
}
@inproceedings{faucris.110424644,
author = {Kastrup, Hans and Thiemann, Thomas},
faupublication = {no},
month = {Jan},
pages = {158-172},
peerreviewed = {unknown},
title = {{Spherically} symmetric gravity and the notion of time in {General} {Relativity}},
year = {1995}
}
@article{faucris.123618704,
abstract = {This new trick might also be of interest for Yang-Mills theories on curved backgrounds.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1907-1921},
peerreviewed = {Yes},
title = {{ON} {THE} {SOLUTION} {OF} {THE} {INITIAL}-{VALUE} {CONSTRAINTS} {FOR} {GENERAL}-{RELATIVITY} {COUPLED} {TO} {MATTER} {IN} {TERMS} {OF} {ASHTEKAR} {VARIABLES}},
volume = {10},
year = {1993}
}
@masterthesis{faucris.111500004,
author = {Lang, Thorsten and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Peakedness} properties of {SU}(3) heat kernel coherent states},
year = {2015}
}
@article{faucris.120317604,
abstract = {Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski to describe quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces. Beside the physical motivations for this approach, it could help designing a quantum state space holding the states we need. In a latter work by Okolow, the description of a theory of Abelian connections within this framework was developed, an important insight being to use building blocks labeled by combinations of edges and surfaces. The present work generalizes this construction to an arbitrary gauge group G (in particular, G is neither assumed to be Abelian nor compact). This involves refining the definition of the label set, as well as deriving explicit formulas to relate the Hilbert spaces attached to different labels. If the gauge group happens to be compact, we also have at our disposal the well-established Ashtekar-Lewandowski Hilbert space, which is defined as an inductive limit using building blocks labeled by edges only. We then show that the quantum state space presented here can be thought as a natural extension of the space of density matrices over this Hilbert space. In addition, it is manifest from the classical counterparts of both formalisms that the projective approach allows for a more balanced treatment of the holonomy and flux variables, so it might pave the way for the development of more satisfactory coherent states. Published by AIP Publishing.},
author = {Lanery, Suzanne and Thiemann, Thomas},
doi = {10.1063/1.4968205},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
peerreviewed = {Yes},
title = {{Projective} loop quantum gravity. {I}. {State} space},
volume = {57},
year = {2016}
}
@article{faucris.109615264,
abstract = {This is the fourth paper in Our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. We now move oil to free field theories with constraints, namely Maxwell theory and linearized gravity. Since the master constraint involves squares of constraint operator valued distributions, one has to be very careful in doing that and we will see that the full flexibility of the master constraint programme must be exploited in order to arrive at sensible results.},
author = {Dittrich, Bianca and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/4/004},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1121-1142},
peerreviewed = {Yes},
title = {{Testing} the master constraint programme for loop quantum gravity: {IV}. {Free} field theories},
volume = {23},
year = {2006}
}
@article{faucris.120318924,
abstract = {We describe a simple dynamical model characterized by the presence of two noncommuting Hamiltonian constraints. This feature mimics the constraint structure of general relativity, where there is one Hamiltonian constraint associated with each space point. We solve the classical and quantum dynamics of the model, which turns out to be governed by an SL(2,R) gauge symmetry, local in time. In classical theory, we solve the equations of motion, find an SO(2,2) algebra of Dirac observables, find the gauge transformations for the Lagrangian and canonical variables and for the Lagrange multipliers. In quantum theory, we find the physical states, the quantum observables, and the physical inner product, which is determined by the reality conditions. In addition, we construct the classical and quantum evolving constants of the system. The model illustrates how to describe physical gauge-invariant relative evolution when coordinate time evolution is a gauge. [S0556-2821(99)02014-7].},
author = {Montesinos, Merced and Rovelli, Carlo and Thiemann, Thomas},
faupublication = {no},
journal = {Physical Review D},
peerreviewed = {unknown},
title = {{SL}(2,{R}) model with two {Hamiltonian} constraints},
volume = {60},
year = {1999}
}
@article{faucris.110411884,
abstract = {There is a gap that has not been filled since the formulation of general relativity in terms of Ashtekar's new variables, namely the treatment of asymptotically flat field configurations that are general enough to be able to define the generators of the Lorentz subgroup of the asymptotic Poincare group. While such a formulation already exists for the old geometrodynamical variables, up to now only the generators of the translation subgroup have been able to be defined, because the function spaces of the fields considered earlier are taken in too special a form. The transcription of the framework from the ADM variables to Ashtekar's variables turns out not to be straightforward, due to the a priori freedom to choose the internal SO(3) frame at spatial infinity, and due to the fact that the non-trivial reality conditions of the Ashtekar framework re-enter the stage when imposing suitable boundary conditions on the fields and the Lagrange multipliers.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
month = {Jan},
pages = {181-198},
peerreviewed = {Yes},
title = {{GENERALIZED} {BOUNDARY}-{CONDITIONS} {FOR} {GENERAL}-{RELATIVITY} {FOR} {THE} {ASYMPTOTICALLY} {FIAT} {CASE} {IN} {TERMS} {OF} {ASHTEKARS} {VARIABLES}},
volume = {12},
year = {1995}
}
@article{faucris.123226884,
abstract = {The volume operator plays a crucial role in the definition of the quantum dynamics Of loop quantum gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the Volume spectrum. While closed formulae for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which ill turn are given in terms of Racah's formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far Hot even numerically the spectrum could be accessed. In this paper, we demonstrate that by means of the Elliot-Biedenharn identify one can get rid of all the 6j symbols for any valence of: the gauge-invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge-invariant 4-vertex. The techniques derived in this paper-could also be of use for the analysis of spin-spin interaction Hamiltonians of many-particle problems in atomic and nuclear physics.},
author = {Brunnemann, Johannes and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/4/014},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1289-1346},
peerreviewed = {Yes},
title = {{Simplification} of the spectral analysis of the volume operator in loop quantum gravity},
volume = {23},
year = {2006}
}
@article{faucris.109133904,
abstract = {In our companion paper we identified a complete set of manifestly gauge-invariant observables for general relativity. This was possible by coupling the system of gravity and matter to pressureless dust which plays the role of a dynamically coupled observer. the evolution of those observables is governed by a physical Hamiltonian and we derived the corresponding equations of motion. Linear perturbation theory of those equations of motion around a general exact solution in terms of manifestly gauge-invariant perturbations was then developed. In this paper we specialize our previous results to an FRW background which is also a solution of our modified equations of motion. We then compare the resulting equations with those derived in standard cosmological perturbation theory (SCPT). We exhibit the precise relation between our manifestly gauge-invariant perturbations and the linearly gauge-invariant variables in SCPT. We find that our equations of motion can be cast into SCPT form plus corrections. These corrections are the trace that the dust leaves on the system in terms of a conserved energy-momentum current density. In turns out that these corrections decay; in fact, in the late universe they are negligible whatever the value of the conserved current. We conclude that the addition of dust which serves as a test observer medium, while implying modifications of Einstein's equations without dust, leads to acceptable agreement with known results, while having the advantage that one now talks about manifestly gauge-invariant, that is measurable, quantities, which can be used even in perturbation theory at higher orders.},
author = {Giesel, Kristina and Hofmann, Stefan and Thiemann, Thomas and Winkler, Oliver},
doi = {10.1088/0264-9381/27/5/055006},
faupublication = {no},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Manifestly} gauge-invariant general relativistic perturbation theory: {II}. {FRW} background and first order},
volume = {27},
year = {2010}
}
@article{faucris.110402644,
abstract = {An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interacting theories and theories with constraints, this is non-trivial, and is normally not the heuristic 'Lebesgue measure' usually used. There have been many determinations of a measure for gravity in the literature, but none for the Palatini or Holst formulations of gravity. Furthermore, the relations between different resulting measures for different formulations of gravity are usually not discussed. In this paper we use the reduced phase technique in order to derive the path-integral measure for the Palatini and Holst formulation of gravity, which is different from the Lebesgue measure up to local measure factors which depend on the spacetime volume element and spatial volume element. From this path integral for the Holst formulation of general relativity we can also give a new derivation of the Plebanski path integral and discover a discrepancy with the result due to Buffenoir, Henneaux, Noui and Roche whose origin we resolve. This paper is the first in a series that aims at better understanding the relation between canonical loop quantum gravity and the spin-foam approach.},
author = {Engle, Jonathan and Han, Muxin and Thiemann, Thomas},
doi = {10.1088/0264-9381/27/24/245014},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Canonical} path integral measures for {Holst} and {Plebanski} gravity: {I}. {Reduced} phase space derivation},
volume = {27},
year = {2010}
}
@inproceedings{faucris.123226444,
abstract = {We use the new canonical variables introduced by Ashtekar which simplifies the analysis tremendously.},
author = {Thiemann, Thomas},
faupublication = {no},
pages = {293-298},
peerreviewed = {unknown},
publisher = {WORLD SCIENTIFIC PUBL CO PTE LTD},
title = {{REDUCED} {PHASE}-{SPACE} {QUANTIZATION} {OF} {SPHERICALLY} {SYMMETRICAL} {EINSTEIN}-{MAXWELL} {THEORY} {INCLUDING} {A} {COSMOLOGICAL} {CONSTANT}},
volume = {3},
year = {1994}
}
@article{faucris.110409904,
abstract = {In this paper, we investigate the properties of gauge-invariant coherent states for loop quantum gravity, for the gauge group U(1). This is done by projecting the corresponding complexifier coherent states defined by Thiemann and Winklerto the gauge- invariant Hilbert space. This being the first step toward constructing physical coherent states, we arrive at a set of gauge- invariant states that approximate well the gauge-invariant degrees of freedom of Abelian loop quantum gravity (LQG). Furthermore, these states turn out to encode explicit information about the graph topology, and show the same pleasant peakedness properties known from the gauge-variant complexifier coherent states. In a companion paper, we will turn to the more sophisticated case of SU(2).},
author = {Bahr, Benjamin and Thiemann, Thomas},
doi = {10.1088/0264-9381/26/4/045011},
faupublication = {no},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Gauge}-invariant coherent states for loop quantum gravity: {I}. {Abelian} gauge groups},
volume = {26},
year = {2009}
}
@article{faucris.110386144,
abstract = {The space of solutions to all constraints turns out to be much larger than that obtained by traditional approaches, however, it is fully included. Thus, by a suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods. The meaning of the remaining 'spurious solutions' is discussed.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1249-1280},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics ({QSD}): {IV}. 2+1 {Euclidean} quantum gravity as a model to test 3+1 {Lorentzian} quantum gravity},
volume = {15},
year = {1998}
}
@inproceedings{faucris.123218084,
author = {Ashtekar, Abhay and Lewandowski, Jerzy and Marolf, Donald and Mourao, José Manuel and Thiemann, Thomas},
faupublication = {no},
month = {Jan},
pages = {60-86},
peerreviewed = {unknown},
title = {{A} manifestly gauge-invariant approach to quantum theories of gauge fields},
year = {1995}
}
@article{faucris.110420024,
abstract = {Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchar model. The main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections. (C) 1995 American Institute of Physics.},
author = {Ashtekar, Abhay and Lewandowski, Jerzy and Marolf, Donald and Mourao, José Manuel and Thiemann, Thomas},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {6456-6493},
peerreviewed = {Yes},
title = {{QUANTIZATION} {OF} {DIFFEOMORPHISM} {INVARIANT} {THEORIES} {OF} {CONNECTIONS} {WITH} {LOCAL} {DEGREES} {OF} {FREEDOM}},
volume = {36},
year = {1995}
}
@article{faucris.123222264,
abstract = {Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tecotl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to solve the difficult fermionic adjointness relations.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1487-1512},
peerreviewed = {Yes},
title = {{Kinematical} {Hilbert} spaces for fermionic and {Higgs} quantum field theories},
volume = {15},
year = {1998}
}
@article{faucris.118263464,
abstract = {Finally, we comment on the status of the Wick rotation transform in the light of the present results and give an intuitive description of the action of the Hamiltonian constraint.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {875-905},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics ({QSD}): {II}. {The} kernel of the {Wheeler}-{DeWitt} constraint operator},
volume = {15},
year = {1998}
}
@article{faucris.109456864,
abstract = {In the two previous papers of this series we defined a new combinatorial approach to quantum gravity, algebraic quantum gravity (AQG). We showed that AQG reproduces the correct infinitesimal dynamics in the semiclassical limit, provided one incorrectly substitutes the non-Abelian group SU(2) by the Abelian group U(1)(3) in the calculations. The mere reason why that substitution was performed at all is that in the non-Abelian case the volume operator, pivotal for the definition of the dynamics, is not diagonizable by analytical methods. This, in contrast to the Abelian case, so far prohibited semiclassical computations. In this paper, we show why this unjustified substitution nevertheless reproduces the correct physical result. Namely, we introduce for the first time semiclassical perturbation theory within AQG ( and LQG) which allows us to compute expectation values of interesting operators such as the master constraint as a power series in h with error control. That is, in particular, matrix elements of fractional powers of the volume operator can be computed with extremely high precision for sufficiently large power of h in the h expansion. With this new tool, the non-Abelian calculation, although technically more involved, is then exactly analogous to the Abelian calculation, thus justifying the Abelian analysis in retrospect. The results of this paper turn AQG into a calculational discipline.},
author = {Giesel, Kristina and Thiemann, Thomas},
doi = {10.1088/0264-9381/24/10/005},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2565-2588},
peerreviewed = {Yes},
title = {{Algebraic} quantum gravity ({AQG}): {III}. {Semiclassical} perturbation theory},
volume = {24},
year = {2007}
}
@article{faucris.122535424,
abstract = {Of course, to show that the entire theory is finite requires more: one would need to know what the physical observables are, apart from the Hamiltonian constraint, and whether they are also finite. However, with the results given in this paper this question can now be answered, at least in principle.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1281-1314},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics ({QSD}): {V}. {Quantum} gravity as the natural regulator of the {Hamiltonian} constraint of matter quantum field theories},
volume = {15},
year = {1998}
}
@misc{faucris.111490764,
author = {Reichert, Thorsten and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
title = {{Angular} {Momentum} and {Quantum} {Gravity}},
year = {2010}
}
@article{faucris.108346964,
abstract = {We quantize the new connection formulation of (D + 1)-dimensional general relativity developed in our companion papers by loop quantum gravity (LQG) methods. It turns out that all the tools prepared for LQG straightforwardly generalize to the new connection formulation in higher dimensions. The only new challenge is the simplicity constraint. While its 'diagonal' components acting at edges of spin-network functions are easily solved, its 'off-diagonal' components acting at vertices are non-trivial and require a more elaborate treatment.},
author = {Bodendorfer, Norbert and Thiemann, Thomas and Thurn, Andreas},
doi = {10.1088/0264-9381/30/4/045003},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{New} variables for classical and quantum gravity in all dimensions: {III}. {Quantum} theory},
volume = {30},
year = {2013}
}
@article{faucris.123223364,
abstract = {We rederive the results of our companion paper, for matching space-time and internal signature, by applying in detail the Dirac algorithm to the Palatini action. While the constraint set of the Palatini action contains second class constraints, by an appeal to the method of gauge unfixing, we map the second class system to an equivalent first class system which turns out to be identical to the first class constraint system obtained via the extension of the ADM phase space performed in our companion paper. Central to our analysis is again the appropriate treatment of the simplicity constraint. Remarkably, the simplicity constraint invariant extension of the Hamiltonian constraint, that is a necessary step in the gauge unfixing procedure, involves a correction term which is precisely the one found in the companion paper and which makes sure that the Hamiltonian constraint derived from the Palatini Lagrangian coincides with the ADM Hamiltonian constraint when Gauss and simplicity constraints are satisfied. We therefore have rederived our new connection formulation of general relativity from an independent starting point, thus confirming the consistency of this framework.},
author = {Bodendorfer, Norbert and Thiemann, Thomas and Thurn, Andreas},
doi = {10.1088/0264-9381/30/4/045002},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{New} variables for classical and quantum gravity in all dimensions: {II}. {Lagrangian} analysis},
volume = {30},
year = {2013}
}
@article{faucris.110411004,
abstract = {This is the second paper concerning gauge-invariant coherent states for loop quantum gravity. Here, we deal with the gauge group SU(2), this being a significant complication compared to the Abelian U(1) case encountered in the previous article (Class. Quantum Grav. 26 045011). We study gauge-invariant coherent states on certain special graphs by analytical and numerical methods. We find that their overlap is Gauss peaked in gauge- invariant quantities, as long as states are not labeled by degenerate gauge orbits, i.e. points where the gauge- invariant configuration space has singularities. In these cases the overlaps are still concentrated around these points, but the peak profile exhibits a plateau structure. This shows how the semiclassical properties of the states are influenced by the geometry of the gauge-invariant phase space.},
author = {Bahr, Benjamin and Thiemann, Thomas},
doi = {10.1088/0264-9381/26/4/045012},
faupublication = {no},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Gauge}-invariant coherent states for loop quantum gravity: {II}. {Non}-{Abelian} gauge groups},
volume = {26},
year = {2009}
}
@article{faucris.110400884,
abstract = {One of the celebrated results of loop quantum gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area, and volume operators. This is an indication that the Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present. Because the aforementioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli's partial and complete observables provides a possible mechanism for turning a non-gauge-invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is, gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that "fundamental discreteness at the Planck scale in LQG" is far from established. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.},
author = {Dittrich, Bianca and Thiemann, Thomas},
doi = {10.1063/1.3054277},
faupublication = {no},
journal = {Journal of Mathematical Physics},
keywords = {geometry;mathematical operators;quantum gravity},
month = {Jan},
peerreviewed = {Yes},
title = {{Are} the spectra of geometrical operators in loop quantum gravity really discrete?},
volume = {50},
year = {2009}
}
@article{faucris.123220504,
abstract = {In this article, the second of three, we discuss and develop the basis of a Weyl quantisation for compact Lie groups aiming at loop quantum gravity-type models. This Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity. Additionally, we conjecture the existence of a new form of the Segal-Bargmann-Hall "coherent state" transform for compact Lie groups G, which we prove for G = U(1)(n) and support by numerical evidence for G = SU(2). The reason for conjoining this conjecture with the main topic of this article originates in the observation that the coherent state transform can be used as a basic building block of a coherent state quantisation (Berezin quantisation) for compact Lie groups G. But, as Weyl and Berezin quantisation for R-2d are intimately related by heat kernel evolution, it is natural to ask whether a similar connection exists for compact Lie groups as well. Moreover, since the formulation of space adiabatic perturbation theory requires a (deformation) quantisation as minimal input, we analyse the question to what extent the coherent state quantisation, defined by the Segal-Bargmann-Hall transform, can serve as basis of the former. Published by AIP Publishing.},
author = {Stottmeister, Alexander and Thiemann, Thomas},
doi = {10.1063/1.4954803},
faupublication = {yes},
journal = {Journal of Mathematical Physics},
peerreviewed = {Yes},
title = {{Coherent} states, quantum gravity, and the {Born}-{Oppenheimer} approximation. {II}. {Compact} {Lie} groups},
volume = {57},
year = {2016}
}
@article{faucris.110421564,
abstract = {Recently the master constraint programme (MCP) for loop quantum gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single master constraint. The MCP is designed to overcome the complications associated with the non-Lie-algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For the case of 3+1 gravity itself, so far only a positive quadratic form for the master constraint operator was derived. In this paper, we close this gap and prove that the quadratic form is closable and thus stems from a unique self-adjoint master constraint operator. The proof rests on a simple feature of the general pattern according to which Hamiltonian constraints in LQG are constructed and thus extends to arbitrary matter coupling and holds for any metric signature. With this result the existence of a physical Hilbert space for LQG is established by standard spectral analysis.},
author = {Thiemann, Thomas},
doi = {10.1088/0264-9381/23/7/003},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2249-2265},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics: {VIII}. {The} master constraint},
volume = {23},
year = {2006}
}
@article{faucris.110368984,
abstract = {We show that the quantization of spherically symmetric pure gravity can be carried out completely in the framework of Ashtekar's self-dual representation. Consistent operator orderings can be given for the constraint functionals yielding two kinds of solutions for the constraint equations, corresponding classically to globally nondegenerate or degenerate metrics. The physical state functionals can be determined by quadratures and the reduced hamiltonian system possesses two degrees of freedom, one of them corresponding to the classical Schwarzschild mass squared and the canonically conjugate one representing a measure for the deviation of the nonstatic field configurations from the static Schwarzschild one. There is a natural choice for the scalar product making the two fundamental observables self-adjoint. Finally, a unitary transformation is performed in order to calculate the triad-representation of the physical state functionals and to provide for a solution of the appropriately regularized Wheeler-DeWitt equation.},
author = {Thiemann, Thomas and Kastrup, Hans},
faupublication = {no},
journal = {Nuclear Physics B},
pages = {211-258},
peerreviewed = {Yes},
title = {{CANONICAL} {QUANTIZATION} {OF} {SPHERICALLY} {SYMMETRICAL} {GRAVITY} {IN} {ASHTEKAR} {SELF}-{DUAL} {REPRESENTATION}},
volume = {399},
year = {1993}
}
@article{faucris.123222484,
abstract = {It is often emphasized that spin-foam models could realize a projection on the physical Hilbert space of canonical loop quantum gravity. As a first test, we analyze the one-vertex expansion of a simple Euclidean spin foam. We find that for fixed Barbero-Immirzi parameter gamma = 1, the one-vertex amplitude in the Kaminski, Kisielowski, and Lewandowski prescription annihilates the Euclidean Hamiltonian constraint of loop quantum gravity [T. Thiemann, Classical Quantum Gravity 15, 839 (1998).]. Since, for gamma = 1, the Lorentzian part of the Hamiltonian constraint does not contribute, this gives rise to new solutions of the Euclidean theory. Furthermore, we find that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.},
author = {Alesci, Emanuele and Thiemann, Thomas and Zipfel, Antonia},
doi = {10.1103/PhysRevD.86.024017},
faupublication = {yes},
journal = {Physical Review D - Particles, Fields, Gravitation and Cosmology},
peerreviewed = {Yes},
title = {{Linking} covariant and canonical loop quantum gravity: {New} solutions to the {Euclidean} scalar constraint},
volume = {86},
year = {2012}
}
@article{faucris.110423104,
abstract = {The quantum symmetry algebra corresponding to the generators of the little group faithfully represents the classical algebra.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1463-1485},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics ({QSD}): {VI}. {Quantum} {Poincare} algebra and a quantum positivity of energy theorem for canonical quantum gravity},
volume = {15},
year = {1998}
}
@article{faucris.110372944,
abstract = {In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only a finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise-smooth finite paths and loops. In particular, we (a) characterize the spectrum of the Ashtekar-Isham configuration space, (b) introduce spin-web states, a generalization of the spin network states, (c) extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism-invariant states and finally (d) extend the 3-geometry operators and the Hamiltonian operator.},
author = {Lewandowski, Jerzy and Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2299-2322},
peerreviewed = {Yes},
title = {{Diffeomorphism}-invariant quantum field theories of connections in terms of webs},
volume = {16},
year = {1999}
}
@article{faucris.122507264,
abstract = {In this paper, we investigate the possibility of approximating the physical inner product of constrained quantum theories. In particular, we calculate the physical inner product of a simple cosmological model in two ways: firstly, we compute it analytically via a trick; secondly, we use the complexifier coherent states to approximate the physical inner product defined by the master constraint of the system. We find that the approximation is able to recover the analytic solution of the problem, which consolidates hopes that coherent states will help to approximate solutions of more complicated theories, like loop quantum gravity.},
author = {Bahr, Benjamin and Thiemann, Thomas},
doi = {10.1088/0264-9381/24/8/011},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2109-2138},
peerreviewed = {Yes},
title = {{Approximating} the physical inner product of loop quantum cosmology},
volume = {24},
year = {2007}
}
@article{faucris.122540044,
abstract = {This is the third paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. In this work, we analyse models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper. These are systems with an SL(2, R) gauge symmetry and the complications arise because non-compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the Spectrum of the master constraint does not contain the point zero. However, the minimum of the spectrum is of order h 2 which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to h normal ordering constants). The physical Hilbert space can then be obtained after subtracting this normal ordering correction.},
author = {Thiemann, Thomas and Dittrich, Bianca},
doi = {10.1088/0264-9381/23/4/003},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1089-1120},
peerreviewed = {Yes},
title = {{Testing} the master constraint programme for loop quantum gravity: {III}. {SL}(2, {R}) models},
volume = {23},
year = {2006}
}
@article{faucris.123221604,
abstract = {Recently, substantial amount of activity in quantum general relativity (QGR) has focused on the semiclassical analysis of the theory. In this paper, we want to comment on two such developments: (1) polymer-like states for Maxwell theory and linearized gravity constructed by Varadarajan which use much of the Hilbert space machinery that has proved useful in QGR, and (2) coherent states for QGR, based on the general complexifier method, with built-in semiclassical properties. We show the following. (A) Varadarajan's states are complexifier coherent states. This unifies all states constructed so far under the general complexifier principle. (B) Ashtekar and Lewandowski suggested a non-Abelian generalization of Varadarajan's states to QGR which, however, are no longer of the complexifier type. We construct a new class of non-Abelian complexifiers which come close to that underlying Varadarajan's construction. (C) Non-Abelian complexifiers close to Varadarajan's induce new types of Hilbert spaces which do not support the operator algebra of QGR. The analysis suggests that if one sticks to the present kinematical framework of QGR and if kinematical coherent states are at all useful, then normalizable, graph-dependent states must be used which are produced by the complexifier method as well. (D) Present proposals for states with mildened graph dependence, obtained by performing a graph average, do not approximate well coordinate-dependent observables. However, graph-dependent states, whether averaged or not, seem to be well suited for the semiclassical analysis of QGR with respect to coordinate-independent operators.},
author = {Thiemann, Thomas},
doi = {10.1088/0264-9381/23/6/013},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {2063-2117},
peerreviewed = {Yes},
title = {{Complexifier} coherent states for quantum general relativity},
volume = {23},
year = {2006}
}
@article{faucris.110403084,
abstract = {The Segal-Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group G with its normalized Haar measure mu(H), the Hall transform is an isometric isomorphism hem L(2)(G, mu(H)) to H(G(C)) boolean AND L(2)(G(C), v), where G(C) the complexification of G, H(G(C)) the space of holomorphic functions on G(C), and v an appropriate heat-kernel measure on G(C). We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group G by (a certain extension of) the space A/g of connections module gauge transformations. The resulting ''coherent state transform'' provides a holomorphic representation of the holonomy C* algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4 dimensions. (C) 1996 Academic Press, Inc.},
author = {Ashtekar, Abhay and Lewandowski, Jerzy and Marolf, Donald and Mourao, José Manuel and Thiemann, Thomas},
faupublication = {no},
journal = {Journal of Functional Analysis},
pages = {519-551},
peerreviewed = {Yes},
title = {{Coherent} state transforms for spaces of connections},
volume = {135},
year = {1996}
}
@article{faucris.123223804,
abstract = {After discussing the formalism at the classical level in a first paper (Lanery, 2017), the present second paper is devoted to the quantum theory. In particular, we inspect in detail how such quantum projective state spaces relate to inductive limit Hilbert spaces and to infinite tensor product constructions (Lanery, 2016, subsection 3.1) [1]. Regarding the quantization of classical projective structures into quantum ones, we extend the results by Okolow (2013), that were set up in the context of linear configuration spaces, to configuration spaces given by simply-connected Lie groups, and to holomorphic quantization of complex phase spaces (Lanky, 2016, subsection 2.2) [1]. (C) 2017 Elsevier B.V. All rights reserved.},
author = {Lanery, Suzanne and Thiemann, Thomas},
doi = {10.1016/j.geomphys.2017.01.011},
faupublication = {yes},
journal = {Journal of Geometry and Physics},
keywords = {Quantum field theory;Projective limits;Algebras of observables;Geometric quantization;Position representation;Holomorphic quantization},
pages = {10-51},
peerreviewed = {Yes},
title = {{Projective} limits of state spaces {II}. {Quantum} formalism},
volume = {116},
year = {2017}
}
@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.111423664,
author = {Zilker, Thomas and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
title = {{Quantum} {Simplicity} {Constraints} and {Area} {Spectrum}},
year = {2010}
}
@article{faucris.122538944,
abstract = {This is the second paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. In this work, we begin with the simplest examples: finite-dimensional models with a finite number of first or second class constraints, Abelian or non-Abelian, with or without structure functions.},
author = {Dittrich, Bianca and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/4/002},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1067-1088},
peerreviewed = {Yes},
title = {{Testing} the master constraint programme for loop quantum gravity: {II}. {Finite}-dimensional systems},
volume = {23},
year = {2006}
}
@article{faucris.120859904,
abstract = {The volume operator plays a pivotal role for the quantum dynamics of loop quantum gravity (LQG). It is essential to construct triad operators that enter the Hamiltonian constraint and which become densely defined operators on the full Hilbert space, even though in the classical theory the triad becomes singular when classical GR breaks down. The expression for the volume and triad operators derives from the quantization of the fundamental electric flux operator of LQG by a complicated regularization procedure. In fact, there are two inequivalent volume operators available in the literature and, moreover, both operators are unique only up to a finite, multiplicative constant which should be viewed as a regularization ambiguity. Now on the one hand, classical volumes and triads can be expressed directly in terms of fluxes and this fact was used to construct the corresponding volume and triad operators. On the other hand, fluxes can be expressed in terms of triads and triads can be replaced by Poisson brackets between the holonomy and the volume operators. Therefore one can also view the holonomy operators and the volume operator as fundamental and consider the flux operator as a derived operator. In this paper we mathematically implement this second point of view and thus can examine whether the volume, triad and flux quantizations are consistent with each other. The results of this consistency analysis are rather surprising. Among other findings we show the following. ( 1) The regularization constant can be uniquely fixed. ( 2) One of the volume operators can be ruled out as inconsistent. ( 3) Factor ordering ambiguities in the definition of triad operators are immaterial for the classical limit of the derived flux operator. The results of this paper show that within full LQG triad operators are consistently quantized.},
author = {Giesel, Kristina and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/18/011},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {5667-5691},
peerreviewed = {Yes},
title = {{Consistency} check on volume and triad operator quantization in loop quantum gravity: {I}},
volume = {23},
year = {2006}
}
@article{faucris.123693064,
abstract = {This paper is the first in a series of seven papers with the title 'quantum spin dynamics (QSD)'.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {839-873},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics ({QSD})},
volume = {15},
year = {1998}
}
@article{faucris.123620244,
abstract = {We derive a canonical algorithm to obtain this holomorphic representation and in particular explicitly compute it for quantum gravity in terms of a Wick rotation transform.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1383-1403},
peerreviewed = {Yes},
title = {{Reality} conditions inducing transforms for quantum gauge field theory and quantum gravity},
volume = {13},
year = {1996}
}
@article{faucris.122512324,
abstract = {Osterwalder and Schrader introduced a procedure to obtain a (Lorentzian) Hamiltonian quantum theory starting from a measure on the space of (Euclidean) histories of a scalar quantum field. In this paper, we extend that construction to more general theories which do not refer to any background, spacetime metric (and in which the space of histories does not admit a natural linear structure). Examples include certain gauge theories, topological field theories and relativistic gravitational theories. The treatment is self-contained in the sense that an a priori knowledge of the Osterwalder-Schrader theorem is not assumed.},
author = {Ashtekar, Abhay and Marolf, Donald and Mourao, José Manuel and Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {4919-4940},
peerreviewed = {Yes},
title = {{Constructing} {Hamiltonian} quantum theories from path integrals in a diffeomorphism-invariant context},
volume = {17},
year = {2000}
}
@article{faucris.123221164,
abstract = {We summarize a recently proposed concrete programme for investigating the (semi)classical limit of canonical, Lorentzian, continuum quantum general relativity in four spacetime dimensions. The analysis is based on a novel set of coherent states labelled by graphs. These fit neatly together with an Infinite Tensor Product (ITP) extension of the currently used Hilbert space. The ITP construction enables us to give rigorous meaning to the infinite volume (thermodynamic) limit of the theory which has been out of reach so far. (C) 2001 Elsevier Science B.V. All rights reserved.},
author = {Sahlmann, Hanno and Thiemann, Thomas and Winkler, Oliver},
faupublication = {no},
journal = {Nuclear Physics B},
pages = {401-440},
peerreviewed = {Yes},
title = {{Coherent} states for canonical quantum general relativity and the infinite tensor product extension},
volume = {606},
year = {2001}
}
@article{faucris.122513644,
abstract = {In the canonical approach to Lorentzian quantum general relativity in four spacetime dimensions an important step forward has been made by Ashtekar, Isham and Lewandowski some eight years ago through the introduction of a Hilbert space structure, which was later proved to be a faithful representation of the canonical commutation and adjointness relations of the quantum field algebra of diffeomorphism invariant gauge field theories by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann. This Hilbert space, together with its generalization due to Baez and Sawin, is appropriate for semi-classical quantum general relativity if the spacetime is spatially compact. In the spatially non-compact case, however, an extension of the Hilbert space is needed in order to approximate metrics that are macroscopically nowhere degenerate. For this purpose, in this paper we apply the theory of the infinite tensor product (ITP) of Hilbert Spaces, developed by von Neumann more than sixty years ago, to quantum general relativity. The cardinality of the number of tensor product factors can take the value of any possible Cantor aleph, making this mathematical theory well suited to our problem in which a Hilbert space is attached to each edge of an arbitrarily complicated, generally infinite graph. The new framework opens access to a new arsenal of techniques, appropriate to describe fascinating physics such as quantum topology change, semi-classical quantum gravity, effective low-energy physics etc from the universal point of view of the ITP. In particular, the study of photons and gravitons propagating on fluctuating quantum spacetimes should now be in reach.},
author = {Thiemann, Thomas and Winkler, Oliver},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {4997-5053},
peerreviewed = {Yes},
title = {{Gauge} field theory coherent states ({GCS}): {IV}. {Infinite} tensor product and thermodynamical limit},
volume = {18},
year = {2001}
}
@article{faucris.122513424,
abstract = {These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. These findings are another step towards establishing that the infinitesimal quantum dynamics of quantum general relativity might, to lowest order in (h) over bar, indeed be given by classical general relativity.},
author = {Thiemann, Thomas and Winkler, Oliver},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {4629-4681},
peerreviewed = {Yes},
title = {{Gauge} field theory coherent states ({GCS}): {III}. {Ehrenfest} theorems},
volume = {18},
year = {2001}
}
@article{faucris.110412544,
abstract = {Barbero has generalized the Ashtekar canonical transformation to a one-parameter scale transformation U(gamma) on the phase space of general relativity. Immirzi has noticed that in loop quantum gravity this transformation alters the spectra of geometrical quantities. We show that U(gamma) is a canonical transformation that cannot be implemented unitarily in quantum theory. This implies that there exists a one-parameter quantization ambiguity in quantum gravity, namely, a free parameter that enters the construction of the quantum theory. The purpose of this paper is to elucidate the origin and the role of this free parameter.},
author = {Thiemann, Thomas and Rovelli, Carlo},
faupublication = {no},
journal = {Physical Review D},
month = {Jan},
pages = {1009-1014},
peerreviewed = {unknown},
title = {{Immirzi} parameter in quantum general relativity},
volume = {57},
year = {1998}
}
@article{faucris.110413644,
abstract = {Loop quantum gravity (LQG) relies heavily on a connection formulation of general relativity such that (1) the connection Poisson commutes with itself and (2) the corresponding gauge group is compact. This can be achieved starting from the Palatini or Holst action when imposing the time gauge. Unfortunately, this method is restricted to D + 1 = 4 spacetime dimensions. However, interesting string theories and supergravity theories require higher dimensions and it would therefore be desirable to have higher dimensional supergravity loop quantizations at one's disposal in order to compare these approaches. In this series of papers we take first steps toward this goal. The present first paper develops a classical canonical platform for a higher dimensional connection formulation of the purely gravitational sector. The new ingredient is a different extension of the ADM phase space than the one used in LQG which does not require the time gauge and which generalizes to any dimension D > 1. The result is a Yang-Mills theory phase space subject to Gauss, spatial diffeomorphism and Hamiltonian constraint as well as one additional constraint, called the simplicity constraint. The structure group can be chosen to be SO(1, D) or SO(D + 1) and the latter choice is preferred for purposes of quantization.},
author = {Bodendorfer, Norbert and Thiemann, Thomas and Thurn, Andreas},
doi = {10.1088/0264-9381/30/4/045001},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{New} variables for classical and quantum gravity in all dimensions: {I}. {Hamiltonian} analysis},
volume = {30},
year = {2013}
}
@article{faucris.110422224,
abstract = {(vii) Equipped with this inner product, the construction of physical observables is straightforward.},
author = {Thiemann, Thomas},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {1207-1247},
peerreviewed = {Yes},
title = {{Quantum} spin dynamics ({QSD}): {III}. {Quantum} constraint algebra and physical scalar product in quantum general relativity},
volume = {15},
year = {1998}
}
@article{faucris.110403744,
abstract = {Spin-foam models are supposed to be discretized path integrals for quantum gravity constructed from the Plebanski-Holst action. The reason for there being several models currently under consideration is that no consensus has been reached for how to implement the simplicity constraints. Indeed, none of these models strictly follows from the original path integral with commuting B fields, rather, by some nonstandardmanipulations one always ends up with non-commuting B fields and the simplicity constraints become in fact anomalous which is the source for there being several inequivalent strategies to circumvent the associated problems. In this paper, we construct a new Euclidian spin-foam model which is constructed by standard methods from the Plebanski-Holst path integral with commuting B fields discretized on a 4D simplicial complex. The resulting model differs from the current ones in several aspects, one of them being that the closure constraint needs special care. Only when dropping the closure constraint by hand and only in the large spin limit can the vertex amplitudes of this model be related to those of the FK gamma. model but even then the face and edge amplitude differ. Interestingly, a non-commutative deformation of the B-IJ variables leads from our new model to the Barrett-Crane model in the case of gamma =infinity.},
author = {Han, Muxin and Thiemann, Thomas},
doi = {10.1088/0264-9381/30/23/235024},
faupublication = {yes},
journal = {Classical and Quantum Gravity},
peerreviewed = {Yes},
title = {{Commuting} simplicity and closure constraints for {4D} spin-foam models},
volume = {30},
year = {2013}
}
@incollection{faucris.123223144,
abstract = {We describe the canonical approach to quantum gravity and report on the status of its most advanced implementation, Loop Quantum Gravity (LQG).},
author = {Thiemann, Thomas},
booktitle = {Approaches to quantum gravity: Toward a new understanding of space, time and matter},
faupublication = {no},
pages = {1113-1129},
peerreviewed = {unknown},
publisher = {WORLD SCIENTIFIC PUBL CO PTE LTD},
title = {{Loop} quantum gravity},
volume = {23},
year = {2008}
}
@article{faucris.120127084,
abstract = {Well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory. The difference of typical configurations of free massive scalar field theories with different masses is studied. We apply the same methods to study the Ashtekar-Lewandowski (AL) measure on spaces of connections. In particular we prove that the diffeomorphism group acts ergodically, with respect to the AL measure, on the Ashtekar-Isham space of quantum connections modulo gauge transformations. We also prove that a typical, with respect to the AL measure, quantum connection restricted to a (piecewise analytic) curve leads to a parallel transport discontinuous at every point of the curve. (C) 1999 American Institute of Physics. [S0022-2488(99)00404-1].},
author = {Mourao, José Manuel and Thiemann, Thomas and Velhinho, Jose},
faupublication = {no},
journal = {Journal of Mathematical Physics},
pages = {2337-2353},
peerreviewed = {Yes},
title = {{Physical} properties of quantum field theory measures},
volume = {40},
year = {1999}
}
@article{faucris.122018424,
abstract = {It is shown - in Ashtekar's canonical framework of General Relativity - that spherically symmetric (Schwarzschild) gravity in four-dimensional space-time constitutes a finite-dimensional completely integrable system. Canonically conjugate observables for asymptotically flat spacetimes are masses as action variables and - surprisingly - time variables as angle variables, each of which is associated with an asymptotic ''end'' of the Cauchy surfaces. The emergence of the time observable is a consequence of the Hamiltonian formulation and its subtleties concerning the slicing of space and time and is not in contradiction to Birkhoff's theorem. The results are of interest as to the concept of time in General Relativity, They can be formulated within the ADM formalism, too. Quantization of the system and the associated Schrodinger equation depend on the allowed spectrum of the masses.},
author = {Kastrup, Hans and Thiemann, Thomas},
faupublication = {no},
journal = {Nuclear Physics B},
pages = {665-686},
peerreviewed = {Yes},
title = {{SPHERICALLY} {SYMMETRICAL} {GRAVITY} {AS} {A} {COMPLETELY} {INTEGRABLE} {SYSTEM}},
volume = {425},
year = {1994}
}
@article{faucris.118959984,
abstract = {Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Isham-Lewandowski (AIL) representation, has been constructed. Recently, several uniqueness results for this representation have been worked out. In the present paper, we contribute to these efforts by showing that the AIL representation is irreducible, provided it is viewed as the representation of a certain C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories. © 2006 IOP Publishing Ltd.},
author = {Sahlmann, Hanno and Thiemann, Thomas},
doi = {10.1088/0264-9381/23/13/010},
faupublication = {no},
journal = {Classical and Quantum Gravity},
pages = {4453-4471},
peerreviewed = {Yes},
title = {{Irreducibility} of the {Ashtekar}-{Isham}-{Lewandowski} representation},
volume = {23},
year = {2006}
}
@masterthesis{faucris.106243324,
author = {Davygora, Yuriy and Thiemann, Thomas},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Kanonische} {Formulierung} der {Gravitationstheorien}},
year = {2009}
}