arXiv subjects

Thomas Thiemann

Publications and source records attributed to Thomas Thiemann.

At least 19 recordsLinked to original sources

arXiv2026

Non-perturbative, background independent canonical quantum gravity in Fock representations

It is commonly believed that a quantum field theory of General Relativity requires a non-perturbative formulation. In addition, the background independence of classical General Relativity supplies a physical selection criterion for suitable Hilbert space representations of the corresponding quantum field theory. In this contribution we show that there exist background independent representations of Fock type within the manifestly non-perturbative, canonical approach to quantum gravity. Mandatory for their existence is the presence of suitable matter fields next to the geometry field. In particular, the excitations of the corresponding Fock vacuum necessarily entangles matter and geometry. In this article we use the constraint quantisation method. We compare the resulting Fock incarnation of background independent, non-perturbative canonical quantum gravity with the well known Loop quantum gravity incarnation. One of the most important differences is that the Fock quantum gravity (FQG) Hilbert space, in contrast to the Loop quantum gravity (LQG) Hilbert space, is separable. This has many advantages when attempting to implement the Hamiltonian constraint, also known as Wheeler-DeWitt constraint, as a densely defined quadratic form.

gr-qc↗

arXiv2026

Quantum Field Theory of Black Hole Perturbations with Backreaction VI. Apparent Horizons, Quasi-Local Mass and Effective Classical Metrics

In a recent series of papers we developed a first-principle and gauge invariant approach to black hole perturbation theory valid to any order. We included back reaction effects to tackle the situation of evaporating black holes and obtained an explicit expression for the dynamics of the reduced phase space to second order. The physics of evaporating black holes is in particular encoded by apparent horizons, an observer dependent generalisation of the event horizon. We determine the shape of the apparent horizon to second order in the perturbations. The area of the apparent horizon is an interesting observable which is expected to decrease in the quantum theory due to Hawking evaporation. We show how the full four dimensional metric can be reconstructed in terms of the reduced phase space variables. In the quantum theory, taking expectation values of this metric, we obtain an effective classical metric, whose causal structure can then be visualised in a quantum corrected Penrose diagram. We conclude with an outlook into the quantisation procedure in the reduced phase space formalism and the implications on the area of the apparent horizon.

gr-qc↗

arXiv2026

Reduced phase space induced decay conditions

The definition of the phase space of field theories in presence of boundaries of Cauchy surfaces requires a choice of boundary conditions or decay behaviour of those fields. Often these conditions are motivated in part by the decay behaviour of the initial data of known exact solutions. In the case of gauge field theories the initial data are not free but are subject to initial value constraints. Still, the decay behaviour is commonly specified for the kinematical, i.e. unconstrained phase space. This can lead to the following practical problem: The constraints are preferably solved for field variables on which they depend only algebraically, i.e. not involving derivatives, as otherwise one would need to solve partial differential equations. However, the specified decay behaviour may prevent from doing that. On the other hand, a precise specification of decay for all kinematical fields appears unnecessary because the decay of gauge degrees of freedom is not observable. Yet, knowledge of their decay is required as one needs to compute Poisson brackets on the kinematical phase space in order to define what gauge invariance means. Thus the interplay between the constraint structure and the decay properties of the kinematical phase space is complex. In this contribution we develop a reduced phase space induced approach to the decay problem. Upon specifying gauge conditions tailored to the algebraic structure of the constraints, these define a split of the kinematical phase space into gauge and true degrees of freedom. Then the decay conditions of the kinematical phase space is systematically parametrised by a choice of decay for just the true degrees of freedom (i.e. the reduced phase space), the decay of the gauge degrees of freedom then follows unambiguously from solving both the constraints and the gauge conditions.

gr-qc↗

arXiv2026

(Quantum) reference frames, relational observables, gauge reduction and physical interpretation

It is mandatory to know how to operationally define and translate a reference frame into mathematics, in order that a physical interpretation of theory calculations in terms of observational data is possible. The situation is particularly challenging for gauge systems such as General Relativity where spacetime coordinates are subject to spacetime diffeomorphisms considered as gauge transformations turning coordinates into non-observables. This motivates the idea of operationally defined (material) reference frames which specify coordinates in terms of matter or geometry reference fields leading to the concept of relational observables, relational reference frames and gauge reduction. Upon quantisation, all fields become operator valued distributions. Now new conceptual and technical questions arise such as: Should one reduce before or after quantisation and how are the reference fields quantised respectively in either route? Is a reference frame itself subject to quantisation and how are different quantum reference frames related? How does the gauge reduction fit into this, i.e. how can it be that a certain reference field is considered a non-observable in one reference frame and an observable in another which upon quantisation even displays fluctuations? How precisely are gauge dependent fields interpreted in terms of the relational observables in a given reference frame? What is the relative dynamics, e.g. how exactly are physical Hamiltonians of two relational reference frames related? The present conceptual work addresses these and related questions in a non-perturbative field theory context of sufficient generality to cover General Relativity coupled to standard matter. A central role is played by the concept of the relational reference frame transformation (RRFT) for which a general formula is derived and its properties are explored.

quant-ph↗

arXiv2026

Quantum Field Theory of Black Hole Perturbations with Backreaction V. Beyond Second Order Perturbations

Black hole perturbation theory beyond second order is not well understood because typically one defines the meaning of gauge invariance order by order which is ambiguous. In this series of works we therefore developed a new approach which disentangles the meaning of gauge invariance from the perturbative order. It is based on the reduced phase space approach to the Hamiltonian formulation of General Relativity and constructs a non-perturbative, albeit implicit, formulation of the dynamics of only observables that are gauge invariant to all orders. To obtain explicit expressions, perturbation theory is then employed, but now only perturbations are considered that are gauge invariant to all orders. There are both spherically symmetric and non-symmetric observables and the formulation takes the (perturbative) backreaction between those fully into account. The formulation has access to both the exterior and interior of the dynamical horizon. In previous papers of this series we have introduced the general formalism and performed consistency checks with second order results obtained in other approaches. The real virtue of our approach starts emerging at higher than second order where we expect differences from previous works both due to backreaction effects and because we work with observables that are gauge invariant to all orders, not only up to a given order. In this paper, we consider the third order. Also new to our approach is that we start from a non-perturbative, namely polynomial, version of the constraints which therefore are finite polynomials in all degrees of freedom before reducing, rather than an infinite series. This allows for an exact and non-perturbative, while implicit, solution of the constraints which does not need to truncate the series and thus is of tremendous technical advantage.

gr-qc↗

arXiv2025

Asymptotically safe canonical quantum gravity: Gaussian dust matter

In a recent series of publications we have started to investigate possible points of contact between the canonical (CQG) and the asymptotically safe (ASQG) approach to quantum gravity, despite the fact that the CQG approach is exclusively for Lorentzian signature gravity while the ASQG approach is mostly for Euclidean signature gravity. Expectedly, the simplest route is via the generating functional of time ordered N-point functions which requires a Lorentzian version of the Wetterich equation and heat kernel methods employed in ASQG. In the present contribution we consider gravity coupled to Gaussian dust matter. This is a generally covariant Lorentzian signature system, which can be considered as a field theoretical implementation of the idealisation of a congruence of collision free test observers in free fall, filling the universe. The field theory version correctly accounts for geometry -- matter backreaction and thus in principle serves as a dark matter model. Moreover, the intuitive geometric interpretation selects a preferred reference frame that allows to disentangle gauge degrees of freedom from observables. The CQG treatment of this theory has already been considered in the past. For this particular matter content it is possible to formulate the quantum field theory of observables as a non-linear $σ$ model described by a highly non-linear conservative Hamiltonian. This allows to apply techniques from Euclidean field theory to derive the generating functional of Schwinger N-point functions which can be treated with the standard Euclidean version of the heat kernel methods employed in ASQG. The corresponding Euclidean action is closely related to Euclidean signature gravity but not identical to it despite the fact that the underlying Hamiltonian is for Lorentzian signature gravity.

hep-th↗

arXiv2024

Relational Lorentzian Asymptotically Safe Quantum Gravity: Showcase model

In a recent contribution we identified possible points of contact between the asymptotically safe and canonical approach to quantum gravity. The idea is to start from the reduced phase space (often called relational) formulation of canonical quantum gravity which provides a reduced (or physical) Hamiltonian for the true (observable) degrees of freedom. The resulting reduced phase space is then canonically quantised and one can construct the generating functional of time ordered Wightman (i.e. Feynman) or Schwinger distributions respectively from the corresponding time translation unitary group or contraction semigroup respectively as a path integral. For the unitary choice that path integral can be rewritten in terms of the Lorentzian Einstein Hilbert action plus observable matter action and a ghost action. The ghost action depends on the Hilbert space representation chosen for the canonical quantisation and a reduction term that encodes the reduction of the full phase space to the phase space of observavbles. This path integral can then be treated with the methods of asymptically safe quantum gravity in its {\it Lorentzian} version. We also exemplified the procedure using a concrete, minimalistic example namely Einstein-Klein-Gordon theory with as many neutral and massless scalar fields as there are spacetime dimensions. However, no explicit calculations were performed. In this paper we fill in the missing steps. Particular care is needed due to the necessary switch to Lorentzian signature which has strong impact on the convergence of ``heat'' kernel time integrals in the heat kernel expansion of the trace involved in the Wetterich equation and which requires different cut-off functions than in the Euclidian version. As usual we truncate at relatively low order and derive and solve the resulting flow equations in that approximation.

hep-th↗

arXiv2024

Observations on representations of the spatial diffeomorphism group and algebra in all dimensions

The canonical quantisation of General Relativity including matter on a spacetime manifold in the globally hyperbolic setting involves in particular the representation theory of the spatial diffeomorphism group (SDG), and/or its Lie algebra (SDA), of the underlying spatial submanifold. There are well known Fock representations of the SDA in one spatial dimension and non-Fock representations of the SDG in all dimensions. The latter are not strongly continuous and do not descend to representations of the SDA. In this work we report some partial results on non anomalous representations of the SDA for both geometry and matter: 1. Background independent Fock representations of the SDA by operators exist in all dimensions. 2. Infinitely many unitary equivalence classes of background dependent Fock representations of the SDA by operators exist in one dimension but these do not extend to higher dimensions. 3. Infinitely many unitary equivalence classes of background dependent Fock representations of the SDA of volume preserving diffeomorphisms by operators exist in all dimensions. 4. Infinitely many unitary equivalence classes of background dependent Fock representations of the SDA by quadratic forms exist in all dimensions. Except for 1. these representations do not descend from an invariant state of the Weyl algebra and 4. points to a new strategy for solving the quantum constraints.

gr-qc↗

arXiv2024

Non-perturbative Quantum Gravity in Fock representations

Perturbative quantum gravity starts from prescribing a background metric. That background metric is then used in order to carry out two separate steps: 1. One splits the non-perturbative metric into background and deviation from it (graviton) and expands the action in terms of the graviton which results in an ifinite series of unknown radius of convergence. 2. One constructs a Fock representation for the graviton and performs perturbative graviton quantum field theory on the fixed background as dictated by the perturbative action. The result is a non-renormalisable theory without predictive power. It is therefore widely believed that a non-perturbative approach is mandatory in order to construct a fundamental, not only effective, predictive quantum field theory of the gravitational interaction. Since perturbation theory is by definition background dependent, the notions of background dependence (BD) and perturbation theory (PT) are often considered as symbiotic, as if they imply each other. In the present work we point out that there is no such symbiosis, these two notions are in fact logically independent. In particular, one can use BD structures while while not using PT at all. Specifically, we construct BD Fock representations (step 2 above) for the full, non-perturbative metric rather than the graviton (not step 1 above) and therefore never perform a perturbative expansion. Despite the fact that the gravitational Lagrangean is a non-polynomial, not even analytic, function of the metric we show that e.g. the Hamiltonian constraint with any density weight can be defined as a quadratic form with dense form domain in such a representation.

gr-qc↗

arXiv2024

Asymptotically safe -- canonical quantum gravity junction

The canonical (CQG) and asymptotically safe (ASQG) approach to quantum gravity share to be both non-perturbative programmes. However, apart from that they seem to differ in several aspects such as: 1. Signature: CQG is Lorentzian while ASQG is mostly Euclidian. 2. Background Independence (BI): CQG is manifesly BI while ASQG is apparently not. 3. Truncations: CQG is apparently free of truncations while ASQG makes heavy use of them. The purpose of the present work is to either overcome actual differences or to explain why apparent differences are actually absent. Thereby we intend to enhance the contact and communication between the two communities. The focus of this contribution is on conceptual issues rather than deep technical details such has high order truncations. On the other hand the paper tries to be self-contained in order to be useful to researchers from both communities. The point of contact is the path integral formulation of Lorentzian CQG in its reduced phase space formulation which yields the formal generating functional of physical (i.e. gauge invariant) either Schwinger or Feynman N-point functions for (relational) observables. The corresponding effective actions of these generating functionals can then be subjected to the ASQG Wetterich type flow equations which serve in particular to find the rigorous generating fuctionals via the inverse Legendre transform of the fixed pointed effective action.

hep-th↗

arXiv2024

Symmetry reduction, gauge reduction, backreaction and consistent higher order perturbation theory

For interacting classical field theories such as general relativity exact solutions typically can only be found by imposing physically motivated (Killing) {\it symmetry} assumptions. Such highly symmetric solutions are then often used as {\it backgrounds} in a {\it perturbative} approach to more general non-symmetric solutions. If the theory is in addition a {\it gauge} theory such as general relativity, the issue arises how to consistently combine the perturbative expansion with the gauge reduction. For instance it is not granted that the corresponding constraints expanded to a given order still close under Poisson brackets with respect to the non-symmetric degrees of freedom up to higher order. If one is interested in the problem of {\it backreaction} between symmetric and non-symmetric dgrees of freedom, then one also must consider the symmetric degrees of freedom as dynamical variables which supply additional terms in Poisson brackets with respect to the symmetric degrees of freedom and the just mentioned consistency problem becomes even more complicated. In this paper we show for a general theory how to consistently combine all of these notions. The idea is to {\it first} perform the {\it exact} gauge reduction on the {\it full} phase space which results in the reduced phase space of observables and physical Hamiltonian respectively and {\it secondly} expand that physical Hamiltonian perturbatively. Surprisingly, this strategy is not only practically feasible but also avoids the above mentioned tensions. We also show how to perform the partial reduction with respect to only the asymmetric constraints but that theory is not quantisable at finite orders unless there is only one symmetric constraint.

gr-qc↗

arXiv2024

Quantum Field Theory of Black Hole Perturbations with Backreaction: I. General framework

In a seminal work, Hawking showed that natural states for free quantum matter fields on classical spacetimes that solve the spherically symmetric vacuum Einstein equations are KMS states of non-vanishing temperature. Although Hawking's calculation does not include backreaction of matter on geometry, it is more than plausible that the corresponding Hawking radiation leads to black hole evaporation which is in principle observable. Obviously, an improvement of Hawking's calculation including backreaction is a problem of quantum gravity. Since no commonly accepted quantum field theory of general relativity is available yet, it has been difficult to reliably derive the backreaction effect. An obvious approach is to use black hole perturbation theory of a Schwarzschild black hole of fixed mass and to quantise those perturbations. But it is not clear how to reconcile perturbation theory with gauge invariance beyond linear perturbations. In a recent work we proposed a new approach to this problem that applies when the physical situation has an approximate symmetry, such as homogeneity (cosmology), spherical symmetry (Schwarzschild) or axial symmetry (Kerr). The idea, which is surprisingly feasible, is to first construct the non-perturbative physical (reduced) Hamiltonian of the reduced phase space of fully gauge invariant observables and only then to apply perturbation theory directly in terms of observables. The task to construct observables is then disentangled from perturbation theory, thus allowing to unambiguosly develop perturbation theory to arbitrary orders. In this first paper of the the series we outline and showcase this approach for spherical symmetry and second order in the perturbations for Einstein-Klein-Gordon-Maxwell theory. Details and generalisation to other matter and symmetry and higher orders will appear in subsequent companion papers.

gr-qc↗

arXiv2024

Quantum Field Theory of Black Hole Perturbations with Backreaction II. Spherically symmetric 2nd order Einstein sector

In this second paper of our series we focus on the classical pure gravity sector of spherically symmetric black hole perturbations and expand the reduced Hamiltonian to second order. To compare our manifestly gauge invariant formalism with established results in the literature we have to translate our results derived in Gullstrand-Painlevé gauge to the gauges used in those works. After several canonical transformations we expectedly find exact agreement with the Hamiltonian given by Moncrief which generates the Regge-Wheeler and Zerilli equations of motion for the linear axial (often denoted odd) and polar (often denoted even) perturbations respectively. This confirms the validity of our method which immediately generalises to higher orders.

gr-qc↗

arXiv2024

Quantum Field Theory of Black Hole Perturbations with Backreaction III. Spherically symmetric 2nd order Maxwell sector

In this paper we extend reduced phase space approach to black hole perturbation theory to Maxwell matter. We expand the resulting reduced Hamiltonian to second order in the graviton and photon perturbations and find that the corresponding equations of motion match the ones derived in the literature. Accordingly our approach reproduces previous results at second order. Its real virtue lies in the fact that it extends to any order in perturbation theory in a manifestly gauge invariant fashion.

gr-qc↗

arXiv2023

Properties of a smooth, dense, invariant domain for singular potential Schroedinger operators

Schrödinger operators often display singularities at the origin, the Coulomb problem in atomic physics or the various matter coupling terms in the Friedmann-Robertson-Walker problem being prominent examples. For various applications it would be desirable to have at one's disposal an explicit basis spanning a dense and invariant domain for such types of Schrödinger operators, for instance stationary perturbation theory or the Raleigh-Ritz method. Here we make the observation, that not only a such basis can indeed be provided but that in addition relevant matrix elements and inner products can be computed analytically in closed form, thus providing the required data e.g. for an analytical Gram-Schmid orthonormalisation.

quant-ph↗

arXiv2023

Quantum gravity in the triangular gauge

Vielbeins are necessary when coupling General Relativity (GR) to fermionic matter. This enhances the gauge group of GR to include local Lorentz transformations. In view of a reduced phase space formulation of quantum gravity, in this work we completely gauge fix that Lorentz gauge symmetry by using a so-called triangular gauge. Having solved the Gauss constraints already classically opens access to new Hilbert space representations which are free of the complications that otherwise arise due to a non Abelian gauge symmetry. In that sense, a connection formulation as being pursued in Loop Quantum Gravity is no longer the only practicable option and other less dimension dependent representations e.g. based on triads and even metrics suggest themselves. These formulations make it easier to identify states representing non-degenerate quantum geometries and thus to investigate the hypersurface deformation algebra which implicitly assumes non-degeneracy.

gr-qc↗

arXiv2023

Hamiltonian Theory: Dynamics

This chapter focuses on the status of the implementation of the dynamics in the canonical version of Loop Quantum Gravity (LQG). Concretely this means to provide a mathematical meaning of the quantum Einstein equations, sometimes called Wheeler-DeWitt equations, to give a physical interpretation and Hilbert space structure to its solutions and to construct a representation of the algebra of observables including a physical Hamiltonian. This is a structural overview intentionally skipping technical details.

gr-qc↗

arXiv2021

Reduced Phase Space Approach to the $U(1)^3$ model for Euclidean Quantum Gravity

If one replaces the constraints of the Ashtekar-Barbero $SU(2)$ gauge theory formulation of Euclidean gravity by their $U(1)^3$ version, one arrives at a consistent model which captures significant structure of its $SU(2)$ version. In particular, it displays a non trivial realisation of the hypersurface deformation algebra which makes it an interesting testing ground for (Euclidean) quantum gravity as has been emphasised in a recent series of papers due to Varadarajan et al. In this paper we consider a reduced phase space approach to this model. This is especially attractive because, after a canonical transformation, the constraints are at most {\it linear} in the momenta. In suitable gauges, it is therefore possible to find a closed and explicit formula for the physical Hamiltonian which depends only on the physical observables. Not surprisingly, that physical Hamiltonian is generically neither polynomial nor spatially local. The corresponding reduced phase space quantisation can be confronted with the constraint quantisation due to Varadarajan et al to gain further insights into the quantum realisation of the hypersurface deformation algebra.

gr-qc↗