arXiv subjects

T. Thiemann

Publications and source records attributed to T. Thiemann.

At least 19 recordsLinked to original sources

arXiv2026

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

A UV complete quantum field theory of general relativity is believed to require a non-perturbative approach. Moreover, background independence of classical general relativity supplies a physical selection for suitable Hilbert space representations of the quantum geometry and matter fields. In this contribution we show that, contrary to common intuition, there exist rigorous, background independent Fock representations available for a non-perturbative canonical quantisation of geometry and suitable matter fields. This is interesting because the Fock Hilbert space is separable while the Hilbert space of other manifestly background independent and non-perturbative canonical quantisation programmes is not. Since non-separability is a source for quantisation ambiguities, such a Fock representation may help to arrive at a significantly more predictive theory. As a simple application we discuss the cosmological truncation and mechanisms for quantum bounces. To make this manuscript concise we focus on the simplest incarnation of this idea. More details and many extensions are supplied in a companion paper.

gr-qc↗

arXiv2025

Rigorous test of the Raleigh-Ritz method for Mexican hat type potentials

Interesting quantum integrable models are rare and one often has to resort to approximation methods. One of these is the Raleigh Ritz method which under certain circumstances allows to approximately compute the lowest energy eigenstate (or ground state) of a given Hamiltonian whose pure point spectrum is bounded from below. The quality of such approximations can then be tested numerically or sometimes by abstract arguments. However, the numerical test is limited by computing power. In order to perform a rigorous test, one would need to have at one's disposal 1. a physically interesting model that is 2. solvable to sufficient extent in order that 3. the exact ground state is known in closed form. In this contribution we show that certain anharmonic potentials of the Mexican hat type belong to this class of models. The corresponding Schroedinger type Hamiltonian can be considered as a crude quantum mechanical toy model Hamiltonian for the Higgs field in the standard model of elementary particle physics.

quant-ph↗

arXiv2025

Hamiltonian renormalisation VIII. P(Phi,2) quantum field theory

In previous works in this series we focussed on Hamiltonian renormalisation of free field theories in all spacetime dimensions. In this paper we address the Hamiltonian renormalisation of the self-interacting scalar field in two spacetime dimensions with polynomial potential, called P(Phi,2). We consider only the finite volume case. The P(Phi,2) theory is one of the few interacting QFT's that can be rigorously constructed non-perturbatively. We find that our Hamiltonian renormalisation flow finds this theory indeed as a fixed point.

hep-th↗

arXiv2025

Hamiltonian renormalisation IX. U(1)**3 quantum gravity

In previous works in this series we focussed on Hamiltonian renormalisation of free field theories in all spacetime dimensions or interacting theories in spacetime dimensions lower than four. In this paper we address the Hamiltonian renormalisation of the U(1)**3 model for Euclidian general relativity in four spacetime dimensions which is self-interacting. The Hamiltonian flow needs as an input a choice of *-algebra and corresponding representation thereof or state on it at each resolution scale. If one uses as input the algebras and states that were used in the recent exact solutions of this model, then one finds that the flow finds as fixed point those exact solution theories.

gr-qc↗

arXiv2023

Smooth, invariant orthonormal basis for singular potential Schroedinger operators

In a recent contribution we showed that there exists a smooth, dense domain for singular potential Schrödinger operators on the real line which is invariant under taking derivatives of arbitrary order and under multiplication by positive and negative integer powers of the coordinate. Moreover, inner products between basis elements of that domain were shown to be easily computable analytically. A task left open was to construct an orthonormal basis from elements of that domain by using Gram-Schmidt orthonormalisation. We perform that step in the present manuscript. We also consider the application of these methods to the positive real line for which one can no longer perform the integrals analytically but for which one can give tight analytical estimates.

quant-ph↗

arXiv2022

Hamiltonian renormalisation VI: Parametrised field theory on the cylinder

Hamiltonian Renormalisation, as defined within this series of works, was derived from covariant Wilson renormalisation via Osterwalder-Schrader reconstruction. As such it directly applies to QFT with a true (physical) Hamiltonian bounded from below. The validity of the scheme was positively tested for free QFT in any dimension with or without Abelian gauge symmetries of Yang-Mills type. The aim of this Hamiltonian renormalisation scheme is to remove quantisation ambiguities of Hamiltonians in interacting QFT that remain even after UV and IR regulators are removed as it happens in highly non-linear QFT such as quantum gravity. While not derived for that case, the renormalisation flow formulae can without change also be applied to QFT without a single true Hamiltonian but rather an infinite number of Hamiltonian constraints. Then a central question is how the constraint algebra reacts to the renormalisation flow. This question should ultimately be addressed in quantum gravity. Before one considers this interacting, constrained QFT it is well motivated to consider a free, constrained QFT where the fixed point is explicitly known. In this paper we therefore address the case of parametrised field theory for which the quantum constraint algebra coincides simultaneously with the hypersurface deformation algebra of quantum gravity (or any other generally covariant theory) and the Virasoro algebra of free, closed, bosonic string theory or other CFT to which the results of this paper apply verbatim. The central result of our investigation is that finite resolution (discretised) constraint algebras {\it must not close} and that anomaly freeness of the continuum algebra is encoded in the convergence behaviour of the renormalisation flow.

gr-qc↗

arXiv2022

Hamiltonian Renormalisation VII: Free fermions and doubler free kernels

The Hamiltonian renormalisation programme motivated by constructive QFT and Osterwalder-Schrader reconstruction which was recently launched for bosonic field theories is extended to fermions. As fermion quantisation is not in terms of measures, the scheme has to be mildly modified accordingly. We exemplify the scheme for free fermions both for compact and non-compact spatial topologies respectively (i.e. with and without IR cut-off) and demonstrate that the convenient Dirichlet or Shannon coarse graining kernels recently advertised in a companion paper lead to a manifestly doubler free flow.

hep-th↗

arXiv2022

Renormalisation, wavelets and the Dirichlet-Shannon kernels

In constructive quantum field theory (CQFT) it is customary to first regularise the theory at finite UV and IR cut-off. Then one first removes the UV cutoff using renormalisation techniques applied to families of CQFT's labelled by finite UV resolutions and then takes the thermodynamic limit. Alternatively, one may try to work directly without IR cut-off. More recently, wavelets have been proposed to define the renormalisation flow of CQFT's which is natural as they come accompanied with a multi-resolution analysis (MRA). However, wavelets so far have been mostly studied in the non-compact case. Practically useful wavelets that display compact support and some degree of smoothness can be constructed on the real line using Fourier space techniques but explicit formulae as functions of position are rarely available. Compactly supported wavelets can be periodised by summing over period translates keeping orthogonality properties but still yield to rather complicated expressions which generically lose their smoothness and position locality properties. It transpires that a direct approach to wavelets in the compact case is desirable. In this contribution we show that the Dirichlet-Shannon kernels serve as a natural scaling function to define generalised orthonormal wavelet bases on tori or copies of real lines respectively. These generalised wavelets are smooth, are simple explicitly computable functions, display quasi-local properties close to the Haar wavelet and have compact momentum supprt. Accordingly they have a built-in cut-off both in position and momentum, making them very useful for renormalisation applications.

hep-th↗

arXiv2022

Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity

A basic assumption in classical GR is that the metric field is nowhere degenerate in spacetime. In particular the induced metric on Cauchy surfaces must be nowhere degenerate. It is only under this assumption that one can derive the hypersurace deformation algebra between the initial value constraints which is absolutely transparent from the fact that the {\it inverse} of the induced metric is needed to close the algebra. This statement is independent of the density weight that one may want to equip the spatial metric with. Accordingly, the very definition of a non-anomalous representation of the hypersurface defomation algebra in quantum gravity has to address the issue of non-degenracy of the induced metric that is needed in the classical theory. In the Hilbert space representation employed in Loop Quantum Gravity (LQG) most emphasis has been layed to define an inverse metric operator on the dense domain of spin network states although they represent induced quantum geometries which are degenerate almost everywhere. It is no surprise that demonstration of closure of the constraint algebra on this domain meets difficulties because it is a sector of the quantum theory which is classically forbidden and which lies outside the domain of definition of the classical hypersurface deformation algebra. Various suggestions for addressing the issue such as non-standard operator topologies, dual spaces (habitats) and density weights have been propposed to address this issue with respect to the quantum dynamics of LQG. In this article we summarise these developments and argue that insisting on a dense domain of non-degenerate states within the LQG representation may provide a natural resolution of the issue thereby possibly avoiding the above mentioned non-standard constructions.

gr-qc↗

arXiv2022

Exact quantisation of U(1)$^3$ quantum gravity via exponentiation of the hypersurface deformation algebroid

The U(1)$^3$ model for 3+1 Euclidian signature general relativity is an interacting, generally covariant field theory with two physical polarisations that shares many features of Lorentzian general relativity. In particular, it displays a non-trivial realisation of the hypersurface deformation algebroid with non-trivial, i.e. phase space dependent structure functions rather than structure constants. In this paper we show that the model admits {\it an exact quantisation}. The quantisation rests on the observation that for this model and in the chosen representation of the canonical commutation relations the density unity hypersurface algebra {\it can be exponentiated on non-degenerate states}. These are states that represent a non-degenerate quantum metric and from a classical perspective are the relevant states on which the hypersurface algebra is representable. The representation of the algebra is exact, with no ambiguities involved and anomaly free. The quantum constraints can be exactly solved using {\it groupoid averaging} and the solutions admit a Hilbert space structure that agrees with the quantisation of a recently found reduced phase space formulation. Using the also recently found covariant action for that model, we start a path integral or spin foam formulation which, due to the Abelian character of the gauge group, is much simpler than for Lorentzian signature general relativity and provides an ideal testing ground for general spin foam models.

gr-qc↗

arXiv2021

Backreaction in Cosmology

In this review, we investigate the question of backreaction in different approaches to cosmological perturbation theory, and with a special focus on quantum theoretical aspects. By backreaction, we refer here to the effects of matter field or cosmological inhomogeneities on the homogeneous dynamical background degrees of freedom of cosmology. We begin with an overview of classical cosmological backreaction which is ideally suited for physical situations in the late time Universe. We then proceed backwards in time, considering semiclassical approaches such as semiclassical or stochastic (semiclassical) gravity which take quantum effects of the perturbations into account. Finally, we review approaches to backreaction in quantum cosmology that should apply to the very early Universe where classical and semiclassical approximations break down. The main focus is on a recently proposed implementation of backreaction in quantum cosmology using a Born-Oppenheimer inspired method.

gr-qc↗

arXiv2019

Quantum Cosmological Backreactions IV: Constrained Quantum Cosmological Perturbation Theory

This is the fourth paper in a series of four in which we use space adiabatic methods in order to incorporate backreactions among the homogeneous and between the homogeneous and inhomogeneous degrees of freedom in quantum cosmological perturbation theory. In this paper, we finally consider the gauge invariant scalar (Mukhanov-Sasaki) and tensor (primordial gravitational wave) inhomogeneous perturbations of General Relativity coupled to an inflaton field which arise from a careful constraint analysis of this system up to second order in the perturbations. The simultaneous quantisation of the homogeneous and inhomogeneous degrees of freedom suggests the space adiabatic perturbation theory as an approximation scheme in order to capture the backreaction effects between these two sets of degrees of freedom. We are confronted with all the challenges at once that we found in the simpler models treated in this series of papers. We are able to compute these effects up to second order in the adiabatic parameter and find significant modifications as compared to earlier derivations of the effective quantum dynamics of the homogeneous sector.

gr-qc↗

arXiv2019

Quantum Cosmological Backreactions I: Cosmological Space Adiabatic Perturbation Theory

The search for quantum gravity fingerprints in currently available cosmological data that have their origin from the Planck era is of growing interest due to major recent progress both in the theoretical modelling as well as the observational precision. Unsurprisingly, the theoretical predictions are very sensitive to the quantum effects that occur close to the classical big bang singularity. It is therefore of substantial interest to describe these effects as precisely as possible. This is the first in a series of papers that aim at improving on the treatment of quantum effects that arise due to backreactions between matter and geometry. The technique we employ is space adiabatic perturbation theory (SAPT) in the form developed in seminal papers by Panati, Spohn and Teufel. SAPT is a generalisation of the more familiar Born Oppenheimer Approximation (BOA) that applies well in systems that allow a split of the degrees of freedom into two sets that propagate on rather different time scales such as the homogeneous and inhomogeneous field modes in cosmology. We will show that this leads to presently neglected correction terms in the quantum Friedman equations. In the present paper we adapt and generalise SAPT to the hybrid approach to quantum cosmology developed by Mena Marugan et al. that allows for a systematic quantum separation of the (in)homogeneous modes. Since SAPT was developed for quantum mechanics rather than quantum field theory, several challenges have to be met.

gr-qc↗

arXiv2019

Quantum Cosmological Backreactions II: Purely Homogeneous Quantum Cosmology

This is the second paper in a series of four in which we use space adiabatic methods in order to incorporate backreactions among the homogeneous and between the homogeneous and inhomogeneous degrees of freedom in quantum cosmological perturbation theory. The purpose of the present paper is twofold. On the one hand, it illustrates the formalism of space adiabatic perturbation theory (SAPT) for two simple quantum mechanical toy models. On the other, it proves the main point, namely that backreactions lead to additional correction terms in effective Hamiltonians that one would otherwise neglect in a crude Born-Oppenheimer approximation. The first model that we consider is a harmonic oscillator coupled to an anharmonic oscillator. We chose it because it displays many similarities with the more interesting second model describing the coupling between an inflaton and gravity restricted to the purely homogeneous and isotropic sector. These results have potential phenomenological consequences in particular for quantum cosmological theories describing big bounces such as Loop Quantum Cosmology (LQC).

gr-qc↗

arXiv2019

Quantum Cosmological Backreactions III: Deparametrised Quantum Cosmological Perturbation Theory

This is the third paper in a series of four in which we use space adiabatic methods in order to incorporate backreactions among the homogeneous and between the homogeneous and inhomogeneous degrees of freedom in quantum cosmological perturbation theory. In this paper we consider a particular kind of cosmological perturbation theory which starts from a gauge fixed version of General Relativity. The gauge fixing is performed using a material reference system called Gaussian dust. The resulting system has no constraints any more but possesses a physical Hamiltonian that drives the dynamics of both geometry and matter. As observable matter content we restrict to a scalar field (inflaton). We then explore the sector of that theory which is purely homogeneous and isotropic with respect to the geometry degrees of freedom but contains inhomogeneous perturbations up to second order of the scalar field. The purpose of this paper is to explore the quantum field theoretical challenges of the space adiabatic framework in a cosmological model of inflation which is technically still relatively simple. We compute the quantum backreaction effects from every energy band of the inhomogeneous matter modes on the evolution of the homogeneous geometry up to second order in the adiabatic parameter. These contributions turn out to be significant due to the infinite number of degrees of freedom and are very sensitive to the choice of Fock representation chosen for the inhomogeneous matter modes.

gr-qc↗

arXiv2014

Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions

We introduce a notion of a weak Poisson structure on a manifold $M$ modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra $\cA \subeq C^\infty(M)$ which has to satisfy a non-degeneracy condition (the differentials of elements of $\cA$ separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields. Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras. As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form $κ$ on a Lie algebra $\g$ and a $κ$-skew-symmetric derivation $D$ a weak affine Poisson structure on $\g$ itself. This leads naturally to a concept of a Hamiltonian $G$-action on a weak Poisson manifold with a $\g$-valued momentum map and hence to a generalization of quasi-hamiltonian group actions.

math.DG↗

arXiv2009

Manifestly Gauge-Invariant General Relativistic Perturbation Theory: I. Foundations

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 n'th 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 realtional 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 generalisation to higher orders. This will also allow us to settle the stability issue in a future publication.

gr-qc↗

arXiv2009

LTB spacetimes in terms of Dirac observables

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 non invariant 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 Kucha{\v r}, 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 article 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.

gr-qc↗