arXiv subjects

Thomas Thiemann

Publications and source records attributed to Thomas Thiemann.

At least 37 records · Page 2Linked to original sources

arXiv2021

Covariant Origin of the $U(1)^3$ model for Euclidean Quantum Gravity

The utility of the U(1)$^3$ model as a test laboratory for quantum gravity has recently been emphasized in a recent series of papers due to Varadarajan et al. The simplification from SU(2) to U(1)$^3$ can be performed simply by hand within the Hamiltonian formulation by dropping all non-Abelian terms from the Gauss, spatial diffeomorphism and Hamiltonian constraints respectively. However, one may ask from which Lagrangian formulation this theory descends. For the SU(2) theory it is known that one can choose the Palatini action, Holst action or (anti-)selfdual action (Euclidian signature) as starting point all leading to equivalent Hamiltonian formulations. In this paper we systematically analyse this question directly for the U(1)$^3$ theory. Surprisingly, it turns out that the Abelian analog of the Palatini or Holst formulation is a consistent but topological theory without propagating degrees of freedom. On the other hand, a twisted Abelian analog of the (anti-)selfdual formulation does lead to the desired Hamiltonian formulation. A new aspect of our derivation is that we work with 1. half-density valued tetrads which simplifies the analysis, 2. without the simplicity constraint (which admits one undesired solution that is usually neglected by hand) and 3. without imposing the time gauge from the beginning. As a byproduct we show that also the non-Abelian theory admits a twisted (anti-)selfdual formulation. Finally we also derive a pure connection formulation of Euclidian GR including a cosmological constant by extending previous work due to Capovilla, Dell, Jacobson and Peldan which may be an interesting starting point for path integral investigations and displays (Euclidian) GR as a Yang-Mills theory with non-polynomial Lagrangian.

gr-qc↗

arXiv2021

On propagation in Loop Quantum Gravity

A rigorous implementation of the Wheeler-Dewitt equations was derived in the context of Loop Quantum Gravity (LQG) and was coined Quantum Spin Dynamics (QSD). The Hamiltonian constraint of QSD was criticised as being too local and to prevent "propagation" in canonical LQG. That criticism was based on an algorithm developed for QSD for generating solutions to the Wheeler-DeWitt equations. The fine details of that algorithm could not be worked out because the QSD Hamiltonian constraint makes crucial use of the volume operator which cannot be diagonalised analytically. In this paper, we consider the U(1)$^3$ model for Euclidean vacuum LQG which consists in replacing the structure group SU(2) by U(1)$^3$ and otherwise keeps all properties of the SU(2) theory intact. This enables analytical calculations and the fine details of the algorithm {\it can} be worked out. By considering one of the simplest possible non-trivial class of solutions based on very small graphs, we show that 1. an infinite number of solutions {\it exist} which are 2. generically {\it not normalisable} with respect to the inner product on the space of spatially diffeomorphism invariant distributions and 3. generically display {\it propagation}. Due to the closeness of the U(1)$^3$ model to Euclidean LQG, it is extremely likely that all three properties hold also in the SU(2) case and even more so in physical Lorentzian LQG. These arguments can in principle be made water tight using modern numerical (e.g. ML or QC) methods combined with the techniques developed in this paper which we reserve for future work.

gr-qc↗

arXiv2020

Asymptotically flat boundary conditions for the $U(1)^3$ model for Euclidean Quantum Gravity

A generally covariant $U(1)^3$ gauge theory describing the $G_N \to 0$ limit of Euclidean general relativity is an interesting test laboratory for general relativity, specially because the algebra of the Hamiltonian and diffeomorphism constraints of this limit is isomorphic to the algebra of the corresponding constraints in general relativity. In the present work, we study boundary conditions and asymptotic symmetries of the $U(1)^3$ model and show that while asymptotic spacetime translations admit well-defined generators, boosts and rotations do not. Comparing with Euclidean general relativity, one finds that exactly the non-Abelian part of the $SU(2)$ Gauss constraint which is absent in the $U(1)^3$ model plays a crucial role in obtaining boost and rotation generators.

gr-qc↗

arXiv2020

Canonical Quantum Gravity, Constructive QFT and Renormalisation

The canonical approach to quantum gravity has been put on a firm mathematical foundation in the recent decades. Even the quantum dynamics can be rigorously defined, however, due to the tremendously non-polynomial character of the gravitational interaction, the corresponding Wheeler-DeWitt operator valued distribution suffers from quantisation ambiguities that need to be fixed. In a very recent series of works we have employed methods from constructive quantum field theory in order to address those ambiguities. Constructive QFT trades quantum fields for random variables and measures thereby phrasing the theory in the language of quantum statistical physics. The connection to the canonical formulation is made via Osterwalder-Schrader reconstruction. It is well known in quantum statistics that the corresponding ambiguities in measures can be fixed using renormalisation. The associated renormalisation flow can thus be used to define a canonical renormalisation programme. The purpose of this article is to review and further develop these ideas and to put them into context with closely related earlier and parallel programmes.

gr-qc↗

arXiv2020

Hamiltonian Renormalisation V: Free Vector Bosons

In a recent proposal we applied methods from constructive QFT to derive a Hamiltonian Renormalisation Group in order to employ it ultimately for canonical quantum gravity. The proposal was successfully tested for free scalar fields and thus a natural next step is to test it for free gauge theories. This can be done in the framework of reduced phase space quantisation which allows using techniques developed earlier for scalar field theories. In addition, in canonical quantum gravity one works in representations that support holonomy operators which are ill defined in the Fock representation of say Maxwell or Proca theory. Thus, we consider toy models that have both features, i.e. which employ Fock representations in which holonomy operators are well-defined. We adapt the coarse graining maps considered for scalar fields to those theories for free vector bosons. It turns out that the corresponding fixed pointed theories can be found analytically.

gr-qc↗

arXiv2019

Hamiltonian diagonalization in hybrid quantum cosmology

We explore the possibility of selecting a natural vacuum state for scalar and tensor gauge-invariant cosmological perturbations in the context of hybrid quantum cosmology, by identifying those variables for the description of the perturbations that display a dynamical behavior adapted to the evolution of the entire cosmology. We make use of a canonical formulation of the whole of the cosmological system in which the perturbative gauge-invariant degrees of freedom are identified as canonical variables. Introducing background-dependent linear canonical transformations on these perturbations and completing them for the entire system, we are able to characterize a generic collection of annihilation and creationlike variables that obey the dynamics dictated by a respective collection of Hamiltonians. We then impose that such Hamiltonians possess no self-interaction terms so that, in a Fock representation with normal ordering, they act diagonally on the basis of $n$-particle states. This leads to a semilinear first-order partial differential equation with respect to the background for the coefficients that define the annihilation and creationlike variables for all Fourier modes, as well as to a very precise ultraviolet characterization of them. Such first-order equation contains, in the imaginary part of its complex solutions, the complicated second-order field equation that typically arises for the time-dependent frequency of the perturbations in the context of quantum field theory in curved spacetimes. We check that the asymptotic knowledge acquired allows one to select the standard vacua in Minkowski and de Sitter spacetimes. Finally, we discuss the relation of our vacuum and the standard adiabatic vacua, and check that our asymptotic characterization of variables with a diagonal Hamiltonian displays the properties that would be desirable for an adiabatic state of infinite order.

gr-qc↗

arXiv2019

Hamiltonian Renormalization III. Renormalisation Flow of 1+1 dimensional free scalar fields: Properties

This is the third paper in a series of four in which a renormalisation flow is introduced which acts directly on the Osterwalder-Schrader data (OS data) without recourse to a path integral. Here the OS data consist of a Hilbert space, a cyclic vacuum vector therein and a Hamiltonian annihilating the vacuum which can be obtained from an OS measure, that is a measure respecting (a subset of) the OS axioms. In the previous paper we successfully tested our proposal for the two-dimensional massive Klein-Gordon model, that is, we could confirm that our framework finds the correct fixed point starting from a natural initial naive discretisation of the finite resolution Hamiltonians, in particular the underlying Laplacian on a lattice, and a natural coarse graining map that drives the renormalisation flow. However, several questions remained unanswered. How generic can the initial discretisation and the coarse graining map be in order that the fixed point is not changed or is at least not lost, in other words, how universal is the fixed point structure? Is the fix point in fact stable, that is, is the fixed point actually a limit of the renormalisation sequence? We will address these questions in the present paper.

gr-qc↗

arXiv2019

Hamiltonian Renormalisation IV. Renormalisation Flow of D+1 dimensional free scalar fields and Rotation Invariance

In this article we extend the test of Hamiltonian Renormalisation proposed in this series of articles to the D-dimensional case using a massive free scalar field. The concepts we introduce are explicitly computed for the D=2 case but transfer immediately to higher dimensions. In this article we define and verify a criterion that monitors, at finite resolution defined by a cubic lattice, whether the flow approaches a rotationally invariant fixed point.

gr-qc↗

arXiv2019

Hamiltonian Renormalisation II. Renormalisation Flow of 1+1 dimensional free scalar fields: Derivation

In the companion paper we motivated a renormalisation flow on Osterwalder-Schrader data (OS-data) consisting of 1. a Hilbert space, 2. a cyclic vacuum and 3. a Hamiltonian annihilating that vacuum. As the name suggests, the motivation was via the OS reconstruction theorem which allows to reconstruct the OS data from an OS measure satisfying (a subset of) the OS axioms, in particular reflection positivity. The guiding principle was to map the usual Wilsonian path integral renormalisation flow onto a flow of the corresponding OS data. We showed that this induced flow on the OS data has an unwanted feature which disqualifies the associated coarse grained Hamiltonians from being the projections of a continuum Hamiltonian onto vectors in the coarse grained Hilbert space. This motivated the definition of a direct Hamiltonian renormalisation flow which follows the guiding principle but does not suffer from the afore mentioned caveat. In order to test our proposal, we apply it to the only known completely solvable model, namely the case of free scalar quantum fields. In this paper we focus on the Klein Gordon field in two spacetime dimensions and illustrate the difference between the path integral induced and direct Hamiltonian flow. Generalisations to more general models in higher dimensions will be discussed in our companion papers.

gr-qc↗

arXiv2019

Hamiltonian Renormalisation I: Derivation from Osterwalder-Schrader Reconstruction

A possible avenue towards a non-perturbative Quantum Field Theory (QFT) on Minkowski space is the constructive approach which employs the Euclidian path integral formulation, in the presence of both ultraviolet (UV) and infrared (IR) regulators, as starting point. The UV regulator is to be taken away by renormalisation group techniques which in case of success leads to a measure on the space of generalised Euclidian fields in finite volume. The IR regulator corresponds to the thermodynamic limit of the system in the statistical physics sense. If the resulting measure obeys the Osterwalder-Schrader axioms, the actual QFT on Minkowski space is then obtained by Osterwalder-Schrader reconstruction. In this work we study the question whether it is possible to reformulate the renormalisation group non-perturbatively directly at the operator (Hamiltonian) level. Hamiltonian renormalisation would be the natural route to follow if one had easier access to an interacting Hamiltonian operator rather than to a path integral, at least in the presence of UV and/or IR cut-off, which is generically the case in complicated gauge theories such as General Relativity. Our guiding principle for the definition of the direct Hamiltonian renormalisation group is that it results in the same continuum theory as the covariant (path integral) renormalisation group. In order to achieve this, we invert the Osterwalder-Schrader reconstruction, which may be called Osterwalder-Schrader construction of a Wiener measure from the underlying Hamiltonian theory. The resulting correspondence between reflection positive measures and Osterwalder-Schrader data consisting of a Hilbert space, a Hamiltonian and a ground state vector allows us to monitor the effect of the renormalisation flow of measures in terms of their Osterwalder-Schrader data which motivates a natural direct Hamiltonian renormalisation scheme.

gr-qc↗

arXiv2019

Holomorphic fractional Fourier transforms

The Fractional Fourier Transform (FrFT) has widespread applications in areas like signal analysis, Fourier optics, diffraction theory, etc. The Holomorphic Fractional Fourier Transform (HFrFT) proposed in the present paper may be used in the same wide range of applications with improved properties. The HFrFT of signals spans a one-parameter family of (essentially) holomorphic functions, where the parameter takes values in the bounded interval $t\in (0,π/2)$. At the boundary values of the parameter, one obtains the original signal at $t=0$ and its Fourier transform at the other end of the interval $t=π/2$. If the initial signal is $L^2 $, then, for an appropriate choice of inner product that will be detailed below, the transform is unitary for all values of the parameter in the interval. This transform provides a heat kernel smoothening of the signals while preserving unitarity for $L^2$-signals and continuously interpolating between the original signal and its Fourier transform.

math-ph↗

arXiv2019

Segal-Bargmann transforms from hyperbolic Hamiltonians

We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schrödinger polarization and study the corresponding evolution of polarized sections. The flow is periodic in imaginary time and the evolution of polarized sections has interesting features. On the time intervals for which the polarization is real or Kähler, the half--form corrected time evolution of polarized sections is given by unitary operators which turn out to be equivalent to the classical Segal-Bargmann transforms (which are usually associated to the quadratic elliptic Hamiltonian $H=\frac12 p^2$ and to the heat operator). At the right endpoint of these intervals, the evolution of polarized sections is given by the Fourier transform from the Schrödinger to the momentum representation. In the complementary intervals of imaginary time, the polarizations are anti--Kähler and the Hilbert space of polarized sections collapses to ${\mathcal H}= \{0\}$. Hyperbolic quadratic Hamiltonians thus give rise to a new factorization of the Segal-Bargmann transform, which is very different from the usual one, where one first applies a bounded contraction operator (the heat kernel operator), mapping $L^2$--states to real analytic functions with unique analytic continuation, and then one applies analytic continuation. In the factorization induced by an hyperbolic complexifier, both factors are unbounded operators but their composition is, in the Kähler or real sectors, unitary. In another paper [KMNT], we explore the application of the above family of unitary transforms to the definition of new holomorphic fractional Fourier transforms.

math-ph↗

arXiv2018

Uniqueness of the Representation in Homogeneous Isotropic LQC

We show that the standard representation of homogeneous isotropic loop quantum cosmology (LQC) is the GNS-representation that corresponds to the unique state on the reduced quantum holonomy-flux $^*$-algebra that is invariant under residual diffeomorphisms $-$ both when the standard algebra is used as well as when one uses the extended algebra proposed by Fleischhack. More precisely, we find that in both situations the GNS-Hilbert spaces coincide, and that in the Fleischhack case the additional algebra elements are just mapped to zero operators. In order for the residual diffeomorphisms to have a well-defined action on the quantum algebra, we have let them act on the fiducial cell as well as on the dynamical variables, thereby recovering covariance. Consistency with Ashtekar and Campiglia in the Bianchi I case is also shown.

gr-qc↗

arXiv2016

Towards the fundamental spectrum of the Quantum Yang-Mills Theory

In this work we focus on the quantum Einstein-Yang-Mills sector quantised by the methods of Loop Quantum Gravity (LQG). We point out the improved UV behaviour of the coupled system as compared to pure quantum Yang-Mills theory on a fixed, classical background spacetime as was considered in a seminal work by Kogut and Susskind. Furthermore, we develop a calculational scheme by which the fundamental spectrum of the quantum Yang-Mills Hamiltonian can be computed in principle and by which one can make contact to the Wilsonian renormalization group, possibly purely within the Hamiltonian framework. Finally, we comment on the relation of the fundamental spectrum to that of pure Yang-Mills theory on a (flat) classical spacetime.

gr-qc↗

arXiv2015

Projective Limits of State Spaces IV. Fractal Label Sets

Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski [Kijowski 1977] to represent quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces. One can thus bypass the need to select a vacuum state for the theory, and still be provided with an explicit and constructive description of the quantum state space, at least as long as the label set indexing the projective structure is countable. Because uncountable label sets are much less practical in this context, we develop in the present article a general procedure to trim an originally uncountable label set down to countable cardinality. In particular, we investigate how to perform this tightening of the label set in a way that preserves both the physical content of the algebra of observables and its symmetries. This work is notably motivated by applications to the holonomy-flux algebra underlying Loop Quantum Gravity. Building on earlier work by Okolow [arXiv:1304.6330], a projective state space was introduced for this algebra in [arXiv:1411.3592]. However, the non-trivial structure of the holonomy-flux algebra prevents the construction of satisfactory semi-classical states. Implementing the general procedure just mentioned in the case of a one-dimensional version of this algebra, we show how a discrete subalgebra can be extracted without destroying universality nor diffeomorphism invariance. On this subalgebra, states can then be constructed whose semi-classicality is enforced step by step, starting from collective, macroscopic degrees of freedom and going down progressively toward smaller and smaller scales.

gr-qc↗

arXiv2015

Projective Loop Quantum Gravity II. Searching for Semi-Classical States

In [arXiv:1411.3592] an extension of the Ashtekar-Lewandowski (AL) state space of Loop Quantum Gravity was set up with the help a projective formalism introduced by Kijowski [Kijowski 1977; see also: arXiv:1304.6330, arXiv:1411.3590]. The motivation for this work was to achieve a more balanced treatment of the position and momentum variables (aka. holonomies and fluxes). Indeed, states in the AL Hilbert spaces describe discrete quantum excitations on top of a vacuum which is an eigenstate of the flux variables (a `no-geometry' state): in such states, most holonomies are totally spread, making it difficult to approximate a smooth, classical 4-geometry. However, going beyond the AL sector does not fully resolve this difficulty: one uncovers a deeper issue hindering the construction of states semi-classical with respect to a full set of observables. In the present article, we analyze this issue in the case of real-valued holonomies (we will briefly comment on the heuristic implications for other gauge groups, eg. $\mathcal{SU}(2)$). Specifically, we show that, in this case, there does not exist any state on the holonomy-flux algebra in which the variances of the holonomies and fluxes 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: as there are too many (uncountably many) non-vanishing commutators between the holonomy and flux operators, the corresponding Heisenberg inequalities force the quantum uncertainties to blow up uncontrollably. A way out would be to suitably restrict the algebra of observables. In a companion paper we take the first steps in this direction by developing a general framework to perform such a restriction without giving up the universality and diffeomorphism invariance of the theory.

gr-qc↗

arXiv2015

Stable coherent states

We analyze the stability under time evolution of complexifier 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 poses stable CCS if the classical evolution of the variable 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 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 to certain regions in the phase space by introducing action-angle coordinates.

gr-qc↗

arXiv2015

Structural aspects of loop quantum gravity and loop quantum cosmology from an algebraic perspective

We comment on structural properties of the algebras $\mathfrak{A}_{LQG/LQC}$ underlying loop quantum gravity and loop quantum cosmology, especially the representation theory, relating the appearance of the (dynamically induced) superselection structure ($θ$-sectors) in loop quantum cosmology to recently proposed representations with non-degenerate background geometries in loop quantum gravity with Abelian structure group. To this end, we review and employ the concept of extending a given (observable) algebra with possibly non-trivial centre to a (charged) field algebra with (global) gauge group.We also interpret the results in terms of the geometry of the structure group G. Furthermore, we analyze the Koslowski-Sahlmann representations with non-degenerate background in the case of a non-Abelian structure group. We find that these representations can be interpreted from two different, though related, points view: Either, the standard algebras of loop quantum gravity need to be extended by a (possibly) central term, or the elementary flux vector fields need to acquire a shift related to the (classical) background to make these representations well-defined. Both perspectives are linked by the fact that the background shift is not an automorphism of the algebras, but rather an affine transformation. Finally, we show how similar algebraic mechanisms, which are used to explain the breaking of chiral symmetry and the occurrence of $θ$-vacua in quantum field theory, extend to loop quantum gravity. Thus, opening a path for the discussion of these questions in loop quantum gravity.

gr-qc↗