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Utilizing the program of expectation values in coherent states and its recently developed algorithmic tools, this letter investigates the dynamical properties of cosmological coherent states for Loop Quantum Gravity. To this end, the Quantum Speed Limit is adapted to Quantum Gravity, yielding necessary consistency checks for any proposal of stable families of states. To showcase the strength of the developed tools, they are applied to a prominent model: the Euclidean part of the quantum scalar constraint. We report the variance of this constraint evaluated on a family of coherent states showing that, for short times, this family passes the Quantum Speed Limit test, allowing the transition from one coherent state to another one., Comment: 6 pages

We consider the Hamiltonian renormalisation group flow of discretised one-dimensional physical theories. In particular, we investigate the influence the choice of different embedding maps has on the RG flow and the resulting continuum limit, and show in which sense they are, and in which sense they are not equivalent as physical theories. We are furthermore elucidating the interplay of the RG flow and the algebras operators satisfy, both on the discrete and the continuum. Further, we propose preferred renormalisation prescriptions for operator algebras guaranteeing to arrive at preferred algebraic relations in the continuum, if suitable extension properties are assumed. Finally, we introduce a weaker form of distributional equivalence, and show how unitarily inequivalent continuum limits, which arise due to a choice of different embedding maps, can still be weakly equivalent in that sense., Comment: 16 pages

In the lattice approach to Loop Quantum Gravity on a fixed graph computations tend to be involved and are rarely analytically manageable. But, when interested in the expectation values of coherent states on the lattice which are sharply peaked on isotropic, flat cosmology several simplifications are possible which reduce the computational effort. We present a step-by-step algorithm resulting in an analytical expression including up to first order corrections in the spread of the state. The algorithm is developed in such a way that it makes the computation straightforward and easy to be implementable in programming languages such as Mathematica. Exemplarily, we demonstrate how the algorithm streamlines the road to obtain the expectation value of the euclidean part of the scalar constraint and as a consistency check perform the analytic computation as well. To showcase further applications of the algorithm, we investigate the fate of the effective dynamics program custom in Loop Cosmology and find that the next-to-leading order corrections can {\it not} be used as corrections for an effective Hamiltonian., Comment: 33 pages; This paper summarizes the results announced at Loops'19 Conference [http://gravity.psu.edu/events/loops19/index-loops19.shtml]. First draft - minor typos can still be found

In the Hamiltonian formulation, it is not a priori clear whether a symmetric configuration will keep its symmetry during evolution. In this paper, we give precise requirements of when this is the case and propose a symmetry restriction to the phase space of the symmetric variables. This can often ease computation, especially when transcending from the infinite dimensional phase space of a field theory to a possibly finite dimensional subspace. We will demonstrate this in the case of gravity. A prominent example is the restriction of full Hamiltonian general relativity to the cosmological configurations of Robertson-Walker type. We will demonstrate our procedure in this setting and extend it to examples which appear useful in certain approaches to quantum gravity., Comment: 69 pages

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., Comment: 23 pages

It is expected that the quantum nature of spacetime leaves its imprint in all semiclassical gravitational systems, at least in certain regimes, including gravitational waves. In this paper we investigate such imprints on gravitational waves within a specific framework: space is assumed to be discrete (in the form of a regular cubic lattice), and this discrete geometry is quantised following Dirac's canonical quantisation scheme. The semiclassical behavior is then extracted by promoting the expectation value of the Hamiltonian operator on a semiclassical state to an effective Hamiltonian. Considering a family of semiclassical states representing small tensor perturbations to Minkowski background, we derive a quantum-corrected effective wave equation. The deviations from the classical gravitational wave equation are found to be encoded in a modified dispersion relation and controlled by the discreteness parameter of the underlying lattice. For finite discretisations, several interesting effects appear: we investigate the thermodynamical properties of these modified gravitons and, under certain assumptions, derive the tensor power spectrum of the cosmic microwave background. The latter is found to deviate from the classical prediction, in that an amplification of UV modes takes place. We discuss under what circumstances such effect can be in agreement with observations., Comment: 9 pages, 3 figures

This article investigates properties of semiclassical Gauge Field Theory Coherent States for general quantum gauge theories. Useful, e.g., for the canonical formulation of Lattice Gauge Theories these states are labelled by a point in the classical phase space and constructed such that the expectation values of the canonical operators are sharply peaked on said phase space point. For the case of the non-abelian gauge group SU(2), we will explicitly compute the expectation value of general polynomials including the first order quantum corrections. This allows asking more precise questions about the quantum fluctuations of any given semiclassical system., Comment: 20 pages

The closed, spatially isotropic FLRW universe (k=+1) is endowed with modifications due to a discrete underlying space-structure. Motivated from Loop Quantum Gravity techniques, a full Thiemann regularisation is performed. The impact of these modifications of the single-graph-sector appearing in the scalar constraint are interpreted as physical quantum gravity effects. We investigate the form of the modified scalar constraint and its analytical approximations for k=+1 spacetimes and assume this effective constraint as the generator of dynamics on the reduced isotropic phase space. It transpires that the system still features a classical recollapse with only marginal discreteness corrections. Moreover, the initial and final singularities are resolved and we present an effective model mirroring the qualitative features of system., Comment: 19 pages, 4 figures

We reexamine a set of existing procedures aimed at recovering the effective description of the dynamics of LQG in the context of cosmological solutions. In particular, the studies of those methods, to which the choice of cuboidal graphs and graph-preserving Hamiltonian is central, result in the formulation of a set of no-go statements, severely limiting the possibility of recovering a physically consistent effective dynamics this way., Comment: 9 pages

The way physics of loop quantum gravity is affected by the underlying quantization ambiguities is an open question. We address this issue in the context of loop quantum cosmology using gauge-covariant fluxes. Consequences are explored for two choices of regularization parameters: $\mu_0$ and $\bar \mu$ in presence of a positive cosmological constant, and two choices of regularizations of the Hamiltonian constraint in loop quantum cosmology: the standard and the Thiemann regularization. We show that novel features of singularity resolution and bounce, occurring due to gauge-covariant fluxes, exist also for Thiemann-regularized dynamics. The $\mu_0$-scheme is found to be unviable as in standard loop quantum cosmology when a positive cosmological constant is included. Our investigation brings out a surprising result that the nature of emergent matter in the pre-bounce regime is determined by the choice of regulator in the Thiemann regularization of the scalar constraint whether or not one uses gauge-covaraint fluxes. Unlike $\bar \mu$-scheme where the emergent matter is a cosmological constant, the emergent matter in $\mu_0$-scheme behaves as a string gas., Comment: 31 pages, 10 figures. References and a figure added. To appear in PRD

Loop quantum cosmology is a symmetry reduced quantization of cosmological spacetimes based on loop quantum gravity. While it has been successful in resolution of various cosmological singularities and connecting Planck scale physics to phenomenology, its connection with loop quantum gravity has remained elusive. It is therefore important to integrate more and more features of the full theory into this framework and understand the reliability of physical predictions. In particular, if one wishes to connect the effective Hamiltonian in loop quantum cosmology to an expectation value of the scalar constraint operator in suitable coherent states for the full theory, one has to go beyond the standard setting of loop quantum cosmology. One possibility is to introduce gauge-covariant fluxes, which become necessary because the presence of a finite regularization parameter causes functions build out of the standard discretized variables to be in general not gauge invariant. Following the construction of gauge-covariant fluxes pioneered by Thiemann in [1], we show that the physics of loop quantum cosmology is affected in a non-trivial way. The bounce turns out to be generically asymmetric with a rescaling of the Newton's constant in the pre-bounce branch. Gauge-covariant fluxes result in a higher order quantum difference equation in comparison to loop quantum cosmology. Even the behavior of matter, which behaves innocuously in loop quantum cosmology, is enriched, resulting in an effective non-minimal coupling. These effects are shown to be common to different choices of regularization parameters., Comment: 28 pages, 7 figures. Figure added to show robustness of asymmetric bounce. To appear in PRD

In the loop quantum gravity context, there have been numerous proposals to quantize the reduced phase space of a black hole, and develop a classical effective description for its interior which eventually resolves the singularity. However, little progress has been made towards understanding the relation between such quantum/effective minisuperspace models and what would be the spherically symmetric sector of loop quantum gravity. In particular, it is not clear whether one can extract the phenomenological predictions obtained in minisuperspace models, such as the singularity resolution and the spacetime continuation beyond the singularity, based on results in full loop quantum gravity. In this paper, we present an attempt in this direction in the context of Kantowski-Sachs spacetime, through the proposal of two new effective Hamiltonians for the reduced classical model. The first is derived using Thiemann classical identities for the regularized expressions, while the second is obtained as a first approximation of the expectation value of a Hamiltonian operator in loop quantum gravity in a semi-classical state peaked on the Kantowski-Sachs initial data. We then proceed with a detailed analysis of the dynamics they generate and compare them with the Hamiltonian derived in General Relativity and the common effective Hamiltonian proposed in earlier literature., Comment: 13 pages, 2 figures; revised version accepted for publication in Phys. Rev. D

We present a detailed analysis of a quantum model for Loop Quantum Cosmology based on strict application of the Thiemann regularization algorithm for the Hamiltonian in Loop Quantum Gravity, extending the results presented previously in our brief report. This construction leads to a qualitative modification of the bounce paradigm. Quantum gravity effects still lead to a quantum bounce connecting deterministically large classical Universes. However, the evolution features a large epoch of de Sitter Universe, with emergent cosmological constant of Planckian order, smoothly transiting into a spatially flat expanding Universe. Moreover, we present an effective Hamiltonian describing the quantum evolution to high accuracy and for which the dynamics can be solved analytically., Comment: 51 pages, 15 figures

We present a gauge-invariant treatment of singularity resolution using loop quantum gravity techniques with respect to local SU(2) transformations. Our analysis reveals many novel features of quantum geometry which were till now hidden in models based on non-gauge-invariant discretizations. Quantum geometric effects resolve the big bang singularity replacing it with a non-singular bounce when spacetime curvature reaches Planckian value. The bounce is found to be generically asymmetric in the sense that pre-bounce and post-bounce branches are not mirrored to each other and effective constants, such as Newton's constant, are rescaled across the bounce. Furthermore, in the vicinity of the bounce, minimally coupled matter behaves as non-minimally coupled. These ramifications of quantum geometry open a rich avenue for potential phenomenological signatures., Comment: 12 pages, 2 figures. Published version in Classical and Quantum Gravity

The quantum nature of the Big Bang is reexamined in the framework of Loop Quantum Cosmology. The strict application of a regularization procedure to the Hamiltonian, originally developed for the Hamiltonian in loop quantum gravity, leads to a qualitative modification of the bounce paradigm. Quantum gravity effects still lead to a quantum bounce connecting deterministically large classical Universes. However, the evolution features a large epoch of de Sitter Universe, with emergent cosmological constant of Planckian order, smoothly transiting into a flat expanding Universe., Comment: 6 pages, 2 figures, revised version accepted for publication in Phys.Rev.Lett., with the addition of fig.2 and subsequent comments

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., Comment: 30 pages

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., Comment: 28 pages

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., Comment: 36 pages

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., Comment: 27 pages

This is the first paper of a series dedicated to LQG coherent states and cosmology. The concept is based on the effective dynamics program of Loop Quantum Cosmology, where the classical dynamics generated by the expectation value of the Hamiltonian on semiclassical states is found to be in agreement with the quantum evolution of such states. We ask the question of whether this expectation value agrees with the one obtained in the full theory. The answer is in the negative. This series of papers is dedicated to detailing the computations that lead to that surprising result. In the current paper, we construct the family of coherent states in LQG which represent flat ($k=0$) Robertson-Walker spacetimes, and present the tools needed to compute expectation values of polynomial operators in holonomy and flux on such states. These tools will be applied to the LQG Hamiltonian operator (in Thiemann regularization) in the second paper of the series. The third paper will present an extension to $k\neq0$ cosmologies and a comparison with alternative regularizations of the Hamiltonian., Comment: 43 pages, 1 figure

The concept of effective dynamics has proven successful in LQC, a loop-inspired quantization of cosmological spacetimes. We apply the same idea of its derivation in LQC to the full theory, by computing the expectation value of the scalar constraint with respect to some coherent states peaked on the phase-space variables of flat Robertson-Walker spacetime. We comment on the relation with effective LQC and find a deviation stemming from the Lorentzian part of the Hamiltonian., Comment: 5 pages, 1 figure. Accepted version in PLB

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., Comment: 51 pages

The Hamiltonian constraint is the key element of the canonical formulation of LQG coding its dynamics. In Ashtekar-Barbero variables it naturally splits into the so called Euclidean and Lorentzian parts. However, due to the high complexity of this operator, only the matrix elements of the Euclidean part have been considered so far. Here we evaluate the action of the full constraint, including the Lorentzian part. The computation requires an heavy use of SU(2) recoupling theory and several tricky identities among n-j symbols are used to find the final result: these identities, together with the graphical calculus used to derive them, also simplify the Euclidean constraint and are of general interest in LQG computations., Comment: 36 pages, minor typos corrected