Canonical Loop Quantum Gravity Research Articles

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 of 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 ingto be worked out. By considering one of the simplest possible non-trivial classes of solutions based on very small graphs, we show that (1) an infinite number of solutions ingexist which are (2) generically not normalisable with respect to the inner product on the space of spatially diffeomorphism invariant distributions and (3) generically display 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.

Relating spin-foam to canonical loop quantum gravity by graphical calculus

The graphical calculus method is generalized to study the relation between covariant and canonical dynamics of loop quantum gravity. On one hand, a graphical derivation of the partition function of the generalized Euclidean Engle-Pereira-Rovelli-Livine (EPRL) spin-foam model is presented. On the other hand, the action of a Euclidean Hamiltonian constraint operator on certain spin network states is calculated by graphical method. It turns out that the EPRL model can provide a rigging map such that the Hamiltonian constraint operator is weakly satisfied on certain physical states for the Immirzi parameter $\beta=1$. In this sense, the quantum dynamics between the covariant and canonical formulations are consistent to each other.

Semiclassical limit of new path integral formulation from reduced phase space loop quantum gravity

Recently, a new path integral formulation of loop quantum gravity (LQG) has been derived in M. Han and H. Liu, Phys. Rev. D 101, 046003 (2020). from the reduced phase space formulation of the canonical LQG. This paper focuses on the semiclassical analysis of this path integral formulation. We show that dominant contributions of the path integral come from solutions of semiclassical equations of motion (EOMs), which reduce to Hamilton's equations of holonomies and fluxes $h(e)$, ${p}^{a}(e)$ in the reduced phase space ${\mathcal{P}}_{\ensuremath{\gamma}}$ of the cubic lattice $\ensuremath{\gamma}$: $\frac{\mathrm{d}h(e)}{\mathrm{d}\ensuremath{\tau}}={h(e),\mathbf{H}}$, $\frac{\mathrm{d}{p}^{a}(e)}{\mathrm{d}\ensuremath{\tau}}={{p}^{a}(e),\mathbf{H}}$, where $\mathbf{H}$ is the discrete physical Hamiltonian. The semiclassical dynamics from the path integral becomes an initial value problem of Hamiltonian time evolution in ${\mathcal{P}}_{\ensuremath{\gamma}}$. Moreover when we take the continuum limit of the lattice $\ensuremath{\gamma}$, these Hamilton's equations reproduce correctly classical reduced phase space EOMs of gravity coupled to dust fields in the continuum, as far as initial and final states are semiclassical. Our result proves that the new path integral formulation has the correct semiclassical limit and indicates that the reduced phase space quantization in LQG is semiclassically consistent. Based on these results, we compare this path integral formulation and the spin foam formulation, and show that this formulation has several advantages including the finiteness, the relation with canonical LQG, and the freedom from cosine and flatness problems.

Graph coherent states for loop quantum gravity

In this article we further investigate the construction of graph coherent states, first introduced in [1], in the context of loop quantum gravity. We specifically investigate the possibility of defining a family of graph coherent states adapted to the canonical loop quantum gravity Hamiltonian. After discussing various aspects of the general framework and the choice of operators, we use the Euclidean part of the Hamiltonian operator to propose a generator of the generalized canonical structure, necessary to define the coherent states. We then apply the construction procedure, leading to a new family of graph coherent states partially adapted to the gravity Hamiltonian in loop quantum gravity.

Effective dynamics from coherent state path integral of full loop quantum gravity

A new routine is proposed to relate Loop Quant Cosmology (LQC) to Loop Quantum Gravity (LQG) from the perspective of effective dynamics. We derive the big-bang singularity resolution and big bounce from the first principle of full canonical LQG. Our results are obtained in the framework of the reduced phase space quantization of LQG. As a key step in our work, we derive with coherent states a new discrete path integral formula of the transition amplitude generated by the physical Hamiltonian. The semiclassical approximation of the path integral formula gives an interesting set of effective equations of motion (EOMs) for full LQG. When solving the EOMs with homogeneous and isotropic ansatz, we reproduce the LQC effective dynamics in $\mu_0$-scheme. The solution replaces the big-bang singularity by a big bounce. In the end, we comment on the possible relation between the $\bar{\mu}$-scheme of effective dynamics and the continuum limit of the path integral formula.

A Brief Review on Canonical Loop Quantum Gravity: The Kinematical Part

In this article, we briefly review the kinematical part of canonical loop quantum gravity. This article starts with tetradic formulation of gravity both in the covariant approach and canonical approach. The next step is to introduce Ashtekar new variables and to apply the Dirac canonical quantization procedure of gravity. By the holonomy representation, one obtains the loop representation of quantum gravity.

Effective line elements and black-hole models in canonical loop quantum gravity

Canonical quantization is often used to suggest new effects in quantum gravity, in the dynamics as well as the structure of space-time. Usually, possible phenomena are first seen in a modified version of the classical dynamics, for instance in an effective Friedmann equation, but there should also be implications for a modified space-time structure. Quantum space-time effects, however, are often ignored in this setting because they are not obvious: they require a careful analysis of gauge transformations and the anomaly problem. It is shown here how modified space-time structures and effective line elements can be derived unambiguously, provided an off-shell anomaly-free system of modified constraints exists. The resulting effective line elements reveal signature change as an inescapable consequence of nonclassical gauge transformations in the presence of holonomy modifications. The general framework is then specialized to black-hole models in loop quantum gravity. In contrast to previous studies, a self-consistent space-time structure is taken into account, leading to a new picture of black-hole interiors.

Canonical methods in classical and quantum gravity: An invitation to canonical LQG

Loop Quantum Gravity (LQG) is a candidate quantum theory of gravity still under construction. LQG was originally conceived as a background independent canonical quantization of Einstein’s general relativity theory. This contribution provides some physical motivations and an overview of some mathematical tools employed in canonical Loop Quantum Gravity. First, Hamiltonian classical methods are reviewed from a geometric perspective. Canonical Dirac quantization of general gauge systems is sketched next. The Hamiltonian formultation of gravity in geometric ADM and connection-triad variables is then presented to finally lay down the canonical loop quantization program. The presentation is geared toward advanced undergradute or graduate students in physics and/or non-specialists curious about LQG.

Exploring the Tomlin-Varadarajan quantum constraints in U(1)3 loop quantum gravity: Solutions and the Minkowski theorem

We explicitly solved the anomaly-free quantum constraints proposed by Tomlin and Varadarajan for the weak Euclidean model of canonical loop quantum gravity, in a large subspace of the model's kinematic Hilbert space, which is the space of the charge network states. In doing so, we first identified the subspace on which each of the constraints acts convergingly, and then by explicitly evaluating such actions we found the complete set of the solutions in the identified subspace. We showed that the space of solutions consists of two classes of states, with the first class having a property that involves the condition known from the Minkowski theorem on polyhedra, and the second class satisfying a weaker form of the spatial diffeomorphism invariance.

Group field theory as the second quantization of loop quantum gravity

We construct a second quantized reformulation of canonical loop quantum gravity (LQG) at both kinematical and dynamical level, in terms of a Fock space of spin networks, and show in full generality that it leads directly to the group field theory (GFT) formalism. In particular, we show the correspondence between canonical LQG dynamics and GFT dynamics leading to a specific GFT model from any definition of quantum canonical dynamics of spin networks. We exemplify the correspondence of dynamics in the specific example of 3d quantum gravity. The correspondence between canonical LQG and covariant spin foam models is obtained via the GFT definition of the latter.

Geometric temperature and entropy of quantum isolated horizons

By reintroducing Lorentz invariance in canonical loop quantum gravity, we define a geometrical notion of temperature for quantum isolated horizons. This is done by demanding that the horizon state satisfying the boundary conditions be a Kubo-Martin-Schwinger state. The exact formula for the temperature can be derived by imposing the reality conditions in the form of the linear simplicity constraints for an imaginary Barbero-Immirzi parameter. Thus, our analysis reveals the connection between the analytic continuation to the Ashtekar self-dual variables and the thermality of the horizon. The horizon thermal equilibrium state can then be used to compute both the entanglement and the Boltzmann entropies. We show that the two provide the same finite answer, which allows us to recover the Bekenstein-Hawking formula in the semi-classical limit. In this way, we shed new light on the microscopic origin of black hole entropy by revealing the equivalence between the near-horizon degrees of freedom entanglement proposal and the state-counting interpretation. The connection with the Connes-Rovelli thermal time proposal for a general relativistic statistical mechanics is worked out.

Constraint algebra in loop quantum gravity reloaded. II. Toy model of an Abelian gauge theory: Spatial diffeomorphisms

In A. Henderson, A. Laddha, and C. Tomlin, preceding article, [Phys. Rev. D 88, 044028 (2013)], we initiated an approach towards quantizing the Hamiltonian constraint in loop quantum gravity by requiring that it generates an anomaly-free representation of constraint algebra of -shell. We investigated this issue in the case of a toy model of a ($2+1$)-dimensional $\mathrm{U}(1{)}^{3}$ gauge theory, which can be thought of as a weak coupling limit of Euclidean three-dimensional gravity. However, in paper I, we only focused on the most nontrivial part of the constraint algebra that involves the commutator of two Hamiltonian constraints. In this paper we continue with our analysis and obtain a representation of full constraint algebra in a loop quantized framework. We show that there is a representation of the diffeomorphism group with respect to which the Hamiltonian constraint quantized in paper I is diffeomorphism covariant. Our work can be thought of as a potential first step towards resolving some longstanding issues with the Hamiltonian constraint in canonical loop quantum gravity.

Deformed general relativity

Deformed special relativity is embedded in deformed general relativity using the methods of canonical relativity and loop quantum gravity. Phase-space dependent deformations of symmetry algebras then appear, which in some regimes can be rewritten as nonlinear Poincar\'e algebras with momentum-dependent deformations of commutators between boosts and time translations. In contrast to deformed special relativity, the deformations are derived for generators with an unambiguous physical role, following from the relationship between canonical constraints of gravity with stress-energy components. The original deformation does not appear in momentum space and does not give rise to nonlocality issues or problems with macroscopic objects. Contact with deformed special relativity may help to test loop quantum gravity or restrict its quantization ambiguities.

The Holst spin foam model via cubulations

Spin foam models are an attempt at a covariant or path integral formulation of canonical loop quantum gravity. The construction of such models usually relies on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional ‘Regge-like’ constraints arising from simplicial triangulations and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a ‘cubulation’ of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: firstly, the structure of its amplitudes differs from the standard spin foam models. Secondly, the tetrad n-point functions admit a ‘Wick-like’ structure. Thirdly, the restriction to simple representations does not automatically occur—unless one makes use of the time gauge, just as in the classical theory.

A dilemma for the emergence of spacetime in canonical quantum gravity

The procedures of canonical quantization of the gravitational field apparently lead to entities for which any interpretation in terms of spatio-temporal localization or spatio-temporal extension seems difficult. This fact is the main ground for the suggestion that can often be found in the physics literature on canonical quantum gravity according to which spacetime may not be fundamental in some sense. This paper aims to investigate this radical suggestion from an ontologically serious point of view in the cases of two standard forms of canonical quantum gravity, quantum geometrodynamics and loop quantum gravity. We start by discussing the physical features of the quantum wave functional of quantum geometrodynamics and of the spin networks (and spin foams) of loop quantum gravity that motivate the view according to which spacetime is not fundamental. We then point out that, by contrast, for any known ontologically serious understanding of quantum entanglement, the commitment to spacetime seems indispensable. Against this background, we then critically discuss the idea that spacetime may emerge from more fundamental entities. As a consequence, we finally suggest that the emergence of classical spacetime in canonical quantum gravity faces a dilemma: either spacetime ontologically emerges from more fundamental non-spatio-temporal entities or it already belongs to the fundamental quantum gravitational level and the emergence of the classical picture is merely a matter of levels of description. On the first horn of the dilemma, it is unclear how to make sense of concrete physical entities that are not in spacetime and of the notion of ontological emergence that is involved. The second horn runs into the difficulties raised by the physics of canonical quantum gravity.

Linking covariant and canonical loop quantum gravity: New solutions to the Euclidean scalar constraint

It is often emphasized that spin-foam models could realize a projection on the physical Hilbert space of canonical loop quantum gravity. As a first test, we analyze the one-vertex expansion of a simple Euclidean spin foam. We find that for fixed Barbero-Immirzi parameter $\ensuremath{\gamma}=1$, the one-vertex amplitude in the Kaminski, Kisielowski, and Lewandowski prescription annihilates the Euclidean Hamiltonian constraint of loop quantum gravity [T. Thiemann, Classical Quantum Gravity 15, 839 (1998).]. Since, for $\ensuremath{\gamma}=1$, the Lorentzian part of the Hamiltonian constraint does not contribute, this gives rise to new solutions of the Euclidean theory. Furthermore, we find that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.

Canonical path integral measures for Holst and Plebanski gravity: I. Reduced phase space derivation

An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interacting theories and theories with constraints, this is non-trivial, and is normally not the heuristic ‘Lebesgue measure’ usually used. There have been many determinations of a measure for gravity in the literature, but none for the Palatini or Holst formulations of gravity. Furthermore, the relations between different resulting measures for different formulations of gravity are usually not discussed. In this paper we use the reduced phase technique in order to derive the path-integral measure for the Palatini and Holst formulation of gravity, which is different from the Lebesgue measure up to local measure factors which depend on the spacetime volume element and spatial volume element. From this path integral for the Holst formulation of general relativity we can also give a new derivation of the Plebanski path integral and discover a discrepancy with the result due to Buffenoir, Henneaux, Noui and Roche whose origin we resolve. This paper is the first in a series that aims at better understanding the relation between canonical loop quantum gravity and the spin-foam approach.

A path integral for the master constraint of loop quantum gravity

In the present paper, we start from the canonical theory of loop quantum gravity and the master constraint program. The physical inner product is expressed by using the group averaging technique for a single self-adjoint master constraint operator. By the standard technique of skeletonization and the coherent state path integral, we derive a path-integral formula from the group averaging for the master constraint operator. Our derivation in this paper suggests there exists a direct link connecting the canonical loop quantum gravity with a path-integral quantization or a spin-foam model of general relativity.

The physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory

A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In this paper we reconsider the implementation of the constraints that defines the model. We define in a simple way the boundary Hilbert space of the theory, introducing a slight modification of the embedding of the SU(2) representations into the ones. We then show directly that all constraints vanish on this space in a weak sense. The vanishing is exact (and not just in the large quantum number limit). We also generalize the definition of the volume operator in the spin-foam model to the Lorentzian signature and show that it matches the one of loop quantum gravity, as in the Euclidean case.

The volume operator in covariant quantum gravity

A covariant spin-foam formulation of quantum gravity has recently been developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In particular, the geometrical observable giving the area of a surface has been shown to be the same as the one in loop quantum gravity. Here we discuss the volume observable. We derive the volume operator in the covariant theory and show that it matches the one of loop quantum gravity, as does the area. We also reconsider the implementation of the constraints that define the model: we derive in a simple way the boundary Hilbert space of the theory from a suitable form of the classical constraints and show directly that all constraints vanish weakly on this space.