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.
We construct a positive complexifier, differentiable almost everywhere on the classical phase space of real triads and $SU(2)$ connections, which generates a Wick Transform from Euclidean to Lorentzian gravity everywhere except on a phase space set of measure zero. This Wick transform assigns an equal role to the self dual and anti-self dual Ashtekar variables in quantum theory. We argue that the appropriate quantum arena for an analysis of the properties of the Wick rotation is the diffeomorphism invariant Hilbert space of Loop Quantum Gravity (LQG) rather than its kinematic Hilbert space. We…
Loop Quantum Gravity (LQG) is a non-perturbative attempt at quantization of a classical phase space description of gravity in terms of $SU(2)$ connections and electric fields. As emphasized recently [1], on this phase space, classical gravitational evolution in $time$ can be understood in terms of certain gauge covariant generalizations of Lie derivatives with respect to a $spatial$ $SU(2)$ Lie algebra valued vector field called the Electric Shift. We present a derivation of a quantum dynamics for Euclidean LQG which is informed by this understanding. In addition to the physically motivated na…
We develop in a companion article the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation which is well-known at zero cosmological constant. In this article, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum …
We propose a new derivation from the full Loop Quantum Gravity (LQG) to the Loop Quantum Cosmology (LQC) improved $\barμ$-scheme effective dynamics, based on the reduced phase space formulation of LQG and a proposal of effective Hamiltonian/action in the full LQG. A key step of our program is an improved regularization of the full LQG Hamiltonian on a cubic lattice. The improved Hamiltonian uses a set of "dressed holonomies" $h_Δ(\mathfrak{s})$ which not only depend on the connection $A$ but also depend on the length of the curve $\mathfrak{s}$. With the improved Hamiltonian, we propose a quan…
The scheme of using the Chern-Simons action to regularize the gravitational Hamiltonian constraint is extended to including the Lorentzian term in the $k=0$ cosmological model. The Euclidean term and the Lorenzian term are thus regularized separately mimicking the treatment of full loop quantum gravity. The new quantum dynamics for the spatially flat Friedmann-Robertson-Walker model with a massless scalar field as an emergent time is studied. By semiclassical analysis, the effective Hamiltonian constraint is obtained, which indicates that the new quantum dynamics has the correct classical limi…
Classical gravitational evolution admits an elegant and compact re-expression in terms of gauge covariant generalizations of Lie derivatives with respect to a spatial phase space dependent $su(2)$ valued vector field called the Electric Shift [1]. A quantum dynamics for Euclidean Loop Quantum Gravity which ascribes a central role to the Electric Shift operator is derived in [2]. $Here\; we\; show\; that\; this \;quantum\; dynamics\; is\; nontrivially\; anomaly$ $free$. Specifically, we show that on a suitable space of off shell states (a) the (non-vanishing) commutator between a pair of Hamilt…
In this paper, the classical and quantum theory of $\mathcal{N}=1$ supergravity in four spacetime dimensions will be studied in the framework of loop quantum gravity. We discuss the canonical analysis of the supergravity Holst action as first introduced by Tsuda. In this way, we also derive a compact expression of the supersymmetry constraint, which plays a crucial role in canonical supergravity theories, akin to the role of the Hamiltonian constraint in non-supersymmetric generally covariant theories. The resulting theory is then quantized using loop quantum gravity methods. In particular, we…
One of remarkable features of loop quantum gravity (LQG) is that it can provide resolutions to both the black hole and big bang singularities. In the mini-superspace approach based on the polymerization procedure in LQG, a quantum corrected black hole metric is constructed. This metric is also known as self-dual spacetime since the form of the metric is invariant under the exchange $r \to a_0/r$ with $a_0$ being proportional to the minimum area in LQG and $r$ is the standard radial coordinate at asymptotic infinity. It modifies the Schwarzschild spacetime by the polymeric function $P$, purely …
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 impleme…
We discuss the quantization of vacuum Bianchi I spacetimes in the modified formalism of loop quantum cosmology recently proposed by Dapor and Liegener. This modification is based on a regularization procedure where both the Euclidean and Lorentzian terms of the Hamiltonian are treated independently. Whereas the Euclidean part has already been dealt with in the literature for Bianchi I spacetimes, the Lorentzian one has not yet been represented quantum mechanically. After a brief review of the quantum kinematics and the quantization of the Euclidean sector, we represent the Lorentzian part of t…
Smolin's generally covariant $G_{\mathrm{Newton}}\rightarrow0$ limit of 4d Euclidean gravity is a useful toy model for the study of the constraint algebra in Loop Quantum Gravity. In particular, the commutator between its Hamiltonian constraints has a metric dependent structure function. While a prior LQG like construction of non-trivial anomaly free constraint commutators for the model exists, that work suffers from two defects. First, Smolin's remarks on the inability of the quantum dynamics to generate propagation effects apply. Second, the construction only yields the action of a single Ha…
To inherit more features of full loop quantum Brans-Dicke theory, the Euclidean and Lorentzian terms of the Hamiltonian constraint are quantized independently in loop quantum Brans-Dicke cosmology. An alternative Hamiltonian constraint operator and its effective expression are obtained in the cosmological model. A residual quantum correction term is found in the effective Hamiltonian constraint, which has no analog in the effective Hamiltonian of the loop quantum cosmology from general relativity. The dynamics driven by this effective Hamiltonian constraint is analyzed in detail. For the physi…
An alternative quantization of the gravitational Hamiltonian constraint of the $k=-1$ Friedmann-Robertson-Walker model is proposed by treating the Euclidean term and the Lorentzian term independently, mimicking the treatment of full loop quantum gravity. The resulting Hamiltonian constraint operator for the $k=-1$ model with a massless scalar field is successfully constructed, and is shown to have the corrected classical limit. Compared to the former quantization schemes in the literature where only the Euclidean term is quantized, the new quantum dynamics of the $k=-1$ model with a massless s…
