Emanuele Alesci, Francesco Cianfrani
Title: Quantum reduced loop gravity
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We propose a new framework for the loop quantization of symmetry-reduced sectors of General Relativity, called Quantum Reduced Loop Gravity, and we apply this scheme to the inhomogeneous extension of the Bianchi I cosmological model (a cosmology that is homogeneous but anisotropic). To explain the meaning of this sentence we need several ingredients that will be presented in the next sections. But let us first focus on the meaning of “symmetry reduction”: this process simply means that a if a physical system has some kind of symmetry we can use it to reduce the number of independent variables needed to describe it. Symmetry then in general allows to restrict the variables of the theory to the true independent degrees of freedom of it. For instance, let us consider a point-like spinless particle moving on a plane under a central potential. The system is invariant under 2-dimensional rotations on the plane around the center of the potential and as a consequence the angular momentum is conserved. The angular velocity around the origin is a constant of motion and the only “true” dynamical variable is the radial coordinate of the particle. Going to the phase space (the space of positions and momenta of the theory), it can be parameterized by the radial and angular coordinates together with the corresponding momenta, but the symmetry forces the momentum associated with the angular coordinate to be conserved. The reduced phase-space associated with such a system is parameterized by the radial coordinate and momentum, from which, given the initial conditions, the whole trajectory of the particle in the plane can be reconstructed. The quantization in the reduced phase space is usually easier to handle than in the full phase space and this is the main reason why it is a technique frequently used in order to test the approaches towards Quantum Gravity, whose final theory is still elusive. In this respect, the canonical analysis of homogeneous models (Loop Quantum Cosmology) and of spherically-symmetric systems (Quantum Black Holes) in Loop Quantum Gravity (LQG) has been mostly performed by first restricting to the reduced phase space and then quantizing the resulting system (what is technically known as reduced quantization). The basic idea of our approach is to invert the order of “reduction” and “quantization”. The motivation will come directly from our analysis and, in particular, from the failure of reduced quantization to provide a sensible dynamics for the inhomogeneous extensions of the homogeneous anisotropic Bianchi I model. Hence, we will follow a different path by defining a “quantum” reduction of the Hilbert space of quantum states of the full theory down to a subspace which captures the relevant degrees of freedom. This procedure will allow us to treat the inhomogeneous Bianchi I system directly at the quantum level in a computable theory with all the ingredients of LQG (just simplified due to the quantum-reduction).
To proceed, let us first review the main features of LQG.
Loop Quantum Gravity
LQG is one of the most promising approaches for the quantization of the gravitational field. Its formulation is canonical and thus it is based on making a 3+1 splitting of the space-time manifold. The phase space is parameterized by the Ashtekar-Barbero connections, and the associated momenta, from which one can compute the metric of spatial sections. A key point of this reformulation is the existence of a gauge invariance (technically known as SU(2) gauge invariance), which together with background independence, lead to the so-called kinematical constraints of the theory (every time there is a symmetry in a theory an associated constraint emerges implying that the variables are not independent and one has to isolate the true degrees of freedom). The quantization procedure is inspired by the approaches developed in the 70s to describe gauge theories on the lattice in the strong-coupling limit. In particular, the quantum states are given in terms of spin networks, which are graphs with "colors" in the links between intersections. An essential ingredient of LQG is background independence. The way this symmetry is implemented is a completely new achievement in Quantum Gravity and it allows to define a regularized expression (free from infinities) for the operator associated with the Hamiltonian constraint asssociated with the dynamics of the theory. Thanks to a procedure introduced by Thiemann, the Hamiltonian constraint can be approximated over a certain triangulation of the spatial manifold. The limit in which the triangulation gets finer and finer gives us back the classical expression and it is well defined on a quantum level over s-knots (classes of spin networks related by smooth deformations). The reason is that s-knots are diffeomorphisms invariant and, thus, insensitive to the characteristic length of the triangulation. This means that the Hamiltonian constraint can be consistently regularized and, by the way, the associated algebra is anomaly-free. Unfortunately, the resulting expression cannot be analytically computed, because of the presence of the volume operator, which is complicated. This drawback appears to be a technical difficulty, rather than a theoretical obstruction, and for this reason our aim is to try to overcome it in a simplified model, like a cosmological one.
Loop Quantum Cosmology
Loop Quantum Cosmology (LQC) is the best theory at our disposal to threat homogeneous cosmologies. LQC is based on a quantization in the reduced phase space, which means that the reduction according with the symmetry is entirely made on a classical level. Once that the classical reduction is made, one then proceeds with a quantization of the degrees of freedom left with LQG techniques. We know that our Universe experiences a highly isotropic and homogeneous phase at scales bigger than 100Mpc. The easiest cosmological description is the one of Friedmann-Robertson-Walker (FRW), in which one deals with an isotropic and homogeneous line element, described by only one variable, the scale factor. A generalization can be given by considering anisotropic extensions, the so-called Bianchi models, in which there are three scale factors defined along some fiducial directions. In LQC one fixes the metric to be of the FRW or Bianchi type and quantizes the dynamical variables. However a direct derivation from LQG is still missing and it is difficult to accommodate in this setting inhomogenities because the theory is defined in the homogeneous reduced phase space.
Inhomogeneous extension of the Bianchi models:
We want to define a new model for cosmology able to retain all the nice features of LQG, in particular a sort of background independence by which the regularization of the Hamiltonian constraint can be carried on as in the full theory. In this respect, we consider the simplest Bianchi model, the type I (a homogeneous but anisotropic space-time), and we define an inhomogeneous extension characterized by scale factors that depend on space. This inhonomogeneous extension contains as a limiting case the homogeneous phase in an arbitrary parameterization. The virtue of these models is that they are invariant under what we called a reduced-diffeomorphism invariance, which is the invariance under a restricted class of diffeomorphisms preserving the fiducial directions of the anisotropies of the Bianchi I model. This is precisely the kind of symmetry we were looking for! In fact, once quantum states are based on reduced graphs, whose edges are along the fiducial directions, we can define some reduced s-knots, which will be insensitive to the length of any cubulation of the spatial manifold (we speak of a cubulation because reduced graphs admit only cubulations and not triangulations). Therefore, all we have to do is to repeat Thiemann's construction for a cubulation rather than for a triangulation. But does it give a good expression for the Hamiltonian constraint?? The answer is no and the reason is that there is an additional symmetry in the reduced phase space that prevents us from repeating the construction used by Thiemann for the Hamiltonian constraint. Henceforth, the dynamical issue cannot be addressed by standard LQG techniques in reduced quantization.
Quantum-Reduced Loop Gravity
What are we missing in reduced quantization? The idea is that we have reduced the gauge symmetry too much and that is what prevents us from constructing the Hamiltonian. We therefore go back and do not reduce the symmetry and proceed to quantize first. We then impose the reduction of the symmetry at a quantum level. Hence, the classical expression of the Hamiltonian constraint for the Bianchi I model can be quantized according with the Thiemann procedure. Moreover, the associated matrix elements can be analytically computed because the volume operator takes a simplified form in the new Hilbert space. Therefore, we have a quantum description for the inhomogeneous Bianchi I model in which all the techniques of LQG can be applied and all the computations can be carried on analytically. This means that for the first time we have a model in which we can explicitly test numerous aspects of loop quantization: Thiemann's original graph changing Hamiltonian, the master constraint program, Algebraic Quantization or the new deparameterized approach with matter fields can all be tested. Such a model is a cuboidal lattice, whose edges are endowed with quantum numbers and with some reduced relations between those numbers at vertices. In two words we have a sort of hybrid “LQC” along the edges with LQG relationships at the nodes, but with a graph structure and diagonal volume! This means that we have an analytically tractable model closer to LQG than LQC and potentially able to threat inhomogeneities and anisotropies at once. Is this model meaningful? What we have to do now is “only” physics: as a first test try to work out the semiclassical limit. If this model will yield General Relativity in the classical regime, then we can proceed to compare its predictions with Loop Quantum Cosmology in the quantum regime, inserting matter fields and analyzing their role, discussing the behavior of inhomogeneities and so on.. We will see..
