Properties of the volume operator in loop quantum gravity: I. Results (original) (raw)
Related papers
Spectral analysis of the volume operator in loop quantum gravity
2006
We describe preliminary results of a detailed numerical analysis of the volume operator as formulated by Ashtekar and Lewandowski [2]. Due to a simplified explicit expression for its matrix elements[3], it is possible for the first time to treat generic vertices of valence greater than four. It is found that the vertex geometry characterizes the volume spectrum.
Spectral Analysis of Volume Operators in Loop Quantum Gravity for Kinematical Case
Journal of Physics: Conference Series
Loop Quantum Gravity has become one of the alternative solutions to quantum gravity. This formulation introduced geometrical operators which successfully used to model that in the quantum scale, the space is actually discretized in the order of Planck length. These operators are area and volume operator. The regularization process of these operators came from the classical definition of area and volume, thus, the eigenvalues of area operator and volume operator are respectively the area and volume of the space. However, there exists two types of volume operator, the Ashtekar-Lewandowski operator and the Rovelli-Smolin operator. The significant difference between these two operators is the fact that Ashtekar-Lewandowski operator is sensitive to the direction of the spin networks link, while Rovelli-Smolin operator is not. This difference will produce different spectral. In this article, we compare the resulting spectral of the two volume operators, where both of them is used to calculate the volume of the monochromatic 4-valent and 6-valent spin network for the kinematical case.
Review on hermiticity of the volume operators in Loop Quantum Gravity
General Relativity and Gravitation, 2019
The aim of this article is to provide a rigorous-but-simple steps to prove the hermiticity of the volume operator of Rovelli-Smolin and Ashtekar-Lewandowski using the angular momentum approach, as well as pointing out some subleties which have not been given a lot of attention previously. Besides of being hermitian, we also prove that both volume operators are real, symmetric, and positive semi-definite, with respect to the inner product defined on the Hilbert space over SU(2). Other special properties follows from this fact, such as the possibility to obtain real orthonormal eigenvectors. Moreover, the matrix representation of the volume operators are degenerate, such that the real positive eigenvalues always come in pairs for even dimension, with an additional zero if the dimension is odd. As a consequence, one has a freedom in choosing the orthonormal eigenvectors for each 2-dimensional eigensubspaces. Furthermore, we provide a formal procedure to obtain the spectrum and matrix representation of the volume operators. In order to compare our procedure with the earlier ones existing in the literature, we give explicit computational examples for the case of monochromatic quantum tetrahedron, where the eigenvalues agrees with the standard earlier procedure.
Semiclassical analysis of the Loop Quantum Gravity volume operator: Area Coherent States
We continue the semiclassical analysis of the Loop Quantum Gravity (LQG) volume operator that was started in the companion paper [23]. In the first paper we prepared the technical tools, in particular the use of complexifier coherent states that use squares of flux operators as the complexifier. In this paper, the complexifier is chosen for the first time to involve squares of area operators. Both cases use coherent states that depend on a graph. However, the basic difference between the two choices of complexifier is that in the first case the set of surfaces involved is discrete, while, in the second it is continuous. This raises the important question of whether the second set of states has improved invariance properties with respect to relative orientation of the chosen graph in the set of surfaces on which the complexifier depends. In this paper, we examine this question in detail, including a semiclassical analysis. The main result is that we obtain the correct semiclassical p...
On the geometry of loop quantum gravity on a graph
We discuss the meaning of geometrical constructions associated to loop quantum gravity states on a graph. In particular, we discuss the "twisted geometries" and derive a simple relation between these and Regge geometries.
Quantum Theory of Geometry II: Volume operators
1997
A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.
Discreteness of area and volume in quantum gravity
We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.
2009
A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Rie-mannian geometry. Operators corresponding to volume of three-dimensional regions are introduced rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-deened operator. Both operators can be completely speciied by giving their action on states labelled by graphs. The two nal results are closely related but diier from one another in that one of the operators is sensitive to the diierential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was rst introduced by Rov-elli and Smolin and De Pietri and Rovelli using a somewhat diierent framework.) The diierence between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumption...
Quantum Theory of Geometry; 2, Volume operators
1997
A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying a...
Discrete spacetime volume for three-dimensional BF theory and quantum gravity
The Turaev-Viro state sum invariant is known to give the transition amplitude for the three dimensional BF theory with cosmological term, and its deformation parameterh is related with the cosmological constant viah = √ Λ. This suggests a way to find the expectation value of the spacetime volume by differentiating the Turaev-Viro amplitude with respect to the cosmological constant. Using this idea, we find an explicit expression for the spacetime volume in BF theory. According to our results, each labelled triangulation carries a volume that depends on the labelling spins. This volume is explicitly discrete. We also show how the Turaev-Viro model can be used to obtain the spacetime volume for (2+1) dimensional quantum gravity.