- Planck-scale discreteness of space
The central physical result obtained from loop quantum gravity is the evidence for a physical quantum discreteness of space at the Planck scale. This is manifested in the fact that certain operators corresponding to the measurement of geometrical quantities, in particular area and volume, have discrete spectra. According to the standard interpretation of quantum mechanics (which we adopt), this means that the theory predicts that a physical measurement of an area or a volume will yield quantized results. In particular, since the smallest eigenvalues are of Planck scale, this implies that there is no way of observing areas smaller than the Planck scale. Space comes, therefore, in “quanta” in the same manner as the energy of an oscillator. The spectra of the area and volume operators have been computed in detail in loop quantum gravity. These spectra have a complicated structure, and they constitute detailed quantitative physical predictions of loop quantum gravity on Planck-scale physics. If we had experimental access to Planck-scale physics, they would allow the theory to be empirically tested in detail.

The discreteness of area and volume is derived as follows. Consider a surface . The physical area of depends on the metric, namely on the gravitational field. In a quantum theory of gravity, the gravitational field is a quantum field operator, and therefore the area of is described by a quantum operator . What is the quantum operator in nonperturbative quantum gravity? It can easily be worked out by writing the standard expression for the area of a surface and replacing the metric with the appropriate function of the loop variables. Promoting these loop variables to operators, we obtain the area operator . The precise construction of this operator requires regularizing the classical expression and then taking the limit of a sequence of operators, in a suitable operator topology [268, 98, 121, 75, 31]. For a complete presentation of the details of this construction, see [261, 294]. The resulting area operator acts as follows on a spin-network state (assuming here for simplicity that is a spin network without nodes on ):

where labels the intersections between the spin network and the surface , and is the spin of the link of crossing the intersection. This result shows that the spin-network states (with a finite number of intersection points with the surface and no nodes on the surface) are eigenstates of the area operator. The corresponding spectrum is labeled by multiplets of positive half integers, with arbitrary , and given by A similar result can be obtained for the volume [268, 183, 184, 32, 98, 175, 286]. The eigenvalues of the volume of a region R turn out to be determined by the intertwiners of the nodes of the spin network contained in R. The two results on area and volume offer a compelling physical interpretation of the spin-network states. These are quantum states in which space is made by a set of “chunks”, or quanta of space, which are represented by the nodes of the spin network, connected by surfaces, which are represented by the links of the spin networks. The intertwiners on the nodes are the quantum numbers of the volume of the chunks, while the spins on the links are the quantum numbers of the area of the surfaces that separate the chunks. See Figure 5.Two comments are in order.

- The reader will wonder why area and volume seem here to play a role more central than length, when classical geometry is usually described in terms of lengths. The reason is that the length operator is difficult to define and has a difficult physical interpretation, see [288]. Whether this is simply a technical difficulty, or it reflects some deep fact, is not clear to me. The basic field of the theory is not the tetrad , which assigns length to line elements, but rather the 2-form , which assigns areas to surface elements. Another way to say it is that the loop representation is based on the theory of quantized angular momentum. Angular momentum is not a vector but a bivector, so it corresponds not to an arrow but to an oriented area element. On this, see Baez’s [49]. On the relation between the field and area, see [247].
- Area and volume are not gauge-invariant operators. Therefore, we cannot directly interpret them
as representing physical measurements, according to the conventional interpretation of quantum
gauge systems. There are three reason, however, to take the discreteness of their spectra as an
indication of the physical discreteness of spacetime.
- A realistic measurement of an area or a volume refers to a surface or a region determined physically, for instance by some physical object. For example, I can measure the area of the surface of a certain table at a certain time. In the dynamical theory that describes the gravitational field, as well as the table (and the clock), the area of the surface of the table is a diffeomorphism-invariant quantity A, which depends on gravitational as well as matter variables. In the quantum theory, will be represented by a diffeomorphism-invariant operator. It is completely plausible to assume that the operator A is the same mathematical operator as the pure gravity area operator. This is because we can gauge fix the matter variables, and use matter location as coordinates, so that non-diff-invariant observables in the pure gravity theory correspond precisely to diff-invariant observables in the matter + gravity theory, as they do in the classical theory [249]. In other words, the fact that these geometrical operators have discrete spectra is true in any gauge.
- The discreteness depends on the commutation structure of the relevant geometrical quantities. This does not change according to the specific versions of the quantities to which it is applied. Compare this with the angular momentum in nonrelativistic quantum theory: the angular momentum is always quantized, and it always has the same eigenvalues, irrespective of whether it is the angular momentum of an atom, a proton or a molecule. This is because the angular momentum observables may be different in the different cases, but their commutation structure remains the same. Similarly, the commutation structure of the components of the area of any physical object can be uniquely dictated by the geometry of the gravitational field, not by specific features of the object.
- Quantum mechanical discreteness is a kinematical property of a system, independent of the system’s dynamics. For instance, the momentum of a particle in a box is quantized, independently from the form of the Hamiltonian of the particle. To make precise sense of this crucial observation in the context of gravity, where dynamics and gauges are mixed up, requires a careful analysis of the way the formalism of quantum theory may be extended in the case of generally covariant systems. This is done in [261]. There discreteness is recognized as always associated to the spectra of the partial observables [260]. This analysis provides a foundation for the claim that area and volume are predicted to be discrete in loop quantum gravity.

- Quantum cosmology
The application of loop quantum gravity to cosmology is one of it most spectacular achievements. The main result is that the initial singularity is controlled by quantum effects. The reason is not difficult to grasp. In the classical theory, the volume of the universe goes continuously to zero at the Big Bang singularity. In the quantum theory, one has transition amplitudes between finite-volume eigenvalues. The singularity is controlled by a mechanics very similar to quantum mechanism that stabilizes the orbit of an electron around the nucleus. This opens up the possibility of studying the physics of the very initial universe and also the physical evolution across the Big Bang. The region around the Big Bang is a region where spacetime enters a genuine quantum regime, which cannot be described in terms of a conventional spacetime manifold, but that can still be described by the quantum theory. For a detailed description of techniques and results of loop cosmology, see the comprehensive Living Review article by Martin Bojowald [73]. For a nice introduction see Abhay Ashtekar [20].

- Black hole singularity
The same techniques applied in quantum cosmology can be utilized to study quantum spacetime in the neighborhood of the classical singularity at the center of a black hole. Again, the singularity is controlled by quantum effects. Again, the region around the classical singularity is a region in which spacetime enters a genuine quantum regime, which cannot be described in terms of a conventional spacetime manifold, but that can still be described by quantum theory [24, 204]. This opens up a new possible “paradigm” [23] for describing the final evolution of a black hole.

- Black hole entropy
Indirect arguments strongly support the idea that a Schwarzschild black hole of (macroscopic) area A behaves as a thermodynamical system governed by the Bekenstein–Hawking entropy [142, 143, 68, 296]

( is the Boltzmann constant; here I put the speed of light equal to one, but write the Planck and Newton constants explicitly). A physical understanding and a first principles derivation of this relation require quantum gravity, and therefore represent a challenge for every candidate theory of quantum theory. The Bekenstein–Hawking expression (34) for the entropy of a Schwarzschild black hole of surface area A can be derived from loop quantum gravity via a statistical mechanical computation [170, 171, 253, 22]. The derivation has been extended to various classes of black holes; see [25] and references therein.This derivation is based on the idea that the entropy of the black hole originates from the microstates of the horizon that correspond to a given macroscopic configuration [297, 85, 84, 58, 57]. Physical arguments indicate that the entropy of such a system is determined by an ensemble of configurations of the horizon with fixed area [253]. In quantum theory these states are finite in number and can be counted [170, 171]. Counting these microstates using loop quantum gravity yields

is defined in Section 6, and is a real number of the order of unity that emerges from the combinatorial calculation (roughly, ). If we choose , we get Equation (34) [270, 92]. Thus, the theory is compatible with the numerical constant in the Bekenstein–Hawking formula, but does not lead to it univocally. The precise significance of this fact is under discussion. In particular, the meaning of is unclear. Jacobson has suggested [159, 160] that finite renormalization effects may affect the relation between the bare and the effective Newton constant, and this may be reflected in . For discussion of the role of in the theory, see [270]. On the issue of entropy in loop gravity, see also [275]. - Low-energy limit: n-point functions
Loop quantum gravity is formulated in a background-independent language. Spacetime is not assumed a priori, but rather it is built up by the states of theory themselves. The relation between this formalism and the conventional formalism of quantum field theory on a given spacetime is far from obvious, and it is far from obvious how to recover low-energy quantities from the full background-independent theory. One would like, in particular, to derive the n-point functions of the theory from the background-independent formalism, in order to compare them with the standard perturbative expansion of quantum general relativity and therefore check that loop quantum gravity yields the correct low-energy limit.

The search for a way to describe the low-energy degrees of freedom, namely “the graviton” in the background-independent formulation of loop quantum gravity, has a long history [157, 158]. To appreciate the difficulty, observe that the n-point functions are intrinsically defined on a background. In fact, they express correlations among the fluctuations of the quantum field around a given background solution. How can one extract this information from the background independent theory?

A strategy for doing so has been introduced in [205]. It is based on the idea of considering a finite region of spacetime and studying the amplitude for having given boundary states around this region. By choosing the boundary states appropriately, one can study the physical configurations that fluctuate around a chosen average internal geometry. In particular, one can recover quantities that converge to the conventional n-point functions in the large-distance limit, by appropriately “adding quanta” to the boundary state that corresponds to an average internal flat geometry. Using this technique, a calculation of the graviton propagator has been completed in [262] to first order and in [69] to second order. Similar calculations have been completed in three dimensions [280]. An improved boundary state has been studied in [180]. See also [102, 202]. The calculation of the complete tensorial structure of the propagator has been recently completed [2].

- Observable effects
The best possibility for testing the theory seems to be via cosmology. However, the investigation of the possibility that quantum gravity effects are observable is constantly under investigation. Various possibilities have been considered, including quantum gravitational effects on light and particle propagation at very long distances [130, 8], which could perhaps be relevant for observations in progress such as AUGER and GLAST, and others. For an overview, see for instance [277, 199].

The MAGIC telescope collaboration has recently reported the measurement of an energy-dependent time delay in the arrival of signals from the active galaxy Markarian 501. The measured phenomenological parameter governing this dependence is on the Planck scale [1]. Energy-dependent time delays in the arrival of signals from far away sources have long been suggested as possible quantum gravity effects [14, 15]. A quantum-gravity interpretation of the MAGIC observation does not appear to be likely at present (see for instance [67]), but the measurement shows that quantum-gravity effects are within the reach of current technology.

- Noncommutative geometry from loop quantum gravity
Laurent Freidel and Etera Livine have shown that the low-energy limit of quantum gravity coupled with matter in three dimensions is equivalent to a field theory on a noncommutative spacetime [116]. This is a remarkable result because it directly connects the study of noncommutative spacetimes with quantum gravity. Work is in progress to understand the extent to which the result is also meaningful in four dimensions.

http://www.livingreviews.org/lrr-2008-5 |
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |