The most remarkable physical result obtained from loop quantum gravity is, in my opinion, 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 spectrum. 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 necessarily yield quantized results. Since the smallest eigenvalues are of Planck scale, this implies that there is no way of observing areas or volumes smaller than Planck scale. Space comes in ``quanta'' in the same manner as the energy of an oscillator. The spectra of the area and volume operators have been computed with much 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 great detail.
A few comments are in order.
The discreteness of area and volume is derived as follows. Consider the area A of a surface . The physical area A 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 we must describe the area of in terms of a quantum observable, described by an operator . We now ask what the quantum operator in nonperturbative quantum gravity is. The result can easily be worked out by writing the standard formula 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 actual construction of this operator requires regularizing the classical expression and then taking the limit of a sequence of operators, in a suitable operator topology. For the details of this construction, see [186, 77, 84, 51]. An alternative regularization technique is discussed in . The resulting area operator acts as follows on a spin network state (assuming here for simplicity that S is a spin network without nodes on ):
where i labels the intersections between the spin network S and the surface , and is the color of the link of S crossing the i - th 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 n, and given by
Shifting from color to spin notation reveals the SU (2) origin of the spectrum:
A similar result can be obtained for the volume [186, 142, 143, 77, 139]. Let us restrict ourselves here, for simplicity, to spin networks S with nondegenerate four-valent nodes, labeled by an index i . Let be the colors of the links adjacent to the i - th node and let label the basis in the intertwiner space. The volume operator acts as follows
where is an operator that acts on the finite dimensional space of the intertwiners in the i - th node, and its matrix elements are explicitly given (in a suitable basis) by ()
See . The volume eigenvalues are obtained by diagonalizing these matrices. For instance, in the simple case a = b, c = d =1, we have
if d = a + b + c, we have
The s-knot states do not represent excitations of the quantum gravitational field over flat space, but rather over ``no-space'', or over the solution. A natural problem is then how flat space (or any other smooth geometry) might emerge from the theory. Notice that in a general relativistic context the Minkowski solution does not have all the properties of the conventional field theoretical vacuum. (In gravitational physics there is no real equivalent of the conventional vacuum, particularly in the spatially compact case.) One then expects that flat space is represented by some highly excited state in the theory. States in that describe flat space when probed at low energy (large distance) have been studied in [23, 217, 49, 99]. These have a discrete structure at the Planck scale. Furthermore, small excitations around such states have been considered in , where it is shown that contains all ``free graviton'' physics, in a suitable approximation.
Recently, Bekenstein and Mukhanov  have suggested that the thermal nature of Hawking's radiation [105, 106] may be affected by quantum properties of gravity (For a review of earlier suggestions in this direction, see ). Bekenstein and Mukhanov observe that in most approaches to quantum gravity the area can take only quantized values . Since the area of the black hole surface is connected to the black hole mass, black hole mass is likely to be quantized as well. The mass of the black hole decreases when radiation is emitted. Therefore emission happens when the black hole makes a quantum leap from one quantized value of the mass (energy) to a lower quantized value, very much as atoms do. A consequence of this picture is that radiation is emitted at quantized frequencies, corresponding to the differences between energy levels. Thus, quantum gravity implies a discretized emission spectrum for the black hole radiation.
This result is not physically in contradiction with Hawking's prediction of a continuous thermal spectrum, because spectral lines can be very dense in macroscopic regimes. But Bekenstein and Mukhanov observed that if we pick the simplest ansatz for the quantization of the area -that the Area is quantized in multiple integers of an elementary area -, then the emitted spectrum turns out to be macroscopically discrete, and therefore very different from Hawking's prediction. I will denote this effect as the kinematical Bekenstein-Mukhanov effect. Unfortunately, however, the kinematical Bekenstein-Mukhanov effect disappears if we replace the naive ansatz with the spectrum (41) computed from loop quantum gravity. In loop quantum gravity, the eigenvalues of the area become exponentially dense for a macroscopic black hole, and therefore the emission spectrum can be consistent with Hawking's thermal spectrum. This is due to the details of the spectrum (41) of the area. A detailed discussion of this result is in , but the result was already contained (implicitly, in the first version) in . It is important to notice that the density of the eigenvalues shows only that the simple kinematical argument of Bekenstein and Mukhanov is not valid in this theory, and not that their conclusions is necessarily wrong. As emphasized by Mukhanov, a discretization of the emitted spectrum could still be originated dynamically.
(k 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. A derivation of the Bekenstein-Hawking expression (46) for the entropy of a Schwarzschild black hole of surface area A via a statistical mechanical computation, using loop quantum gravity, was obtained in [134, 135, 176].
This derivation is based on the ideas that the entropy of the hole originates from the microstates of the horizon that correspond to a given macroscopic configuration [213, 63, 64, 37, 38]. Physical arguments indicate that the entropy of such a system is determined by an ensemble of configurations of the horizon with fixed area . In the quantum theory these states are finite in number, and can be counted [134, 135]. Counting these microstates using loop quantum gravity yields
(An alternative derivation of this result has been announced from Ashtekar, Baez, Corichi and Krasnov .) is defined in section 6, and c is a real number of the order of unity that emerges from the combinatorial calculation (roughly, ). If we choose , we get (46) [189, 70]. 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  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 . On the issue of entropy in loop gravity, see also .
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