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 [17
]. 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:
where
are half integers. For the full spectrum, see [17
] (connection representation) and [84] (loop representation).
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 [51]. 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
For more details, and the full derivation of these formulas, see [51, 203]
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 [125
], where it is shown that
contains all ``free graviton'' physics, in a suitable
approximation.
Recently, Bekenstein and Mukhanov [46] 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 [193]). Bekenstein and Mukhanov observe that in most approaches to
quantum gravity the area can take only quantized values [97]. 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 [43], but the result was already contained (implicitly, in the
first version) in [17]. 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.
Indirect arguments [105, 106, 45, 210] strongly support the idea that a Schwarzschild black hole of (macroscopic) area A behaves as a thermodynamical system governed by the Bekenstein-Hawking entropy
(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 [176]. 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 [11].)
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 [127] 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 [189
]. On the issue of entropy in loop gravity, see also [194].
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Loop Quantum Gravity
Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |