The quantum of area?
John Baez
November 28, 2003
One of the key predictions of loop quantum gravity is that the area of a
surface can only take on a discrete spectrum of values. In particular,
there is a smallest nonzero area that a surface can have. We could
call this the "quantum of area".
So far, calculations working strictly within the framework of loop
quantum gravity have been unable to determine the quantum of area. But
in 2002, thanks to work of
Olaf
Dreyer and Lubos Motl,
two very different methods of calculating the quantum of
area have been shown to give the same answer: 4 ln(3) times the Planck
area. Both use information about classical
black holes. It is still completely mysterious why they give the
same answer. It could be a misleading coincidence, or it could be an
important clue. Either way, the story is worth telling.
Here's a quick, nontechnical version of the story:
And here's a somewhat longer version with a few equations:
Both versions cite a bunch of papers available online;
here are links to those, in alphabetical order by author.
They make a good way to learn about the subject!

Abhay Ashtekar, John Baez, Alejandro Corichi and Kirill Krasnov,
Quantum geometry and black hole entropy.

Abhay Ashtekar, Alejandro Corichi and Kirill Krasnov,
Isolated horizons: the classical phase space.

Abhay Ashtekar, John Baez, and Kirill Krasnov,
Quantum geometry of isolated horizons and black hole entropy.

Fernando Barbero,
Real Ashtekar variables for Lorentzian signature
spacetimes.

Jacob Bekenstein and Vladimir Mukhanov,
Spectroscopy of the quantum black hole.

Alejandro Corichi,
On quasinormal modes, black hole entropy, and
quantum geometry.
 Olaf Dreyer,
Quasinormal modes, the area spectrum, and
black hole entropy.

Shahar Hod,
Bohr's correspondence principle and the area spectrum
of quantum black holes.

Shahar Hod,
Gravitation, the quantum, and Bohr's correspondence principle.

Shahar Hod,
Kerr black hole quasinormal frequencies.

Giorgio Immirzi,
Quantum gravity and Regge calculus.

Kirill Krasnov,
On quantum statistical mechanics of a Schwarzschild black
hole.

Lubos Motl,
An analytical computation of asymptotic Schwarzschild quasinormal
frequencies.

Lubos Motl and Andrew Neitzke, Asymptotic black hole
quasinormal frequencies.

Carlo Rovelli, Loop quantum gravity.

Carlo Rovelli and Lee Smolin,
Discreteness of area and volume in quantum gravity.
Warning: in the summer of 2004, Domagala
and Lewandowski found a mistake in the
AshtekarBaezCorichiKrasnov paper. Their corrected calculations,
done with the help of Meissner, give a new value of the quantum of area!
This casts the work of Dreyer in doubt:
So, the mystery continues....
© 2004 John Baez
baez@math.removethis.ucr.andthis.edu