Natural Science Forum / Physics / Research / March 2005

According to standard lore, the quantization of a diffeomorphism-
invariant theory like general relativity can not have local degrees
of freedom. Hence the best description of quantum gravity we can
hope for is in terms of holographic data living on the boundary of
the universe. However, it is difficult to reconcile this viewpoint
with a positive cosmological constant, as described e.g. by Witten
in http://www.arxiv.org/abs/hep-th/0106109 . The problem is that de
Sitter space (positive CC) has no boundary at spatial infinity, so
the only asympotia are at temporal +- infinity, and the Hilbert
space does not allow for any observables at all.

Witten then attempts to find some way to circumvent this no-go
theorem, IMO not very convincingly. However, taking his argument at
face value, we are lead to an important conclusion: THERE CAN BE NO
HOLOGRAPHIC QUANTUM THEORY OF GRAVITY WITH POSITIVE COSMOLOGICAL
CONSTANT. It is thus the sign, rather than the magnitude, which is
the big conceptual problem.

One may think of various ways to resolve this paradox:

1. Experimental error. However, the CC = (2.13 +- 0.1) 10^-3 eV
is quite far away from zero.

2. Decree that the entire observable universe is a dS bubble living
inside a bigger, unobservable, AdS multiverse. This idea seems
somewhat difficult to test experimentally.

3. Don't quantize gravity at all. This is a logical possibility,
since no clearcut quantum gravity effects have been seen, but it
leads to severe conceptual problems and nobody really believes it.

4. Decree that spacetime is only a large-distance, emergent concept.
This seems to be Witten's position, though I don't really understand
how it can help with the CC, which is a very-large-distance concept.

5. Find a LOCAL quantum theory of gravity, i.e. throw out
holography. This seems to me as the obvious solution, since both
general relativity and QFT are local theories in appropriate senses.
The positive CC is then not a problem (not an immediate one anyway),
because the degrees of freedom live locally in the bulk rather than
at temporal infinity.

Of course, to make QG into a local theory, we need diffeomorphism
anomalies, which elevate the diffeomorphism symmetry from a mere
redundancy of the description to a genuine symmetry. The positive
CC can thus be construed as strong evidence for diff anomalies.