The Quantum Structure of Space and Time (293 pages)

Emergent Spacetime 189

indices are used in the above equation is a gauge choice, equivalent to the choice of

a particular horizon volume.

A similar analysis leads to a guess about the description of particle states in dS

space. If, using ideas from quantum field theory, we ask for the maximal entropy

states in dS space which contain no black holes whose radius goes to infinity with

R, then we find that they are made from massless particles (or other conformal

field theory degrees of freedom) with a typical momentum of order The

entropy of such states scales like (RMP)~/'. These are the only states of the dS

theory which are kept in the Poincare invariant R 4 00 limit. We can view the full

entropy of dS space as built up from (RMp>'/' independent horizon volumes[2],

each filled with such maximal entropy particle states.

In terms of the matrix $?, we model these degrees of freedom as follows. Write

the block decomposition

1 2 3 ... M

M 1 2 ... M-1 '1' ),

2 3 4 ...

where each block is an independent M x M matrix, with M N N1/2. The integer

labels on the blocks refer to a given horizon volume, of which there are of order M.

In a future publication [13] I hope to show, using Matrix Theory ideas, similar to

those described in [l], that the degrees of freedom in a single block correspond, in
the large M limit to those of a single 4 dimensional supergraviton. The integer M
will be the longitudinal momentum of the supergraviton in units of the cutoff.
If this idea works, it is clear that corrections to the commutator [Po, Qa] for the
super-Poincare algebra will be of order N-4, which implies the scaling law for the

gravitino mass postulated in [3].

The holographic approach to quantum gravity is thus a promising generaliza-
tion of string theory which is applicable to cosmological backgrounds. Much work
remains to be done to refine its principles, make more explicit contact with string
theory, find non-perturbative equations for S-matrices in asymptotically flat space,
and solve the quantum theory of de Sitter space.

Bibliography

[l] T. Banks, won Neumann algebras and the holographic description of llD SUGRA,

manuscript in preparation.

[2] T. Banks, W. Fischler and S. Paban, JHEP 0212 (2002) 062 [arXiv:hep-th/0210160].

[3] T. Banks, arXiv:hep-th/0007146. ; T. Banks, Int. J. Mod. Phys. A 16 (2001) 910.

[4] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, Phys. Rev. D 55 (1997) 5112

[arXiv:hep-th/9610043].

[5] W. Fischler and L. Susskind, arXiv:hep-th/9806039.

[6] R. BOUSSO, JHEP 9907 (1999) 004 [arXiv:hep-th/9905177].

[7] R. Bousso, Class. Quant. Grav. 17 (2000) 997 [arXiv:hep-th/9911002].