Emergent Spacetime 185which is the most general rule giving a finite number of states per pixel and co-
variant under the transverse SO(d - 2) little group of the null vector &@@. It
breaks the projective invariance of the CP equation to a 22 (for each pixel), which
we treat as a gauge symmetry and identify with fermion parity (-l)F, enforcing
the usual connection between spin and statistics. This quantization rule imple-
ments the Bekenstein Hawking relation between quantum entropy, and area. The
logarithm of the dimension of the Hilbert space of the irreducible representation
of this algebra is the area of a fundamental pixel, in d dimensional Planck units.
Compact dimensions lead to an enlarged pixel algebra, incorporating charges for
Kaluza-Klein Killing symmetries, their magnetic duals, and wrapped brane config-
urations. Note that it is precisely these quantum numbers which remain unchanged
under the topology changing duality transformations of string/M-theory. We will
ignore the complications of compactification in the brief description which follows.
After performing a Klein transformation using the (-l)F gauge symmetry, the
operator algebra of an entire causal diamond takes the form,
[Sa(n), Sb(m)]+ = dabbn.
The middle alphabet labels stand for individual pixels. More generally we say
that the geometry of the holographic screen is pixelated by replacing its algebra
of functions by a finite dimensional algebra, and these labels stand for a general
basis in that algebra. If we use finite dimensional non-abelian function algebras,
we can have finite causal diamonds with exact rotational invariance, which would
be appropriate for describing the local physics in asymptotically symmetric space-
times.
The Sa(n) operators should be thought of as transforming in the spinor bundle
over the holographic screen. Informally, we can say that the algebra of operators of
a pixel on the holoscreen, is described by the degrees of freedom of a massless super-
particle which exits the holoscreen via that pixel. In a forthcoming paper[l], I will
describe how an infinite dimensional limit of such a construction can reproduce the
Fock space of eleven dimensional supergravity. The basic idea is to find a sequence
of algebras which converges to
A11 R[o,i] 8 M(S9),
where R[o,l~ is the unique, hyperfinite Type 11, von Neumann factor, and M(S9)
the algebra of measurable functions on the sphere. We take our quantum operator
algebra to be operator valued linear functionals S(q), q E S[A11], which are invariant
under inner automorphisms of All. S is the spinor bundle over the algebra. The
projectors in R[o,ll are characterized, up to inner automorphism, by their trace,
which is a real number between 0 and 00. A general invariant linear functional is
determined by its value on a finite sum of projectors. Thus, the quantum algebra
consists of finite collections of operators of the form Sa(pi)qa(i2i) where pi is a
positive number and Kli a direction on Sg. These parametrize a null momentum