Approaches to Quantum Gravity

192 T. Banks
Since the holographic screen is central to these ideas, it is natural to take its
geometry for a given causal diamond to be the primordial dynamical variable
of Quantum Gravity. Consider an infinitesimal area element on the screen. The
Cartan–Penrose (CP) equation gives us a way to specify the holographic screen
element associated with this area, in terms of a pure spinor. This is a commuting
classical spinor satisfying
¯ψγ μ ψγ μ ψ = 0.
The equation implies that ¯ψγ μ ψ is a null vector,n μ , and the non-vanishing components
of ¯ψγ μ 1...μ k
ψ,forallk lie in a d − 2 hyperplane transverse to n μ .We
will call ñ μ the reflected null vector transverse to the same surface and satisfying
n μ ñ μ =−2. Actually, the CP equation is homogeneous and the rescaling ψ → λψ
is considered a gauge equivalence at the classical level. The classical CP equation
specifies only the local orientation of the holoscreen and of null directions passing
through it. The non-vanishing components of the pure spinor
S = γ μ n μ γ ν ñ ν ψ,
transform like an SO(d − 2) spinor under rotations transverse to n μ .
Thus, the full conformal structure of the holoscreen is encoded in an element,
S a (σ ), of the spinor bundle over the holoscreen. S a are the real components of this
spinor, and represent the independent components of a covariant spinor satisfying
the CP equation.
As might be expected from the Bekenstein–Hawking formula, the classical
notion of area is only obtained after quantization of the spinor variables. If k
specifies a pixel on the holoscreen, then we quantize S a (k) by postulating
[S a (k), S b (k)] + = δ ab .
This rule is SO(d −2) invariant, and assigns a finite number of states to the pixel. It
also breaks the projective invariance of the CP equation down to the Z 2 , S a →−S a .
We will keep this as a gauge invariance of the formalism, which will turn out to be
fermion parity, (−1) F .
The S a operators for independent pixels should commute, but we can use this
gauge invariance to perform a Klein transformation and cast the full operator
algebra of the holoscreen as
[S a (k), S b (l)] + = δ ab δ kl .
We have used the word pixel, and discrete labeling to anticipate the fact that the
requirement of a finite number of states forces us to discretize the geometry of
the holoscreen of a finite area causal diamond. The labels k, l run over a finite
set of integers. Note that the operator algebra is invariant under a larger group