Approaches to Quantum Gravity

String theory, holography and Quantum Gravity 193of transformations than SO(d − 2). It is not reasonable to associate every linearcombination of S a (n) with pixels, or small areas on the holoscreen. Rather, weshould think of the discretization of the holoscreen topology to occur through thereplacement of its algebra of functions with a finite dimensional algebra. Differentlinear combinations of the S a (k) correspond to operators associated with differentbases of the finite function algebra. If the function algebra were abelian, we wouldhave a standard geometrical discretization of the surface (e.g. a triangulation ofa two surface) and we could choose a special basis for the algebra consisting offunctions which were non-vanishing on only a single pixel.At least in the case where we want to preserve exact continuous symmetries, 6this will lead us into the simple case of non-commutative geometry, called fuzzygeometry. The function algebras for spherical holoscreens will be finite dimensionalmatrix algebras. Here the notion of a pixel is only an approximate one,similar to the localization of quantum Hall states within a Larmor radius of a point.If we go to the particular basis where S a (n) represents a single pixel, we see aconnection between this formalism and supersymmetry. The algebra of operatorsfor a pixel is precisely the supersymmetry algebra for a massless supermultipletwith fixed momentum. We thus claim that the degrees of freedom specifying theorientation of a pixel on the holographic screen of a causal diamond are the states ofa massless superparticle which emerges from (or enters into) the diamond throughthat pixel. In an asymptotically flat space, the limit of large causal diamonds shouldapproach null infinity. The number of degrees of freedom becomes infinite, and inparticular, we expect the pixel size to shrink to zero, relative to the area of theholoscreen. Thus, we should associate the pixel with a particular outgoing nulldirection (1,)at null infinity. We will see later that the overall scale of the masslessmomentum can also be encoded in the algebra of operators. This observationis, I believe, an indication that the formalism automatically generates supersymmetrictheories in asymptotically flat space. Indeed, when the SUSY algebra islarge enough to force us to include the gravitino in the multiplet, we already knowthat the dynamics must be exactly supersymmetric. The conjecture that all asymptoticallyflat theories of Quantum Gravity must be Super Poincaré invariant is calledCosmological SUSY Breaking (CSB)[17].11.2 Dynamical constraintsThe time evolution operator describing dynamics for a given observer cannot be agauge invariant operator. In a generally covariant theory there cannot be a canonical6 If we are trying to model causal diamonds in a space-time with an asymptotic symmetry group, it is reasonableto restrict attention to diamonds which are invariant under as much of that group as possible. Remember thatthe choice of finite causal diamonds is a gauge choice.