Dimensional Reduction in Quantum Gravity

The requirement that physical phenomena associated with gravitational collapse should be duly reconciled with the postulates of quantum mechanics implies that at a Planckian scale our world is not 3+1 dimensional. Rather, the observable degrees of freedom can best be described as if they were Boolean variables defined on a two-dimensional lattice, evolving with time. This observation, deduced from not much more than unitarity, entropy and counting arguments, implies severe restrictions on possible models of quantum gravity. Using cellular automata as an example it is argued that this dimensional reduction implies more constraints than the freedom we have in constructing models. This is the main reason why so-far no completely consistent mathematical models of quantum black holes have been found. † Essay dedicated to Abdus Salam With the request to write a short paper in honor of Abdus Salam I am given the opportunity to contemplate some very deep questions concerning the ultimate unification that may perhaps be achieved when all aspects of quantum theory, particle theory and general relativity are combined. One of these questions is the dimensionality of space and time. At first sight our world has three spacelike dimensions and one timelike. This was used a a starting point of all quantum field theories and indeed also all string theories as soon as they invoke the Kaluza Klein mechanism. Also when we quantize gravity perturbatively we start by postulating a Fock space in which basically free particles roam in a three plus one dimensional world. Naturally, when people discuss possible cut-off mechanisms, they think of some sort of lattice scheme either in 3+1 dimenisional Minkowski space or in 4 dimensional Euclidean space. The cut-off distance scale is then suspected to be the Planck scale. Unfortunately any such lattice scheme seems to be in conflict with local Lorentz invariance or Euclidean invariance, as the case may be, and most of all also with coordinate reparametrization invariance. It seems to be virtually impossible to recover these symmetries at large distance scales, where we want them. So the details of the cut-off are kept necessarily vague. The most direct and obvious physical cut-off does not come from non-renormalizability alone, but from the formation of microscopic black holes as soon as too much energy would be accumulated into too small a region. From a physical point of view it is the black holes that should provide for a natural cut-off all by themselves. This has been this author’s main subject of research for over a decade. A mathematically consistent formulation of the black hole cut-off turns out to be extremely difficult to find, and in this short note I will explain what may well be the main reason for this difficulty: nature is much more crazy at the Planck scale than even string theorists could have imagined. One of my starting points has been that quantum mechanics itself is not at all a mystery to me. The emergence of a Hilbert space with a Copenhagen interpretation of its inner products is a quite natural feature of any theory with the following characteristics at a local scale: the system must have discrete degrees of freedom at tiny ditance scales, and the laws of evolution must be reversible in time. With discrete degrees of freedom one can construct Hilbert space in a quite natural way by postulating that any state of the physical degrees of freedom corresponds to an element of a basis of this Hilbert space[1]. Reversibility in time is required if we wish to see a quantum superposition principle: the norm of all states is then preserved, even if they are quantum superpositions of these basis elements.