Approaches to Quantum Gravity

204 T. BanksThus, there are of order (RM P ) 3/2 field theoretic degrees of freedom in a horizonvolume. If the field theory has a particle description this corresponds to of order(RM P ) 3/2 particles.This description gets the counting right, but conflicts with the experimental factthat we can excite momenta much higher than this cut off in the laboratory. We willsee that the fermionic pixel variables suggest a more flexible way for the particleinterpretation to emerge from the formalism.We should also note that, although we will continue to concentrate on thedescription appropriate to a given causal diamond, this estimate allows us to understandhow the global coordinate description of dS space might emerge in the largeR limit. The total entropy of dS space is of order (RM P ) 2 . This means that there areenough degrees of freedom to account for (RM P ) 1/2 commuting copies of the fieldtheory variables allowed in a given horizon volume. In global coordinates, at earlyand late times, the number of independent horizon volumes seems to grow withoutbound. However, if we imagine filling each of those volumes with a generic fieldtheoretic state, then the extrapolation into either the past or the future leads to aspace-like singularity before the minimal volume sphere is reached. We interpretthis as saying that the general field theoretic state in very late or very early timedS space, does not correspond to a state in the quantum theory of dS space. Onlywhen the absolute value of the global time is small enough that there are at√most(RM P ) 1/2 Mhorizon volumes, does a generic field theory state (with cut-off PR )correspond to a state in Quantum Gravity. At later times, most horizon volumesmust be empty. As RM P → ∞, these restrictions become less important. Theconventional formalism of quantum field theory in dS space is the singular limitRM P →∞with R kept finite in units of particle masses. If particle masses inPlanck units do not approach constant values at RM P →∞, then this limit doesnot make any sense. In particular, if SUSY is restored in this limit, the splittings insupermultiplets do not approach constant values.Here is the way to reproduce the field theory state counting in terms of fermionicpixel variables. Write the fermionic matrix in terms of blocks of size M × M + kwith k = 0, 1andM ∼ √ N. The states in a given horizon volume are associatedwith fermionic variables along a block diagonal, as follows⎛⎞1 2 3 ... MM 1 2 ... M − 1M − 1 M 1 ... M − 2... ... ... ... ....... ... ... ... ...⎜⎟⎝ 3 4 5 ... 2 ⎠2 3 4 ... 1