Approaches to Quantum Gravity

Consistent discretizations as a road to Quantum Gravity 381data that satisfy the constraint identically at level n, the quadratic equation for thelapse has a vanishing independent term and therefore the solution is that the lapseN vanishes (the non-vanishing root will be large and would imply a large time evolutionstep that puts us away from the continuum generically). To construct initialdata one therefore considers a set for which the constraint vanishes and introducesa small perturbation on one (or more) of the variables. Then one will have evolution.Notice that one can make the perturbation as small as desired. The smallerthe perturbation, the smaller the lapse and the closer the solution will be to thecontinuum.For concreteness, we choose the following initial values for the variables, M =2, q 1 0 = 0, q2 0 = (√ 3 − ) sin( π 4 ), P1 q,0 = 1, P1 q,0 = (√ 3 − ) cos( π 4 ).We choose the parameter to be the perturbation, i.e. = 0 corresponds to anexact solution of the constraint, for which the observable A = 1/2 (seebelowforits definition). The evolution scheme can easily be implemented using a computeralgebra program like Maple or Mathematica.Before we show results of the evolution, we need to discuss in some detail howthe method determines the lapse. As we mentioned it is obtained by solving thequadratic equation (20.7). This implies that for this model there will be two possiblesolutions and in some situations they could be negative or complex. One canchoose either of the two solutions at each point during the evolution. This ambiguitycan be seen as a remnant of the re-parameterization invariance of the continuum.It is natural numerically to choose one “branch” of the solution and keep with it.However, if one encounters that the roots become complex, we have observed thatit is possible to backtrack to the previous point in the iteration, choose the alternateroot to the one that had been used up to that point and continue with the evolution.A similar procedure could be followed when the lapse becomes negative. It shouldbe noted that negative lapses are not a problem per se, it is just that the evolutionwill be retraced backwards. We have not attempted to correct such retracings,i.e. in the evolutions shown we have only “switched branches” whenever the lapsebecomes complex. This occurs when the discriminant in the quadratic equation(20.7) changes sign.We would like to argue that in some sense the discrete model “approximates” thecontinuum model well. This, however, turns out to be a challenging proposition inre-parameterization invariant theories. The first thing to try, to study the evolutionof the quantities as a function of n is of course meaningless as a grounds to comparewith the continuum. In the discrete theory we do not control the lapse, thereforeplots of quantities as a function of n are meaningless. To try to get more meaningfulinformation one would like to concentrate on “observables”. In the continuum theory,these are quantities that have vanishing Poisson brackets with the constraints(also sometimes known as “perennials”). Knowing these quantities as functions of