Approaches to Quantum Gravity

382 R. Gambini and J. Pullinphase space allows one to know any type of dynamical physical behavior of thesystem. One can use them, for instance, to construct “evolving constants” [27].The existence of perennials in the continuum theory is associated with symmetriesof the theory. If such symmetries are not broken by the discretization process,then in the discrete theory one will have exact conserved quantities that correspondto the perennials of the continuum theory. The conserved quantities will be givenby discretizations of the perennials of the continuum. It should be noted that in thecontinuum theory perennials as functions of phase space are defined up to the additionof multiples of the constraints. There are therefore infinitely many versions ofa given perennial. When discretized these versions are inequivalent (since in thediscrete theory the constraints of the continuum theory do not hold exactly) andonly one of these versions will correspond to an exact conserved quantity of thediscrete theory.In this model there are two independent perennials in the continuum. One ofthem becomes straightforwardly upon discretization an exact conserved quantityof the discrete theory,Another perennial is given byO 1 = p 1 q 2 − p 2 q 1 . (20.8)O 2 = (p 1 ) 2 − (p 2 ) 2 + (q 1 ) 2 − (q 2 ) 2 . (20.9)This quantity is not an exact conserved quantity of the discrete model, it is conservedapproximately, as we can see in figure 20.1. We at present do not knowhow to find an exact conserved quantity in the discrete theory that corresponds toa discretization of this perennial (plus terms proportional to the constraint). In theend, this will be the generic situation, since in more complicated models one willnot know exact expressions either for the perennials of the continuum theory orthe constants of motion of the discrete theory. Notice also that in the continuum,in order to recover physical information about the system, one generically needsthe two perennials plus combinations involving the constraints. In the discrete theorythese combinations will not be exactly preserved. Therefore even if we foundexact conserved quantities for both perennials in the discrete theory, the extractedphysics would still only be approximate, and the measure of the error will be givenby how well the constraint of the continuum theory is satisfied in the discrete theory.It is in this sense that one can best say that the discrete theory “approximatesthe continuum theory well”.Figure 20.1 depicts the relative errors throughout evolution in the value of thesecond perennial we discussed. Interestingly, although in intermediate steps of theevolution the error grows, it decreases later.