Uniform discretizations: the continuum limit of consistent discretizations

1. Uniform discretizations: the continuum limit of consistent discretizations Jorge Pullin Horace Hearne Institute for Theoretical Physics Louisiana State University With Rodolfo Gambini Miguel Campiglia, Cayetano Di Bartolo

2. General context: As is by now well known, the kinematical Hilbert space in loop quantum gravityis well under control, and to a certain extent it is unique. In this Hilbert space, diffeomorphisms are well defined but are not weakly continuous, that is, the infinitesimal generators of diffeomorphisms cannot berepresented. There exist proposals for the Hamiltonian constraint, but they act on the spaceof diffeomorphism invariant states. There appears to be a general conviction that one cannot define a satisfactoryHamiltonian constraint in a space where one could check the “off shell” constraint algebra [H,H]~qC. This has led several of us to seek alternatives to the Dirac quantization procedureto apply in the case of gravity. An example of this point of view is the “Phoenixproject” of Thomas and collaborators. Our point of view is to attempt to define the continuum theory as a suitable limitof lattice theories that do not have the problem of the constraint algebra but that nevertheless provide a correspondence principle with the continuum theory.

3. Most people here have heard me talk about “consistent discretizations”. This isa technique for discretizing constrained theories, in particular general relativity. One starts from the classical action of the continuum theory and discretizes theunderlying manifold. One then works out the equations of motion for the resultingdiscrete action. Three things happen generically: a) The resulting equations of motion are consistent, they can all be solved simultaneously. b) Quantities that used to be Lagrange multipliers in the continuum becomedynamical variables and are determined by the evolution equations. c) The resulting theory has no constraints, what used to be constraints in the continuum theory become evolution equations. The last point is very attractive from the point of view of quantizing the theories. But… Point b) proved unsettling to a lot of people, since it implied there was not a clearway of taking the continuum limit.

4. Today I would like to present a class of consistent discretizations that have theproperty that the continuum limit is well defined. We call them “uniform discretizations” and they are defined by the following canonical transformationbetween instants n and n+1, Where A is any dynamical variable and H is a “Hamiltonian”. It is constructedas a function of the constraints of the continuum theory. An example could be, (More generally, any positive definite function of the constraints that vanisheswhen the constraints vanish and has non-vanishing second derivatives at theorigin would do) Notice also that parallels arise with the “master constraint programme”. These discretizations have desirable properties. For instance H is automaticallya constant of the motion. So if we choose initial data such that H<e, suchstatement would be preserved upon evolution.

5. Graphically, Initial data So if we choose initial data such that H<e then the constraints remain boundedthroughout the evolution and will tend to zero in the limit e->0. We can also show that in such limit the equations of motion derived from Hreproduce those of the total Hamiltonian of the continuum theory. For thiswe take H0=d2/2 and define The evolution of a dynamical variable is given by One obtains in the limit, Constraint surface

6. The constants of motion of the discrete theory become in the continuum limitthe observables (“perennials”) of the continuum theory. Conversely, every perennial of the continuum theory has as a counterpart a set of constants ofthe motion of the discrete theory that coincide with it as a function of phase spacein the continuum limit. We therefore see that in the continuum limit we recover entirely the classicaltheory: its equations of motion, its constraints and its observables (perennials). An important caveat is that the proof of the previous page assumed the constraintsare first class. If they are second class the same proof goes through but one hasto use Dirac brackets. This is important for the case of field theories where discretization of space may turn first class constraints into second class ones. In this case one has two options: either one works with Dirac brackets, whichmay be challenging, or one works with ordinary Poisson brackets but takes thespatial continuum limit first. It may occur in that case that the constraints becomefirst class. Then the method is applicable and leads to a quantization in whichone has to take the spatial continuum limit first in order to define the physical space of states.

7. To quantize the discrete theory one starts by writing the classical evolution equations One then defines a kinematical space of states Hk as the space of functionsof N real variables y(q) that are square integrable. We define operators Quantization: This guarantees we will recover the classical evolution up to factor orderings,providing a desirable “correspondence principle”.

8. At a classical level, since H is the sum of squares of the constraints, one has thatthe constraints are satisfied iff H=0. Quantum mechanically we can thereforeimpose the necessary condition Uy=y in order to define the physical space ofstate Hphys. More precisely, states y in Hphys are functions in the dual of a subspace of sufficiently regular functions () of Hkin such that This condition defines the physical space of states without having to implementthe constraints of the continuum theory as quantum operators. We see similaritieswith the “master constraint”. The operators U allow to define the “projectors” onto the physical space of states of the continuum theory by,

9. To conclude, let us consider a simple, yet rather general example. Consider a totally constrained mechanical system with one constraint f(q,p)=0. We would like to show that the projector we construct with our technique coincideswith that of the group averaging procedure, that is, (The definition of the projector given assumes continuum spectrum, a slightly different definition can be introduced for cases with discrete spectrum).

10. We have analyzed several examples up to now: The example of the previous slide can be easily extended to the case of N Abelian constraints and in particular immediately can be applied to the formulation of 2+1 gravity of Noui and Perez. We also studied the case of a finite number of non Abelian constraints (forinstance the case of imposing the generators of SU(2) as constraints). In thiscase we proved that the method reproduces the results of the standard Diracquantization and the group averaging approach. In the case of a non-compact group SO(2,1), the discrete theories exist andcontains very good approximations of the classical behavior but the continuum limit does not seem to exist. This parallels technical problems associated with the spectrum of H not containing zero that appear in the master constraint andother approaches as well. The last example suggests a point of view: the continuum limit is a desirableconsistency check, but one could work with the discrete theories close to thecontinuum limit, which in particular automatically solves the problem of time sincethe theories are unconstrained and one can work out a relational description withvariables that are not constants of the motion.

11. Summary • The uniform discretizations allow to controlthe continuum limit classically and quantum mechanically. • They allow to define the physical space ofa continuum theory without defining theconstraints as operators. • There are interesting parallels with the “master constraint” programme but also important differences. • Our next task is to subject the technique tothe same battery of tests that Thomas and collaborators developed for their program,in particular to work out field theories ■