Uniform discretizations: the continuum limit of consistent discretizations

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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
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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,HqC. 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.
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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.
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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 n1,
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
Hlte, suchstatement would be preserved upon
evolution.
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So if we choose initial data such that Hlte then
the constraints remain boundedthroughout the
evolution and will tend to zero in the limit
e-gt0.
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 H0d2/2 and define
The evolution of a dynamical variable is given by
One obtains in the limit,
Constraint surface
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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.
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Quantization
This guarantees we will recover the classical
evolution up to factor orderings,providing a
desirable correspondence principle.
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At a classical level, since H is the sum of
squares of the constraints, one has thatthe
constraints are satisfied iff H0. Quantum
mechanically we can thereforeimpose the
necessary condition Uyy 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,
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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).
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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 21 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.
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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