Transcript Transparencies

Totally constrained systems
with discrete evolution parameter
Rodolfo Gambini
In collaboration with Jorge Pullin
During the last years we have developed a canonical procedure for the
treatment of discrete constrained systems. Even if one considers that the
final theory should be continuous, which is not obvious, discretizations
play a crucial role at the intermediate steps. In fact they are required for
numerical computations and provide a regularization for quantum field
theory and probably for quantum gravity.
Discretizations present considerable difficulties:
In Numerical Relativity the constraints are not preserved by the
discrete evolution and the description is inconsistent.
At the quantum level, Regge calculus and the spin foam formalism
lead to problems related with the integration measure and the
recovery of the continuum limit.
Finally in Loop Quantum Gravity some of the ambiguities of the dynamics
may be traced back to the discrete theory.
When this approach, usually called consistent discretizations, is applied
to G.R., one does not fix a gauge but nevertheless the theory is
constraint free. This allows to solve many of the hard conceptual problems
of general relativity. At the classical level it preserves the constraints with
a high degree of approximation and at the quantum level the theory has the
same kinematics than loop quantum gravity.
Discrete Canonical formalism.
Relations that define a type 1 canonical transformation
Canonical formulation for constrained discrete
dynamical systems
The final evolution equations are obtained by substituting
the Lagrange multipliers.
Notice that here the Lagrange multipliers were determined
without imposing any gauge fixing. Notice that more
precisely what has been determined is
For a completely parameterized theory there is no explicit
dependence on epsilon, which may be fixed arbitrarily. Once
the time interval (or the lattice spacing) is chosen, the
lapse is determined.
When
 (qn , P )  0
B
q
n
nB  0
And one recovers the continuum limit.
“Dirac’s” canonical approach to general discrete systems.
=0
Primary constraints
V and U arbitrary functions
Consistency:
By using a Type II, III, or IV transformation one can show
that this evolution is canonical, preserves the Poisson brackets
and the constraint surface. This is equivalent to what happens
in the continuum, where consistency may be achieved by
determining the complete constraint surface and a total
Hamiltonian that preserves the Poisson structure
Finally one can recognize the second class constraints and
impose them strongly. While some symmetries of the continuum
are broken by the discretization others are preserved.
The procedure has been extended to quantum field
theories and reproduces, for usual gauge theories
as Yang-Mills, the standard results obtained by using
transfer matrix techniques on a lattice.
Class.Quant.Grav.19:5275-5296,2002
C.Di Bartolo, R. Gambini and J.Pullin
The procedure also provides a very simple description
of B-F theories on a lattice.
General relativity may be discretized in
several ways. One can start from the standard ADM
formalism or use Ashtekar’s variables. I will discuss later
on the use of Regge Calculus.
.
In the case of general relativity the gauge invariance
and the invariance under spatial diffeomorphisms may be
exactly preserved and the final quantum theory may be
in principle treated in terms of loops making use of the
Ashtekar Lewandowski measure.
At the classical level, the procedure is computationally
intensive in a generic situation, requiring the simultaneous
solution of ten coupled non linear equations at each
lattice point.
We have recently completed the classical study of the
Gowdy models and shown that the constraints are preserved
with a good degree of precision during evolution.
R.G and J. Pullin Phys. Rev. Lett. 90 021301 2003
R.G. M. Ponce and J. Pullin Phys Rev. D 72 024031 2005
4 Issues
1) Existence of global descriptions: Is it possible to cover the
complete classical orbits?
2) Canonical description for geometric actions of G.R. like
the Regge action within the consistent discretization approach.
3) The Issue of time and a description of the evolution in
terms of conditional probabilities.
4) Unitary implementation of classical canonical transformations.
An example of a totally constrained system without a global time
variable.
q1
q2
n
q1
An example of application: The Regge Action.
The continuum Regge action in D=2 and D=3.
S ( g )   d x g (   .R)
D
In 2 dimensions:
(n,m)
d
2
x g R 4
  2(1  h)
In two dimensions the classical action is not
determined by the initial and final one
dimensional complex that does not have any
information about the intrinsic curvature, and it
needs to be modified.
One usually considers the discrete version of
the Polyakov string which has additional
information about the embedding of the lattice.
In this case, besides the Regge length
variables li [n, m]
we need the coordinates
of the points in the target space X  [n, m]
For reasons of simplicity I discuss here the 2 dimensional case. The analysis
of the 3D system follows similar lines.
P n l 2  f (l1n , l 2 n , l 3n , l1n1 )  0
P
n
l3
 g (l1n , l 2 n , l 3n , l1n1 )  0
P
n
l1
 h(l1n , l 2n , l 3n )
l1 and l2 play the role of Lagrange multipliers and are fixed in terms of the
n
canonical pair l1n , P l1
by the previous eqs.
They also determine the evolution equation:
l1n1  F (l1n , l 2n , l 3n )
And finally from the definition of the canonical momentum at level n+1 we get
The remaining evolution equation.
P n1l1  G(l1n , l 2n , l 3n , P n l1 )
Each prism is decomposed in three
tetrahedra:
(1) In ABA’D, DEE’D, DEAA’
(2) In ABB’D, AB’D’D, A’FDD’
The quantization of the system
determines the Euclidean or unitary
evolution operator.
D  2  U (l1n , l1n1 )
D  3  U (l1n , l 2n , l 3n , l1n1 , l 2n1 , l 3n1 )
The consistent discretization scheme uniquely determines the integration
measure in the Path Integral. This measure differs from the standard
measures usually considered in Regge calculus:

 n dl1n dl 2n dl 3n  (l1n , l 2n , l 3n ) exp i  LR (l1n , l 2n , l 3n , l1n1 )
n
While here:

 n dl1n  (l1n ) exp i  L'R (l1n , l1n1 )
n
We haven’t checked yet if the resulting transition amplitudes present the
same pathologies than the standard Regge amplitudes.
A solution to the problem of time.
Standard quantum mechanics presupposes the existence of an externally
defined classical variable called t. The other variables, x play a very different
role and are represented by operators. This is clearly an approximation
that requires the existence of a classical clock external to the system,
and will not be very useful in the context of closed systems where
everything behaves quantum mechanically, as the Universe
close to the Big Bang.
Page and Wooters proposed to treat all variables quantum mechanically and
use one of them as a clock as long as it behaves semi-classically.
However, in standard canonical quantum gravity and other totally constrained
systems the idea runs into problems. The clock variable must change during
evolution, and therefore it cannot commute with the constraints. This implies
that it will not be well defined on the physical space of states annihilated by
the constraints. If one tries to work in the kinematical space, the wave
functions corresponding to physical states are distributional and cannot be
used to construct a probabilistic interpretation.
Thus, up to present, the description of the time evolution in totally
constrained systems has always involved a classical clock variable.
Either by fixing a gauge or by introducing Rovelli’s evolving constants.
Relational Time.
The elimination of the constraints associated with the reparameterizaton
Invariance simplify all the quantization process and allows to treat in a
simpler way old conceptual problems as the issue of time.
Notice that the evolution variable n does not have any intrinsic meaning
and it is not associated with any dynamical variable. We shall introduce
time via conditional probabilities. In many simple models with discrete
evolution, the physical and the kinematical spaces coincide.
For instance in a cosmological model with two degrees of freedom A, Φ:
Loss of coherence.
It will be convenient to introduce conditional probabilities for a state given
in terms of a density operator.
Let us now assume that the clock and the system are weakly interacting
in such a way that:
and assume the existence of a semi-classical regime for the variable
chosen as time in a given initial state of the clock.
That means that n and time are strongly correlated.
is the probability that the measurement t
corresponds to the value n.
Now, one can define a time dependent density operator
Such that the conditional probability takes the usual form
We have therefore ended with the standard probability expression with
an effective density matrix in the Schroedinger picture. It is evident from
its definition that exact unitarity is lost, since we end up with a
statistical mixture of states associated with different n’s.
This fact leads to a modification of the Schroedinger equation for any
quantum system whose evolution is described with real clocks.
I will not analyze here the physical consequences of this result.
R.G., R. Porto, J.Pullin: NJP6,45(2004)
Quantum unitary implementations of the canonical transformations.
The relational procedure that we have introduced here allowed us to describe
the evolution in terms of conditional probabilities defined in terms of a
quantum clock variable.
However, as it is well known, and was extensively discussed by
Arlen Anderson many canonical transformations do not lead to unitary
transformations or isometric transformations at the quantum level.
This is the case when one trays to recover the standard quantum
mechanics starting from a totally constrained system with a quantum
relational time.
We shall see that for this kind of systems the implementation of the
canonical transformations at the quantum level by unitary transformations
requires restricting the kinematical Hilbert space.
p2
2
S   { p0 q  p q     N ( p0 
 q 
)}
2M
2m
0


Substituting the Lagrange multiplier N in the evolution equations
one gets the canonical transformations connecting level n and n+1
The step of the evolution is governed by C a quantity that vanishes in
the continuum limit.
p

H0 

 V (q)  U ( )
2 M 2m
2
2
V
U
 M
q

N [
]  [
V 2
U 2
m( )  M ( )
q

mp
]2 
2 Mm( H 0  p0 )
V 2
U 2
m( )  M ( )
q



The choice of the sign
n  (3  2 M n
σ is made to insure that
the sign of Δ is preserved by the evolution.
At the turning point of the macroscopic particle one needs to
change the branch of the lapse.
O1  P
q0
O3  P
O2  P  q
q
 2 pn )
0
P 0
O4    q
m

The 4 independent perennials of the continuum theory are constants
of the motion of the discrete counterpart.
q0
q0
qq
q
Quantization: How to find a unitary implementation of the evolution.
We start by choosing a polarization. In order to simplify the determination
of the evolution operator we choose to work in a kinematical space H  with
square integrable wave-functions
( p , , u)
n
such that:
0

^ q0
P  n ( p0 ,  , u )  p0 n ( p0 ,  , u )

P  n ( p0 ,  , u )   n ( p0 ,  , u )



u n ( p0 ,  , u )  (3  2M  n  2 p n ) n ( p0 ,  , u )  u n ( p0 ,  , u )
The classical evolution given by canonical transformations needs
to be implemented at the quantum level by unitary operators.
 n1 ( p0 ,  , u )   dp'0 d ' du 'U ( p0 ,  , u | p'0 ,  ' , u ' ) n ( p'0 ,  ' , u ' )
And as usual U can be interpreted as the transition amplitudes between
 q0   
eigenstates of the operators
at levels n and n+1.
P , P ,u
in the Heisenberg picture
U ( p0 ,  , u | p'0 ,  ' , u ' )  n  1, p0 ,  , u | p'0 ,  ' , u ' , n 
To determine U one needs to impose the evolution equations. For instance:

0
 n  1, p0 ,  , u | q

0
n1


q
 q n  ( (u )  2M  n  P n ) /  | p'0 ,  ' , u ' , n 
And similarly for the other variables. Solving these equations one gets U.
However, U is unitary only in the space H ph expanded by the negative
eigenvalues of
2
 

  ( p0 
  2i ) ph ( p0 ,  .u )   ph
( p0 ,  .u )
2m
u
Notice that here H ph is a subspace of the kinematical space!


ph
H ph  H 
With
Δ<0
By construction the evolution
preserves H ph As usual only the

observables such that A H ph  H ph are well defined in the physical space.
O  P  q
q
0
But now, not only the constants of the motion
2


q0
O4   P / m 1
3
are observable, the operator associated to the position of the macroscopic

O P

O P
 
variable
is also observable because [ q,  ]  .0 This fact is basic
for a relational description of the evolution.
q
Relational time and continuum behavior.
One can recover the standard quantum mechanical description
using q as the clock variable and working in the continuum regime where;
q 
0
(O2    2M )

O2    2M P
  O4  (
)

m
The quantum continuum regime is reached by considering states
such that:
1) The mean value of the operator that measure the step of the evolution
is small.
 C  /   1
2) The evolution preserves the continuum regime.
3) The measurement of any admissible dynamical variable preserves
this regime.
We have shown that these conditions are compatible and one can
define a time dependent density operator such that:
P ( | q 0 ) 
  

Tr[  (q 0 ) |    |]
Tr[  (q 0 )]
Where ρ satisfies the standard Schroedinger equation plus corrections.
CONCLUSIONS
In the last year we have made several advances within the consistent
discretizations approach.
We now know how to approach the complete continuum orbit y a totally
constrained system.
We have shown that the Regge action may be treated within this
approach leading to a non trivial constraint structure. This seems to be
the most natural way of treating general relativity in a discretized scheme.
We have made progress in the treatment of the issue of time in terms of
conditional probabilities and discussed several applications.
We have learn to recover continuum the quantum mechanics behavior
from a discrete formulation.
The main open problem is the issue of the continuum limit in the case
of totally constrain systems with infinite degrees of freedom.