Carlip S. Quantum Gravity in 2+1 Dimensions
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106 6 The
connection representation
analogous to (6.15). The action of the mapping class group on the rj can
be shown to be
T
:
(^,^)-,(^
+ ^^±). (6.20)
In contrast to the case of a vanishing cosmological constant, the modular
group now mixes positions and momenta. The generators of modular
transformations can be represented as operators
=
exp
{
(ft)
2
+
(ft)
2
~
(K)
2
~
(^]
}
(6.21)
and the requirement of invariance becomes a set of differential equations.
Using the properties of SL(2,R) representations, it has recently been
shown that some invariant wave functions exist, but the known solutions
are nonnormalizable, and the general characteristics of invariant wave
functions are not yet understood [70].
Another reaction to the problem of imposing modular invariance is to
simply drop the requirement. This might seem too drastic a step: the
modular transformations are, after all, diffeomorphisms, and we could
reasonably insist that any theory of quantum gravity behave well under
diffeomorphisms. But we shall see below that even without an explicit
demand of modular invariance, the relationship between the connection
representation and ADM quantization can teach us a good deal about
the modular group.
6.3
Relating quantizations
Quite apart from the technical difficulties with the modular group, the
quantum theory constructed above exhibits a fundamental conceptual
problem: it has no visible dynamics. We have built an archetypical
'frozen time' theory, in which the observables are constants of motion and
the evolution is completely hidden. This feature is not peculiar to 2+1
dimensions, but is a generic problem in quantum gravity. The natural
candidate for the Hamiltonian in quantum gravity is the Hamiltonian
constraint Jf of chapter 2, but at least for a closed universe, we have seen
that Jf is zero on physical states. We can attempt to restore dynamics by
introducing an explicit time-slicing, as in chapter 5, but as we have seen,
such a procedure violates at least the spirit of general covariance.
This 'timeless' nature of quantum gravity already exists in the classi-
cal formalism: the description of (2+l)-dimensional gravity in terms of
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6.3 Relating quantizations 107
geometric structures exhibits the same lack of explicit dynamics. In the
classical theory, however, this problem causes no great concern. Given a
set of holonomies, we can determine a classical spacetime, and given that
spacetime, we can answer any dynamical question we care to ask: for
example, how do the moduli of a slice of constant TrK evolve with time?
One of the principle achievements of (2+l)-dimensional quantum gravity
has been to show that this procedure can be carried over to a quantum
theory.
As usual, we shall concentrate on the topology [0,1] x T
2
. Suppose first
that the cosmological constant A vanishes. In equation (4.31), we con-
structed a function T(T;X,fi,a,b) whose physical meaning was 'the value
of the modulus on the slice of constant mean curvature TrK =
—
T in a
universe with a geometric structure characterized by the holonomies A, fi,
a, and b\ along with similar functions
p(T;X,
//,a,b) and H
re
d(T;A,fi,a,b).
We can now turn these functions into operators acting on the Hilbert
space
Jtfconn
of the connection representation [52, 68, 69]. If we choose the
orderings
p(T) = -iT (a -
H
red
(T) = ^, (6.22)
then the modular transformations (6.16) induce the standard modular
transformations of T and p. Moreover, it may be checked that the
commutation relations (6.13) in the connection representation induce the
ADM commutation relations (5.3), and that the operators
T
and p satisfy
the 'Heisenberg equations of motion'
ijf =
[*,
Hred],
^ =
\P,
Hredl (6-23)
The ADM moduli and momenta may thus be viewed as 'time'-dependent
operators in the connection representation, obeying standard quantum
mechanical equations of motion. Given a state tp(A
?)
u), we can thus
determine, for example, the probability that the slice with a given value
of TrK has a certain modulus.
Operators such as T(T) are not exactly functions of time: after all, we
have completely solved the constraints, and there is no time coordinate left
in the connection representation. Indeed, for each value of the parameter
T, T(T) and p(T) are constants of motion. These operators can best
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108 6 The
connection
representation
be understood as examples of Rovelli's 'evolving constants of motion',
operators that are time-independent but nevertheless provide information
about dynamics [230, 231]. We shall return to the question of how to
interpret such operators at the end of this chapter.
We can learn more about these operators - and in the process, gain
some insight into the problem of modular invariance - by diagonalizing
T(T)
and its adjoint to obtain 'Schrodinger picture' wave functions [53].
A long but straightforward computation shows that the function
(6.24)
is a simultaneous eigenfunction of the operators x and T^ of equation
(6.22),
with
xK(T;x,x;X,n)
= xK{T;x,x;X,n),
^,x;X,n)
= xK{T;x,x\X,n).
(6.25)
The n and
X
dependence of K is determined by the eigenvalue equations,
while the prefactor is fixed by the normalization
(6.26)
The factor of T2
2
on the right hand side of this expression converts the
ordinary delta function into a delta function with respect to the inner
product (5.34). Note that the integration in (6.26) is over the entire l-\i
plane, not over some fundamental region for the modular group.
The function
K(T;T,T;
A,/i)
can be viewed as the transformation matrix
between the connection representation and the ADM representation of
the preceding chapter. It has the following important properties:
1.
Under the modular transformations (3.10) of
T,
K transforms as
1 1
S :
\y
T
:
K(T;T,T;l,ti) -* K(T;T + IT + Ul,fi)
,T;A,/I-A).
(6.27)
Comparing equation (5.29), we can identify these transformations as
those of an automorphic form of weight —1/2, combined with the
inverse of the transformations (6.16) of
X
and \i.
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6.3
Relating quantizations
109
2.
The connection representation Hamiltonian (6.22) acts on
K as
^,t;X,n).
(6.28)
Moreover, the weight —1/2 ADM Hamiltonian of chapter 5 acts
as
A_
l/2
K(T;x,x;A,
t
i)
=
(iT^=)
K(T;T,T;X,H),
V
dTj
(6.29)
which is the square of the Schrodinger equation (5.37).
We can now write an arbitrary connection representation wave function
A as
a
superposition
d
2
r
K(TVAM%T)
=
(ip
9
K)
9
(6.30)
where property 1 suggests that we should require
xp
to be an automorphic
form
of
weight —1/2. The coefficient
xp,
which we shall interpret
as a
'Schrodinger picture' wave function, has been written as
a
function
of T,
but the
T
dependence is not arbitrary, since the left-hand side of equation
(6.30) is independent
of
T. Indeed, we can use equation (6.26) to solve for
V>(T,T,
T); differentiating with respect
to T
and using (6.29), we find that
If)
^
(T
'
T>T)
=
A
-
which is the square of the Schrodinger equation (5.37) for an automorphic
form of weight —1/2.
To determine the region
of
integration
& in
(6.30),
let
us consider the
normalization of wave functions:
<VlV>
=
(6.32)
r
r
d
2
r
d
2
r
f
>
dMfi
/
_-^K(T;T,T;A^)K(T;T
/
,T
/
;A,
i
u)v(T,T,T)^(T
/
,T
/
,T)
d
2
r
d
2
r
f
_ C d
2
r
—
^r
T
2
2
^(t-T0^,T,T)^(T
/
,f
/
,T)= / _|tp(
T
,T,r)|
2
,
where
I
have used
the
orthogonality relations (6.26)
for K. But
xp
is
an automorphic form,
so
this last integral will
be
normalized
if & is a
fundamental region for action of the modular group on Teichmiiller space,
such as the 'keyhole region' described
in
appendix A.
We can now return
to
the question
of
how the modular group acts
in
the connection representation. A wave function defined by equation (6.30)
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110 6 The connection representation
is not modular invariant. On the contrary, if g is an arbitrary element of
the modular group, tp(/l,/x) and ip(ghgfi) are orthogonal. For example,
(xp\Sxp)
=
r r d
2
r dV -
/T'X
1
/
2
,T',
T\ (6.33)
d
2
x
d
2
x
r
- Sx') -
\T'J
where I have used the transformation properties (6.27) of K. But S maps
x
1
from IF to an adjacent fundamental region, and except at isolated
points, fundamental regions for the mapping class group are disjoint, so
- Sx') = 0, (6.34)
and the inner product (6.33) thus vanishes. The same argument holds if
S is replaced by T, or by any nontrivial combination of S and T, that is,
by any element of the modular group.
A similar argument shows that if y(A,/i) and x(/l,ju) are any two wave
functions defined by (6.30), with integrals over the same fundamental
region J% then
(vl?z)=0 (6.35)
for any element g of the modular group. Viewed as an integral transform,
the mapping (6.30) thus takes the weight —1/2 ADM Hilbert space #'~
1
/
2
of chapter
5
into an infinite number of orthogonal subspaces of
the
Hilbert
space
J^conn,
which are interchanged by elements of the modular group.
A version of modular invariance for the connection representation would
thus be the restriction of
J^
con
n
to one of these subspaces.
As an automorphic form of weight —1/2, ip(x\x
f
, T) is essentially a
spinor on moduli space. It was shown in reference [53] that the second-
order Klein-Gordon-like equation (6.31) may be converted to a first-order
'Dirac equation' for a two-component spinor
| ) (6-36)
on moduli space, where the lower component is essentially a time-reversed
ADM wave function. The appearance of such time-reversed solutions
should not be surprising. Classically, a solution of the Einstein equations
with the topology [0,1] x T
2
can describe either an expanding or a
collapsing universe. As we saw in the preceding chapter, the Schrodinger
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6.3 Relating
quantizations
111
equation in the ADM formalism requires
a
choice
of
sign in the definition
of the square root A
n
,
and that choice amounts
to a
restriction
to
either
purely expansion
or
pure collapse.
In
the connection representation,
the
same ambiguity has reappeared
in a
slightly different guise.
Are weight —1/2 automorphic forms really necessary
in
the connection
representation?
The
first indication
of
their importance came from
the
transformation properties (6.27)
of
the eigenfunctions
of
T
and
T^.
But
the definition (6.22)
of
these moduli operators involves
an
implicit choice
of operator ordering, which
is by no
means unique.
It
may
be
shown
that different, though perhaps less natural, choices
of
ordering can lead
to automorphic forms
of
arbitrary weight [55]. Equivalently,
at
least
in
this simple context, the choice of inner product in the Hilbert space
J^
C
onn
is ambiguous,
and
different choices lead
to
different adjoints
ft". For
example,
if
one chooses the inner product
{y>\x)
=
J
dMi4P
l/2
V>T(P
l/2
X),
(6-37)
where
p is
the momentum operator
of
equation (6.22),
it
may
be
shown
that the corresponding ADM wave functions are weight zero forms, that
is,
ordinary automorphic functions.
By
finding
a
series
of
canonical
transformations that link
the
ADM representation
and the
connection
representation, Anderson
has
found
a
similar, although
not
identical,
measure [7]. The connection representation has not resolved the ambiguity
described
in the
preceding chapter,
but
has merely transformed
it to an
ambiguity
in
the choice
of
operator ordering.
So
far, we
have interpreted
the
function
K(T;T,T;A,ii)
as a
trans-
formation matrix between
the
connection representation
and the
ADM
representation, as manifested by the transformation (6.30). But this func-
tion
has
another interpretation
as
well:
for
fixed values
of
T,
T,
and T,
it may
be
viewed
as the
state
|T,T,
T) of a
torus universe with modulus
T
at
time
T,
expressed
in the
connection representation. This dual role
is familiar from ordinary quantum mechanics, where
the
function
e
ipx
is
both the transformation matrix between the position and the momentum
representations and the position representation wave function for the state
\p).
As
in
ordinary quantum mechanics, the matrix element
(T',T',T'|T,T,:T)
= \
J
(6.38)
thus gives an amplitude for
a
spatial geometry determined by the modulus
T
at
time
T
to evolve
to a
geometry determined by the modulus
x
1
at
time
V.
This amplitude has been worked out explicitly by Ezawa, who shows
that
it is
peaked
at
precisely
the
classical trajectory (3.70)
[107].
The
quantum theory thus approximates the classical behavior, as
it
should.
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112 6 The connection representation
A similar set of constructions is possible when A < 0. Starting with
equation (4.38), we can define operators
-l
KdCt) = j sin | (fftf - Pfo-). (6.39)
Again, it may be checked that the modular transformations (6.16) induce
the standard transformations oft and
p,
that the connection representation
commutators (6.13) induce the correct ADM commutators, and that
T
and
p satisfy the appropriate Heisenberg equations of motion. If we take our
states in the connection representation to be functions of rj and r^~, the
transformation matrix analogous to (6.24) is then
K(7;t,t;r
2
+
,r
2
-) = g (sin
j^'^
{$*"
+
r^e"*") x (6.40)
exp
f
{t! [(r+)
2
-
(r
2
-)
2
]
-1
2
[(r
2
+)
2
+
2r
2
+
r
2
-
sec
|
+
(r
2
~)
2
]
cot |} .
A rather long computation now shows that the transformation properties
(6.27) again hold, with S and T now represented by the operators of
equation (6.21). Moreover, the analog of (6.29) continues to be valid:
=
(i-sin—-=)
l € dtj
(641)
which by (4.40) is again the square of the Schrodinger equation of the
preceding chapter [70].
6.4 Ashtekar variables
The connection representation of the preceding sections was based on a
particular set of classical solutions, those with nondegenerate metrics. For
the torus, the classical phase space was even more restricted: we excluded
those geometries on [0,1] x T
2
, discussed briefly in chapter 3, for which
the T
2
slices were not spacelike. By loosening these restrictions, we can
obtain new quantum theories that are not quite the same as those we have
seen so far. In particular, the Ashtekar approach to (2+l)-dimensional
quantum gravity relaxes the requirement of nondegeneracy of the metric,
thus enlarging the classical phase space and the quantum Hilbert space.
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6.4 Ashtekar variables 113
In the connection representation developed in the last two sections, the
phase space T* Jf was a space of flat metrics and spin connections. This
flatness originates in the constraints of chapter 2,
=
\&
\d
iej
a
-
djef
+
e
abc
(co
ib
e
jc
-
co
jb
e
ic
)]
= 0,
2
(6
(6.42)
,*]
= ^Fij = y>
[dicoj
a
-
djCOi
a
+
e
abc
co
ib
co
jc
]
= 0, (6.43)
where D\ denotes the
SO (2,1)
gauge-covariant derivative. The Ashtekar
constraints, on the other hand, take a slightly different form. The phase
space variables are again a spin connection and a triad, or more commonly
a spin connection and a densitized triad,*
E
ia
= e
ij
ej
a
, (6.44)
which clearly carries exactly the same information as e. The constraints
are now written in a form identical to that of the Ashtekar constraints in
3+1 dimensions [11, 35],
D
t
E
ia
= 0,
(6.45)
E'aFfj
= 0,
(6.46)
e
abc
E
ib
EJ
c
F?j
= 0. (6.47)
The constraint (6.45) is clearly equivalent to (6.42). When the spatial
metric gtj =
e\
a
ej
a
is nondegenerate, it is not hard to show that the con-
straints (6.46)-(6.47) are equivalent to (6.43). Indeed, using the definition
of E
ia
and the fact that Ff
}
=
e\f£
a
>
we can rewrite these constraints as
eutf"
= 0
€abce
kl
e
k
b
ei
c
^
a
= 0
(6.48)
and the equivalence follows from some elementary algebra.
The Ashtekar constraints (6.45)-(6.47) admit additional solutions, how-
ever, for which the spatial metric is degenerate and the curvature Ffj does
not vanish. The full space of solutions is not completely understood, but
portions have been investigated. Barbero and Varadarajan have argued
that the full phase space has an infinite number of degrees of freedom
[25].
Let us define a 'flat patch' of a surface Z as a region in which the
constraints (6.42)-(6.43) are satisfied, and a 'null patch' as a region in
which the metric is degenerate, the constraints (6.45)-(6.47) are satisfied,
*
It is
traditional
in
papers using Ashtekar variables
to use a
tilde over
a
letter
to
denote
a density
of
weight
one.
Cambridge Books Online © Cambridge University Press, 2009
114 6 The
connection representation
and curvature satisfies
<€
a
<€
a
= 0. Then if initial data are chosen so that Z
consists of a collection of disjoint flat patches separated by null patches,
this pattern is preserved by evolution, and the holonomies in each of the
flat patches become independent observables. The consequences for the
quantum theory are not yet understood, although some work has been
done in the loop representation and in minisuperspace models [26, 109].
At a minimum, the (2+l)-dimensional model serves as a warning that
the choice of how to treat degenerate metrics in quantum gravity can
drastically affect the outcome.
6.5 More pros and cons
The quantum theory developed in this chapter has managed to avoid
some of the conceptual difficulties of the reduced phase space ADM
quantization of chapter 5. In particular, we have not had to choose any
particular time slicing, and have thus preserved general covariance. This
manifest covariance comes at a price: the physical interpretation of the
resulting 'frozen time' formalism, and in particular the identification of
dynamical behavior, becomes much less obvious. But we have seen in at
least one example that the problems of interpretation can be overcome,
and that dynamical questions can be answered through the construction
of appropriate operators.
Operators of the sort we have studied here are not quite 'dynamical'.
The modulus
T(T),
for example, should not be thought of as a single,
time-dependent operator. Rather, for each value of the parameter T,
T(T)
is defined throughout spacetime, and commutes with the constraints - it
is a constant of motion, or in Kuchaf's terms, a 'perennial'
[172].
We
must interpret
T(T)
as a one-parameter family of operators, indexed by a
number T, that are defined everywhere.
Nevertheless, we can use the correspondence principle and our knowl-
edge of the classical theory to give
T(T)
its dynamical interpretation as
the operator that represents the modulus of the slice TrK = — T. The
rather mysterious role of time may become clearer when we compare the
quantum theory to the classical theory of chapter 4. There, too, the space-
time geometry was described completely in terms of constants of motion,
and no explicit time coordinate was ever introduced. Cosgrove has called
the resulting quantum theory 'quantization after evolution': the dynamics
is completely solved classically and reexpressed in terms of correlations
among time-independent constants of motion, which can then be studied
quantum mechanically [81]. This point of view is closely related to Rov-
elli's description of quantum mechanics in terms of 'evolving constants of
motion' [230, 231].
Cambridge Books Online © Cambridge University Press, 2009
6.5 More pros and cons 115
We can now return to a question raised in the preceding chapter:
are the quantum theories corresponding to different choices of time-
slicing equivalent? In this chapter, we considered only a particular set
of 'dynamical' operators, those corresponding to the York time-slicing
TrK = — T. But nothing prevents us from looking at a different choice
of slicing. Given any classical choice of time, with slices labeled by some
parameter t, we can construct the corresponding classical observables x(t)
and p(t) and convert them into operators in the connection representation.
Just as we did for the York slicing, we can then diagonalize x(t) and its
adjoint and transform to a 'Schrodinger picture'.
By construction, though, this 'Schrodinger picture' is equivalent to
the original 'Heisenberg picture' of the connection representation. In
particular, a transition amplitude between slices t\ and tf depends only
on the initial operator
T(U)
and its adjoint, the final operator x(tf) and its
adjoint, and a time-independent wave function
\p(X,ix).
Provided that the
operator orderings in
T(£,-), %tf\ and their adjoints are fixed, the choice of
intermediate slicing cannot affect the amplitude.
This last proviso, however, emphasizes a remaining arbitrariness in the
connection representation. Dynamical quantities like
T(T)
are classically
well-determined, but the process of constructing operators is fraught with
ordering ambiguities. In particular, the 'natural' ordering with respect
to one choice of time-slicing may not be the same as the 'natural' or-
dering with respect to another. This problem reflects a genuine physical
uncertainty: when we measure a classical gravitational field, we do not
know exactly which quantum operators we are observing. This problem
is already present in ordinary quantum mechanics, although it is usually
ignored because we 'know' the right variables. But we know only because
of the results of experiments, and these are sorely lacking in quantum
gravity.
A similar ambiguity arises in the choice of polarization, that is, the
splitting of the phase space into 'positions' and 'momenta'. We saw briefly
in section 2 that this choice can, at a minimum, affect the representation of
the modular group. There is no proof that different choices of polarization
lead to equivalent quantum theories, and there is no obvious way to choose
a 'correct' polarization, although the mapping class group might offer some
guidance.
There is another difficulty with the connection representation, which is in
some sense 'merely technical' but which has the potential to cause serious
problems. To construct the dynamical operators needed for a physical
interpretation, we must know the classical solutions of the equations of
motion; more specifically, we need a parametrization of the classical
solutions in terms of a set of constants of motion. Equation (6.22) for
the operator T, for example, was obtained from the classical equation
Cambridge Books Online © Cambridge University Press, 2009