Carlip S. Quantum Gravity in 2+1 Dimensions
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116 6 The connection representation
(4.31),
which in turn depended on our explicit knowledge of the general
solution of
the
vacuum
field
equations. For (3+l)-dimensional gravity, this
is clearly impractical; even for (2+l)-dimensional gravity on a manifold
with a spatial topology of genus g > 1, the general solution for the metric
is not known explicitly in terms of holonomy variables. The obvious
approach to this problem is to try to develop a perturbative formalism for
operators, based on a perturbative approach to the classical theory. So
far, however, such a 'perturbative covariant canonical quantization' has
not been investigated.
Cambridge Books Online © Cambridge University Press, 2009
7
Operator algebras and loops
In its most basic form, quantization is the process of replacing the Poisson
brackets of a classical theory with commutators of operators acting on
some Hilbert space,
{*,j,}->i[a,5>]. (7.i)
Many of the ambiguities in quantization arise because this is not, strictly
speaking, possible: even for the simplest classical theories, factor-ordering
difficulties prevent us from simultaneously making the substitutions (7.1)
for all classical variables x and y
[263].
The usual procedure is therefore
to pick out a set Sf of 'elementary classical variables', which are to have
unambiguous quantum analogs, and to construct the remaining operators
from elements of £f. The set Sf must be small enough to close under
Poisson brackets, and it must be large enough that the remaining classical
variables can be constructed from sums and products of
its
elements, or as
limits of such sums and products. But apart from these basic requirements,
there is no known 'right' way to choose the elementary classical variables.
In chapter 5, we took as our elementary variables the moduli of the
spatial metric at TrK = — T, along with their conjugate momenta. In
chapter 6, we chose a set of coordinates fi
a
on the space of
SO
(2,1)
holonomies, along with their conjugate momenta. But there is another
obvious candidate for 9*\ the
ISO (2,1)
holonomies themselves, or a
suitable subset of these holonomies. As we saw in chapter 4, the Poisson
algebra of the holonomies will lead to commutators that are considerably
more complicated that the simple canonical brackets (5.3) or (6.13), but
we can nevertheless look for an operator representation.
The literature contains two fundamental approaches to the quantization
of
the
algebra of
holonomies.
The first, developed by Nelson and Regge, is
most easily implemented for the case of a negative cosmological constant
117
Cambridge Books Online © Cambridge University Press, 2009
118
7
Operator algebras
and
loops
A =
-I//
2
;
the group of holonomies is then SL(2,R) x SL(2,R) [205, 206,
207,
208, 209]. The second, initiated by Ashtekar and his collaborators,
starts with the case A = 0, and invokes a transformation from the
holonomy representation to a dual 'loop representation' [10, 11, 13].
7.1
The
operator algebra
of
Nelson
and
Regge
The approach to (2+l)-dimensional quantum gravity initiated by Nelson
and Regge is based on the algebra of holonomies described in chapter 4.
The program can be summarized as follows:
1.
Choose a sufficiently small but complete (or overcomplete) set Sf
of holonomies R^ = \Trp[y
a
], whose classical algebra closes under
Poisson brackets (4.73). If the set is overcomplete, find a complete
set of classical relations Fp = 0.
2.
Represent the holonomies as operators R^
G
&, with a commutator
algebra obtained, up to possible higher-order terms in h, by the
substitution (7.1), and with the symmetric ordering
xy-^\(xy
+ yx) (7.2)
for products of holonomies occurring in the commutator algebra.
The set
&?
must remain closed under this algebra.
3.
If the set £f is overcomplete, find operator orderings for the relations
such that
[p}0
(7.3)
for all holonomies in &.
4.
Represent the operator algebra as an algebra acting on some Hilbert
space of states, restricted to the subspace of states annihilated by
the Pp.
5.
Find the action of the mapping class group on this space of states,
and restrict to invariant or suitably covariant states.
As usual, this program is simplest for spacetimes with the topology
[0,1] x T
2
, and the steps 1-5 have been largely completed for the case of
a negative cosmological constant. For our elementary set of holonomies,
we take the set {R^R^R^} described in section 7 of chapter 4, with
the relations F
±
= 0. The operator algebra corresponding to the brackets
Cambridge Books Online © Cambridge University Press, 2009
7.1 The
operator algebra
of Nelson and Regge 119
(4.74) can then be determined by direct substitution of operators for
classical variables; one finds that
R^-R^e-
16
— Rj-RYe^
1
®
=
+2i
sin
6
R^
cxnd cyclical permutations
(7.4)
with
tantf-
—, (O)
while the relators F- become
F
±
= \-\<m
2
e-e
±m
±)
2
+
2e
±w
cos
ek±R±R±.
(7.6)
It is straightforward to check that the F
±
do, in fact, commute with the
R*.
This algebra is not a Lie algebra, but it is related to the Lie algebra
of the quantum group
U
q
(sl(2))
x U
q
(sl{2)\ where q = exp4i'0 and the
F
±
are related to the cyclically invariant g-Casimir. The commutation
relations (7.4) are invariant under the modular transformations
A I A I A I A I A
_J_
A
_J_
A I A I A
_J_
A I
C . pi v
DIE DIE
. Dl
DIE
v
DIE l?X
_i nl nt ox
O . iVj —>
£<2
, i\
2
—> ivj , A
12
—^ ^i ^2 ' ^2 ^1 ~~ ^12
A
_|_
A I A I A I A I A I A I A I A I A I
T
. Dl . D± Dl . DX D± . pi D± _|_ D± D± DIE
• "*M "^
^12'
^2 ^2
?
^12
^12^2
' ^2 ^12 ~ ^1 '
(7.7)
which reduce to the standard transformations in the classical limit.
To obtain a quantum theory, we must now represent this algebra as
an algebra of operators acting on a suitable Hilbert space. To obtain an
explicit representation, let us define
A +
Rf =
sec 6
cosh ^~
&r =
sec 0
cosh
~^-
r
±
+ r
±
R±
2
=
sec 9
cosh
1
^
2
. (7.8)
It is then easily checked that the relations (7.4) hold provided
Pf,^]=±8i0, [r
a
+
,r^]=0, (7.9)
and that the relators F- are identically zero.
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120
7
Operator algebras
and
loops
For small
0,
that is, small cosmological constant,
the
commutators
(7.9)
reduce
to
those found
in the
connection representation
of
chapter
6, up to
terms
of
order
9
3
.
Moreover,
for any
value
of 0,
equation
(7.9) can be in-
terpreted
as a set of
connection representation commutators appropriately
rescaled. Hence
the
results
of
chapter
6 can be
carried over directly
to the
Nelson-Regge quantization.
In
particular,
the
representations discussed
in
chapter
6
automatically give rise
to
representations
of the
operator alge-
bra (7.4).
We
have reduced
the
Nelson-Regge approach
to the
connection
representation,
but
with
a
small adjustment:
the
coupling constant
has
been slightly deformed.
The results
of
chapter
6
also give
us a
physical interpretation
of the
holonomies
R^. For
instance,
we can use the
holonomies
to
construct
operators representing
the ADM
modulus:
as in
equations (6.39)
and
(4.40),
we can
define
H
red
Ct)
= esc
j (f^ - rpi).
(7.10)
Equation
(7.9)
then gives
the
commutators
[x\p]
= [t,f] =
-16tf0,
[x,p]
= [*t,£t] = 0
(7.11)
and
the
Heisenberg equations
of
motion
JA tA
\P,
H
red
]
=
-M9 J,
[T,
H
red
]
=
-8i/0^, (7.12)
which differ from
the
corresponding equations
in ADM
quantization
by
terms
of
order
O(h
3
),
small when
the
cosmological constant
A is
small.
An important advantage
of
this algebraic approach
is
that
it may be
generalizable
to
more complicated spatial topologies.
For a
surface
2 of
genus
g > 1,
consider
the set of
closed curves £,-,
i = 1,2,...,
2g
+
2,
shown
in figure
7.1.
These curves
can be
constructed explicitly
in
terms
of the
standard generators
At and B
t
of n\(L)
described
in
appendix
A, and
they
satisfy
the
three relations
•
tlg+2
= 1
l.
(7.13)
If we
cut E
along
the
£,-,
the
surface splits into four pieces, each
a
(2g
+
2)-
sided polygon,
as
shown
in
figure
7.2 for the
case
of
genus
2.
Cambridge Books Online © Cambridge University Press, 2009
7.1 The operator algebra of Nelson and Regge
121
Fig. 7.1. A genus two surface can be cut along the curves
t\,...,t6,
shown here
as dark lines, to form four hexagons as shown in figure 7.2. (Only the top halves
of
t2
9
£4, and te are shown in this figure.)
Fig. 7.2. Four hexagons of this type are formed by cutting open the surface in
figure
7.1.
The curves U, d\$, and diA are eight of the fifteen whose holonomy
algebra has been studied by Nelson and Regge.
Nelson and Regge consider the collection of (g + l)(2g + 1) curves
dij = tfti+i...tj-i, which can be represented as line segments connecting
vertices i and j of a polygon such as that of figure 7.2. Any two curves in
this set intersect at most twice, and the classical Poisson algebra (4.73) of
holonomies R^[dij] closes. Moreover, the quantum algebra generalizing
(7.4) may again be shown to close, providing a starting point for the
quantum theory.
Cambridge Books Online © Cambridge University Press, 2009
122 7
Operator algebras
and
loops
The observables R-^ldij] form a highly overcomplete set, of course -
in all, only 12g
—
12 of them should be independent - and the general
form of the relators analogous to (7.6) is not known, although the rather
complicated classical analogs are understood. For the case of genus 2,
however, the model is more tractable. As a 'configuration space', one may
choose the six hyperbolic angles
xpf-
defined by*
/i],
cosht/;^ =
(7.14)
It is evident from figure 7.2 that the curves t\
9
£3, and £5 do not intersect,
so the
xpf-
commute with one another. Nelson and Regge have shown
explicitly how to reconstruct the remaining operators R-[dij\ from the
\pf and their conjugate momenta pu providing the genus 2 analog of
equation (7.8). Moreover, the inner product on the resulting Hilbert space
of functions of the xpf- is largely determined by the requirement that
the R^idtj] be Hermitian, and the operators that generate the mapping
class group are known explicitly. We do not yet know how to construct
time-dependent observables such as the ADM moduli, but the approach
appears to be promising.
7.2 The
connection representation revisited
We turn next to another set of holonomy variables, the functions T°
and T
1
of chapter 4. As we saw in that chapter, these provide a con-
venient set of gauge-invariant and diffeomorphism-invariant functions on
phase space, which have particularly simple Poisson brackets when the
cosmological constant vanishes. These variables are the natural (2+1)-
dimensional analogs of the connection variables in Ashtekar's approach
to (3+l)-dimensional gravity [11], although the (3+l)-dimensional version
is rather more complicated: in 3+1 dimensions, T° and T
1
depend on
loops, not just on homotopy classes of
loops,
and they no longer commute
with the constraints.
Our goal is to promote the algebra (4.84)-(4.86) of Poisson brackets to
an operator algebra, with operators T°[y] and T
{
[y] acting on a suitable
Hilbert space. One approach to this problem is fairly simple, at least
in principle, but will turn out not give us anything new. As usual, we
consider a spacetime with the topology [0,1] x 2, with 2 a closed surface.
We would like to find a representation - technically, a ^representation -
* Conventions
for the
angle
0,
here given
by
(7.5),
differ among papers
in
this
field.
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7.2 The
connection representation
revisited
123
of
the
abstract algebra
o
[yi],f
o
[
r2
]]=o (7.15)
'bx], T°[y
2
]]
= ~
X>(Pi;yi,72) (f
°[
n
•«•
y
2
]
-
f°[yi
-,-
y
2
-
1
]
?,; yi,y
2
)
(f
l
[
yi
-ai] -
f Hyi •
i
y
2
-
1
]
with (T
0
)*
= T° and
(T
1
)*
= T
1
,
subject
to the
relations
°[y
2
]
= ±
(f°[yi
•
72]
+ T°
(
1
f
2
v 7
(7.16)
and
T
X
[0]
=0,
T
ll
[7]=
f
1
^"
1
],
r
J
[yi
-72] =
r
1
^-^!].
(7.17)
Note first that
the
functions
T°
depend
on the
spin connection alone,
and
are
therefore defined
on
the space of equivalence classes
of
flat
SO
(2,1)
connections, that
is, the
moduli space
Jf of
hyperbolic structures
on E.
Indeed,
by
equation (4.81),
T° is
just
the
trace
of the
homomorphism
po:7ri(S)^SO(2,l) (7.18)
of chapter
4, and the
trace
is
invariant under
the
equivalence relation
in
(4.15).
Just
as in
chapter
6, we can
therefore choose
our
Hilbert space
^conn
to be a
suitable space
of
functions
on Jf - or a
space
of half-
densities
if
we wish
to
avoid
the
problem
of
choosing
a
measure
- and we
can represent
the T°[y] as
acting
by
multiplication.
Moreover, when viewed
as a
function
of
the triad
and
spin connection,
the variable T
l
[a>,e]
can be
regarded
as a
cotangent vector
to Jf at the
point representing co, with
a
direction determined
by
e. Recall that
if
co
is
a flat connection
and e
satisfies
the
Einstein field equations (2.62), then
to
first order
in t,
co
+ te is
also
a
flat connection. Equation (4.82)
can
thus
be rewritten
as
d
T
1
[cD,e]=2~T°[co
dt
t=o
(7.19)
clearly defining
a
cotangent vector
to Jf. As in
section
1 of
chapter
6, the
physical symplectic structure
is not
quite
the
natural symplectic structure
Cambridge Books Online © Cambridge University Press, 2009
124 7 Operator algebras and loops
of the cotangent bundle T*" Jf, but the difference is calculable, and once
again allows us to represent
T
l
[(o,e]
as a derivative acting on
3tf
C
onn>
The result of this quantization is not new: it is precisely the connection
representation of chapter 6. For the torus, for instance, homotopy classes
of curves are labeled by two winding numbers m and n, and T° and T
1
can be computed explicitly from the triad and spin connection derived in
chapter 3: by equation (4.87),
A A
T°[m,n] = cosh ^^,
f
i
[m
9
n]=-(m&
+ ^^
2
^ (7.20)
If we now impose the connection representation commutators
A A i
[%d\
= —
[A,
b]
= —-, (7.21)
2
of chapter 6, it is easily checked that the T° and T
1
commutation
relations (7.15) are satisfied. Conversely, given a Hilbert space upon
which the T°[m,n] act by multiplication, the commutators (7.15) may
be used to derive the connection representation commutators. Note that
the identities (7.16) and (7.17) are also automatically satisfied, since they
simply express properties of SL(2, R) traces.
7.3 The loop representation
There is another way to look at the operator algebra of T°[y] and T
1
^]
which leads to an important new approach to quantum gravity, the loop
representation. So far, we have been thinking of the operators T as a set
of functions of the triad and spin connection, indexed by loops y. Our
wave functions arejhus functional of
co,
or, more precisely, functions on
the moduli space Jf of flat spin connections. This dependence is clear in
the expressions (7.20) for T° and T
1
on the torus. However, we could
equally well have chosen to view the T operators as functions of loops
- or in 2+1 dimensions, of homotopy classes of loops - indexed by e
and
co.
Wave functions would then most naturally be functions of loops
or sets of loops. This change of viewpoint is rather like the decision in
ordinary quantum mechanics to view a wave function
e
ipq
as a function
on momentum space, indexed by q, rather than a function on position
space, indexed by p. This analogy will be strengthened below, when we
derive a 'loop transform' as a sort of generalized Fourier transform.
The starting point for the loop representation is again the algebra
(7.15) and the identities (7.16)—(7.17). We would now like to find a
representation of this algebra that makes no explicit reference to the triad
Cambridge Books Online © Cambridge University Press, 2009
7.3 The loop representation 125
or spin connection. There are a number of equivalent approaches to this
problem, but perhaps the simplest is due to Ashtekar [11]
."I"
We begin by
introducing a vacuum state |0), defined by the condition that
T
1
[y]|0>=0 (7.22)
for all homotopy classes [y]. We can act on this state by arbitrary
products of T° and T
1
. By using the algebra (7.15)—(7.16), however, we
can move any factor T
1
to the left, where it will annihilate the vacuum
state.
Similarly, any product T
0
[yi]T°[y2] can be reduced by (7.15)—
(7.16) to a sum of single copies of T°. By repeated application of these
rearrangements, we can finally express any state as a linear combination
with \[
7i
]) = f°[
yi
]\0). (7.23)
The use of [yj\ to label the state |[y,-]) is suggestive, but also slightly
misleading: not all distinct homotopy classes give rise to different states.
For example, by (7.17), |[a]) = {[or
1
]) and |[a
•
jff]) = |[jB
•
a]). Ashtekar
has suggested the term 'equitopy' to describe the equivalence relation
[?i] ~ M if T°[yi](co) = T°[y2](co) for all flat spin connections co.
(7.24)
We should think of the states \[yi\) as being labeled by equitopy classes
of curves.
Even with this caveat, the set of states |[yj) is overcomplete. For
example, by (7.16) and (7.17),
|[a-jJ-y]> + |[a-y-
1
-/J"
1
]>-|[a-yi8]>-|[a-r
1
-y"
1
]>=0.
(7.25)
Identities of this type determine a subspace defined by the relation
° whenever £
Cl
T°[
yi
-] = 0. (7.26)
We can now define the Hilbert space for the loop representation as the
quotient space
3tf\
OO
p
= Jtf"/ ~
9
where Jf' is the space generated by the
\[yi\) and ~ is the equivalence relation (7.26). The inner product on ^i
OO
p
can be defined by demanding that the |[yj) be orthonormal. The actions
of T° and T
1
on
3tf\oo
V
follow from the definition of the basis states and
the algebra (7.15)-(7.17):
(7.27)
This approach is similar to that of DePietri and Rovelli in 3+1 dimensions [86].
Cambridge Books Online © Cambridge University Press, 2009