Download Nonperturbative quantum geometries

Nuclear Physics B299 (1988) 295-345
North-Holland, A m s t e r d a m
NONPERTURBATIVE Q U A N T U M GEOMETRIES
Ted JACOBSON
Department of Physics, Brandeis Universi(v, Waltham MA 02254*, USA
and Department of Physics, University of California, Santa Barbara, CA 93106, USA
Lee SMOL1N
Department of Physics, Yale University, New Haven, CT 0651] *, USA
and Institute for Theoretical Physics; University of California, Santa Barbara, CA 93106, USA
Received 18 August 1987
Using the self-dual representation of quantum general relativity, based on Ashtekar's new
phase space variables, we present an infinite dimensional family of q u a n t u m states of the
gravitational field which are exactly annihilated by the hamiltonian constraint. These states are
constructed from Wilson loops for Ashtekar's connection (which is the spatial part of the left
handed spin connection). We propose a new regularization procedure which allows us to evaluate
the action of the hamiltonian constraint on these states.
Infinite linear combinations of these states which are formally annihilated by the diffeomorphism constraints as well are also described. These are explicit examples of physical states of the
gravitational field - and for the compact case are exact zero eigenstates of the hamiltonian ot
q u a n t u m general relativity. Several different approaches to constructing diffeomorphism invariant
states in the self dual representation are also described.
The physical interpretation of the states described here is discussed. However, as we do not yet
know the physical inner product, any interpretation is at this stage speculative. Nevertheless, this
work suggests that q u a n t u m geometry at Planck scales might be much simpler when explored in
terms of the parallel transport of left-handed spinors than when explored in terms of the three
metric.
1. Introduction
In this paper we present a large class of exact solutions to the hamiltonian
constraint of quantized general relativity. These solutions are found by using a new
representation for quantum gravity based on a set of variables for canonical general
relativity recently formulated by Ashtekar [1, 2]. In this representation, called the
self-dual representation, the form of the hamiltonian constraint is very much
simpler than in the usual metric representation. Whereas in the metric representation, no explicit wave functionals which are annihilated by the hamiltonian con* Present address.
0550-3213/88/$03.50 ©Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
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T. Jacobson, L Smolin / Quantum geometries
straint have ever been found, we will show that in the self-dual representation an
infinite dimensional space of such solutions may be constructed.
Physical states of quantum gravity must, in addition to being annihilated by the
hamiltonian constraint, be invariant under the action of spatial diffeomorphisms.
While most of the solutions to the hamiltonian constraint we have found are not
diffeomorphism invariant, we are able to give a construction of a class of these
solutions which are annihilated by all of the constraints. This construction relies on
a functional integral over the space of three metrics, and so, in contrast to the
results on solutions to the hamiltonian constraint, must be considered at this time a
formal result. However, we believe that taken together, these results signal that with
the self-dual representation the construction of explicit and regulated physical states
of the gravitational field is a task which is now within our grasp.
Indeed, the solutions we will describe seem to indicate, by their form as well as
simply by their existence, that quantum general relativity is endowed with a wealth
of structure at the nonperturbative level. These solutions take a form which could
not possibly have been guessed from a knowledge of the ordinary perturbation
theory (although it might be argued, in retrospect, that hints were present in the
strong coupling expansion [3].)
Thus, the existence of these solutions bears on the old question of whether the
perturbative renormalizability of general relativity is an indication that the theory
does not yield a sensible quantum theory, or is only an indication that the
perturbation theory breaks down at short distances. Proponents of the latter view
have long argued that nonperturbative effects could dominate the behavior of the
theory at Planck scales and shorter [4]. The existence of the solutions we are about
to describe lends support to this view, because it tells us that the space of exact
solutions to the hamiltonian constraint of general relativity has structure not
indicated by the perturbation theory.
Before we proceed to describe the new solutions, let us recall what it would mean
to quantize general relativity, from the hamiltonian point of view.
We begin with the basic structure of hamiltonian general relativity, at the classical
level [5]. The arena for hamiltonian general relativity is a three-dimensional manifold, Z, on which are defined a three-metric, qab and its conjugate momentum, pah
(pah is constructed from a linear combination of the extrinsic curvatures of Z in the
four-dimensional spacetime ~ ) . The evolution of these canonical variables is
subject to four constraints at each point of ~, the three diffeomorphism constraints,
~a ---
D b p ab
(1.1)
and the hamiltonian constraint,
G
¢4 R
J~'= - ~ ( p~bp~b- ½(p:)2) - G
l
(1.2)
T. Jacobson, L. Smolin / Quantumgeometries
297
The three diffeomorphism constraints generate diffeomorphisms in Z while the
hamiltonian constraint generates deformations in the imbedding of Z in Jh' which
arise from infinitesimal reparametrizations of the fourth, time, coordinate.
Now, it is important to note that locally, the canonical transformations generated
by the hamiltonian constraint are indistinguishable from the effect of evolution of
the dynamical variables. As a consequence the hamiltonian contains, besides a
linear combination of the constraints, only a boundary term. The hamiltonian thus
is written,
g = fz[N~-t-Na~a] + fo. d2SbN(Oaqb(.-Obqac)~a~'.
(1.3)
The Lagrange multiplier fields, N and N a, are called, respectively, the lapse and
the shift. In the classical theory they parametrize the slicing of the four manifold JCl
into a family of three surfaces.
A quantization of general relativity, from the hamiltonian point of view [7], would
then consist of the following elements.
(i) A choice of a representation space 50 for the wave functionals, together with a
representation of the dynamical variables as operators on 50 such that the canonical
commutation relations are satisfied.
The conventional choice, up till now, has been the metric representation in which
50 consists of functionals of the three-metric, '/'[q~h], and the operators qab and p~,b
are defined as,
q~bq'[ q~b ] -= qab ( X ) ~[ qab ] ,
h
(1.4)
8
P"bg'[qab]- i ~qab(X~ vt"[q"b]"
(1.5)
(ii) A choice of operator ordering, together with a regularization procedure, which
makes the action of the constraints ~ and ~ well defined on elements of the
representation space 5 °. These choices must be made in such a way that the algebra
of constraints remains consistent. In particular, there must be no anomalous
c-number terms.
Here it is important to point out an important difference between the case of
general relativity and the better understood case of Yang-Mills theory. In the case of
general relativity the classical algebra of the constraints is an open algebra, in the
sense that, while the algebra is first class, the structure "constants" are actually
functions of the dynamical variables. This makes the problem of consistently
realizing the algebra more difficult than in Yang-Mills theory [8]*.
* This problem is discussed in sect. 6.
T. Jacobson, L. Smolin / Quantum geometries
298
(iii) A physical
satisfy [19]
Hilbert space 5:Phys C 5"
which consists of states ~b~ 5: which
aug+ = 0,
(1.6)
~.~k = 0.
(1.7)
A nontrivial physical Hilbert space can exist only in the case that the algebra of
the quantum mechanical constraints closes.
(iv) An inner product (~k[q~) defined on the physical Hilbert space, 5:Phy~. It is
important to stress that one need not have any inner product on the larger
representation space 5:. Even if an inner product exists on the larger space, it is
unlikely to be directly related to the physical inner product on the physical Hilbert
space. This is because physical states, by virtue of satisfying the constraints, will be
non-normalizable under most inner products defined on the larger space 5:.
Because the physical inner product defined only on the space of states which are
annihilated by the constraints, there is no need or reason to require that the
constraints themselves be hermitian. The one condition that we must impose is that
the physical inner product be chosen such that the remaining, boundary, piece of
the hamiltonian is hermitian*.
Until now, most work on the hamiltonian approach to quantum gravity has been
based on the metric representation. This work has been inconclusive, for two related
reasons: (i) A choice of operator ordering and regularization procedure was not
found such that the algebra of constraints closes. (ii) Although it is possible to find
many functionals '/'[Gh] which are (under one choice of operator ordering) annihilated by the diffeomorphism constraints, no states q'[Gb] were ever found which are
annihilated by the hamiltonian constraint**. In the metric representation, this latter
condition takes, up to operator ordering, the form,
~q (qacqba--½qahqca) 3qah3qc~a + G
'/'[ qab ] = 0 ,
(1.8)
where R is the Ricci scalar of the three metric qab* In a recent paper [9] Kuchar" has given examples of finite dimensional systems with first class
constraints in which one can show directly that it is incorrect to demand that the constraints be
hermitian with respect to an apparently natural inner product on the space of unconstrained
wavefunctions. He showed that results obtained by demanding that the constraints be hermitian were
inconsistent with the (presumably correct) results obtained by quantizing the fully gauge fixed
system.
** We might note that a formal solution to the constraint equations, based on a functional integral, has
been proposed by Hartle and Hawking [10], Whether or not there is a choice of operator ordering
together with a regularization procedure such that the proposed functional integral is an exact
solution to all of the constraints remains an open issue.
T. Ja~vbson, L. Smolin / Quantum geometries'
299
It is clear from the complicated form of this constraint why solutions are so hard
to find. Any solution would require that each point of Z the effect of the "kinetic
energy" term ~2/6q2 be precisely balanced against the effect of the "potential
energy" term 1/~-R.
In the self-dual representation introduced by Ashtekar [1, 2], which is the subject
of this paper, the form of the hamiltonian constraint is very much simpler. Without
going into details, which are described in the next section, this is accomplished in
the following way.
The idea is to condense the two terms in (1.8) into a single term homogeneous in
the conjugate momentum by introducing a new variable, A~, which contains
information about both the three metric q~b and its conjugate momentum, pab. Aia
is a certain connection, defined on spinor fields on the three manifold Z. It has the
form,
l '
i
A ~i _-_ f~(i Oe) + ~22
lH~,
(1.9)
where F~(Oe) is the spin connection of the triad e ai a n d / / ~ involves a certain linear
combination of the momenta pub. The index i takes values over the three generators
of SU(2).
Note that A~ is a complex coordinate on the real phase space (e~, p~). It plays a
role in the theory somewhat like that played by the variable z = q + ip employed in
the Bargmann, or coherent state, representation of the harmonic oscillator [11]. Like
wavefunctions ~b(z) in the Bargmann representation, wave functionals '/~[A~] are
functionals on the phase space rather than on the configuration space. The condition that a wave function depend on only half the variables in the phase space is
then realised in the same manner as in the Bargmann representation - by requiring
that ~[A'~] be a holomorphic functional of A~.
Now there is a local SU(2) (--SO(3)) invariance in the formalism which arises
i In the hamiltonian formulation these SU(2)
from the freedom to rotate the triad e~.
rotations are generated by three new constraints, Ni. When we quantize general
relativity in this representation we must add the condition that the wave functionals
are annihilated by these constraints,
=0.
(1.10)
Thus, we are interested, to begin with, in the space of holomorphic, gauge
invariant functionals of a complexified SU(2) connection Aia. These constitute the
representation space 5" on which we wish to represent the constraints of general
relativity.
We shall argue in sect. 3 that any holomorphic gauge invariant functional of Ai,
depends on the connection only via the trace of its holonomy map H: Z,°x~ C,
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T. Jacobson, L. Smolin / Quantum geometries
where Y'x is the loop space of ~J. On a loop 7: S 1 ---'2J, H is defined by
H(7) =
TrPexp~dsA~(y(s))9"(s)C,
(1.11)
where the r i are the Pauli matrices (divided by 2i).
Note that, for each 7, H ( 7 ) is a gauge invariant holomorphic functional of A'~.
Thus, to each loop 7 there corresponds a state H ( y ) of the representation space 5#.
We will be interested in constructing a space of states out of functions of the
H ( y ) ' s . For the purposes of this paper, we will consider a particular space of such
functions which will be defined as follows.
First of all, the state
10> = 1
(1.12)
will be called the "ground" state*. Then, given N closed curves, V~, a = 1 . . . . . N in
X, we will define an N loop state,
N
I{Y~}) = F [ H ( y ~ ) .
(1.13)
a=l
The totality of all states formed by linear combinations of states of the form
(1.12) and (1.13) will be called o ~ ( £ o ) _ the " F o c k space" associated to the loop
space of I;.
The main results of this paper concern the action of the hamiltonian constraint on
states in ~ ' ( . f ' x ) . It is possible to obtain exact results for this problem because,
when acting on functionals of A~, and in the presence of the SU(2) constraint, Ni,
the hamiltonian constraint takes the simple form**,
~2
( ~ ( X ) = EijkFakb(X) ~ A i a ( x ) ~ A J ( x ) ,
(1.14)
where Ffb is the curvature of the connection A~.
Of course, since ~ ( x ) contains a product of operators at a single point, x, its
action on states must be defined in terms of a regularization procedure. It is
therefore important to mention at this stage that because we are working in a
representation which is nondiagonal in the metric
the conventional regularization procedures, familiar from ordinary quantum field theory, are not applicable in
this context.
qab,
* Although the name is convenient, for reasons that will be discussed, we are not making the assertion
that [0) is actually the ground state of the theory.
* * Given a particular choice of operator ordering, which will be discussed below.
T. Jacobson, L. Smolin / Quantum geometries
301
In addition, the quantities that we need to define are not, as they are in usual
q u a n t u m field theory, expectation values. They are constraint equations which state
that a product of operators annihilates some particular state. In particular, although
regularization is required to define the action of the operators on the states,
renormalization is not required. Instead, we show that, in the limit that the regulator
is removed, the operator annihilates the state. Since no renormalization or subtraction is performed this limit can be studied topologically. It does not require for its
definition any parametrization in terms of a physical scale.
This is actually fortunate, because, as we are working in a representation which is
not diagonal in the metric, it is not at all clear how to work a physical scale into the
definition of the regularization procedure. As we shall see, any operator which
measures metric information in the self-dual representation involves products of
functional derivatives at a single point. Such an operator cannot itself be defined
without a regularization procedure. Thus, in the representation in which we are
working, no operator which measures metric information can be used in the
definition of a regularization procedure. Without this, it is not possible to define a
physical renormalization scale to parametrize the regulated expressions.
In this paper we will propose, and study, two regularization procedures which
permit us to define the action of the constraint (1.14) on states in ~.~(£Ps). One of
these is a modified point splitting procedure, the use of which, however, raises
certain technical issues which are, as yet, unresolved. The other, which is unconventional, but without obvious technical flaws, is a regularization procedure in which
states of the form (1.13) are approximated by states based on flux tubes with finite
cross sections.
The main results of this paper are then the following*.
(i) All states of the form (1.13) are annihilated by the hamiltonian constraint
(1.14), as long as the curves "/~ are differentiable, and without intersection.
(ii) When an intersection point occurs, certain additional conditions must be
satisfied for a state to be annihilated by the hamiltonian constraint. In particular,
states of the form (1.13) containing simple intersections must be superposed in a
particular fashion with states involving different routings through the intersection in
order to be annihilated by C~(x).
In the next section we give a brief summary of the new variables of Ashtekar, and
describe the self-dual representation of the quantum theory. In sect. 3 the basic
solutions, for the case of differentiable curves without intersections, are derived.
Here the flux tube regularization is introduced and employed. In sect. 4 the same
results are then studied using a point splitting regularization of the operator.
Sect. 5 of this paper is devoted to the solutions which invol;ce intersecting loops.
In sect. 6 we discuss issues of operator ordering and the consistency of the
* These and other results described here were discussed by the authors at the June 1986 Santa Cruz
meeting on mathematical general relativity [12].
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T. Jacobson, L. Smolin / Quantum geometries
constraint algebra, with particular attention to the effects of the open character of
the algebra.
Sect. 7 is devoted to the problem of constructing states in the self-dual representation which are diffeomorphism invariant. We give two constructions of physical
states - states annihilated by all of the constraints of quantum gravity. The first
construction is based on a functional integral over the space of metrics. As such the
construction is formal, and further work must be done to show that a regularization
procedure exists which defines the integrals while preserving the diffeomorphism
invariance of the limit. The second construction, which was proposed originally by
Ashtekar [23, 25], is a set of asymptotic physical states, which are defined in the
context of asymptotically flat boundary conditions. In addition, in this section we
describe four other ways to construct diffeomorphism invariant states, which may
be useful in the future.
Finally, the paper closes with a discussion of where we now stand with respect to
the four steps of canonical quantization outlined above. Directions of current and
future research are discussed, and some remarks are made about the relation of our
states to classical general relativity.
2. The self-dual representation of quantum gravity
In this section we begin the work of this paper with a review of the definitions of
Ashtekar's new variables, followed by an introduction to the self-dual representation of quantum general relativity. Those wanting a more detailed description of the
variables can find it in the papers listed in refs. [1] and [2]. In addition, an alternate
approach to the new variables, based on an action principle, is described in refs. [13]
and [14].
2.1. THE CLASSICAL PHASE SPACE
The usual phase space of general relativity consists of pairs (q~,b, pea) defined on
a three dimensional manifold Z. (latin indices a, b . . . . . h are three dimensional
abstract indices throughout this paper.) In the asymptotically flat case these are
required to satisfy certain asymptotic fall off conditions. These are given in [2], they
will not concern us here.
Ashtekar's new variables are defined on an extended phase space which is
essentially an SU(2) bundle built on the phase space of general relativity. These
SU(2) rotations correspond to the possibility of making local rotations in a two
component spinor space built on X. Fields in this spin space are denoted as k , and
#B, with indices A, B,... denoting two component abstract indices.
Because we will be working in a context in which the metric q,b on Z is not
always sharply defined, it is perhaps more proper to call two component fields such
as kA "prespinors". These are defined, without reference to a metric on 2~. Instead
T. Jacobson, L. Smolin / Quantum geometries
303
two, a priori, structures are imposed directly [1, 2]. These structures are:
(i) A fixed symplectic spinor metric eA~ and its inverse eAB which satisfy
e AR = - e RA,
e'4ReAR
= 2.
(2.1)
Indices are raised and lowered according to the conventions
/*8 =/laeAB,
XA = eABXB.
(2.2)
(ii) A hermitian conjugate operation t: ?tA ~ )t~ which satisfies the following
four conditions:
(O~)k A
-I-•/.tA)t =
a}k~
-I-~/ZtA ,
(X'A) t = - X A ,
X*A?tA> 0,
(2.3)
(2.4)
equality implying, XA = 0,
(XA/*8) t = X~p)B .
(2.5)
(2.6)
The variables which define this extended phase space then consist of the soldering
form o ~ , together with its conjugate momentum, M ~ B. The soldering form is
defined to be a map from hermitian symmetric two-index spinors UA B t o real vector
fields v ~, given by
v ~ = Oj~BVAB = - T r o ~ v .
(2.7)
The three-metric and its conjugate momentum are then defined in terms of these
variables by the relations,
qab = O~BobAB = _ T r o t , oh,
(2.8)
p~b = _ Tr M("o b).
(2.9)
The elements of this extended phase space may be equivalently represented in
terms of a triad notation. To do this we introduce a fixed basis of hermitian,
symmetric spinors, riB. In this paper we choose the r~B such that in a fixed spinor
basis (in, ia*) their components are equal to 1 / 2 i times the usual Pauli matrices*.
* N o t e that the r ~ are then a n t i h e r m i t i a n matrices. This is a little confusing, since they are the
c o m p o n e n t s of an abstract spinor r~B which is h e r m i t i a n in the sense of the definition we have j u s t
given. This is an unfortunate accident of convention.
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T. Jacobson, L. Smolin / Quantum geometries
Then we may write,
OffB = o{*'rjB,
(2.10)
M d 8 = Mbi riaB .
(2.11)
The fields oi" are then a triad of vector fields on G. We will use both this triad
notation and the original spinorial notation in this paper, since sometimes one is
more convenient than the other.
Ashtekar introduces on this extended phase space (of, Mbj ) a complex coordinate, given by a certain complexified SU(2) connection, A~ [1, 2]*, which is defined
by
• i
A'o- F~+ ~i~1 zH.,
(2.12)
where F] is the metric compatible, torsion free spin connection determined from the
condition that it leave of covariantly constant, and Hia is defined by the equation,
G
(2.13)
Thus, F~ contains only information about the three-metric, q,b (or rather its
square root off), while H~ contains information also about the conjugate momentum p,b. In this sense Aia is analogous to the Bargmann coordinate z = q + ip on
the phase space of the harmonic oscillator. This is an analogy which will be very
useful when we come to the quantum theory.
Ai, actually has a direct physical interpretation in classical general relativity. For
solutions to the Einstein equations, A~ is in fact the four-dimensional spin connection for left-handed spinors, restricted to the embedded hypersurface 2~ in the
evolved spacetime [1,2]. Thus W, is a potential for the self-dual part of the
curvature.
One of the principle results of Ashtekar's work is that the Ai~ are a complete set of
commuting variables in that they satisfy
{Aia(x),A~(y))
=0,
(2.14)
where the brackets denote the Poisson bracket on the extended phase space
(aft, Mbj),
( f , g} = 2
f~[6M~
8f 6of
~g 6M~
8g 8of8f] "
(2.15)
Of course A~, like z, has a nonvanishing Poisson bracket with its complex conjugate.
* T h i s is a m o d i f i c a t i o n o f a c o n n e c t i o n o r i g i n a l l y i n t r o d u c e d b y S e n [15].
T. Jacobson, L. Smolin / Quantum geometries'
305
A second result of Ashtekar's work is that A'~ has simple Poisson brackets with
the densitized frame fields
oai=v~oai.
(2.16)
These are,
(2.17)
These relations may be thought of as analogous to the relation (z, q } = - i . Of
course, the 8fl(x) commute among themselves.
We then want to describe the real (extended) phase space of general relativity in
terms of the coordinates (Wa, ~b). This would be analogous to describing the phase
space of the harmonic oscillator by the pair (z, q). Note that in the case of the
harmonic oscillator it is necessary to impose conditions to pick out the phase space
of the real (as opposed to the complex valued) harmonic oscillator. These conditions
are q = ~ and z + £ = 2q. To specify the phase space of real, as opposed to complex,
general relativity, we also need to impose two conditions. One of them is that 62B
be hermitian, it may also be expressed by requiring the 6y to be real. The second
condition is, from (2.12),
A~ + X~ = 2F,~.
(2.18)
The constraints of general relativity take a very simple form when written in
terms of the covariant derivative,
~ k ___a,kOa +
GgjkA~
(2.19)
and the corresponding curvature (setting G = 1),
Fa~b =
oijk Aj Ak
20taA~l + ~ "'a"b,
(2.20)
where 0~ is an internally flat derivative operator that acts on spatial scalars with
internal indices. Note that N~ does not know how to act on general tensors. The
constraints of general relativity then become [1, 2],
C~( x ) -- eiJkF~hfa'fbi = O,
(2.21)
cg~( x ) -- F],~6 hi= O,
(2.22)
f~i( x ) = ~ 6 " i = O.
(2.23)
We call (2.21), (2.22) and (2.23) respectively the hamiltonian, diffeomorphism and
gauge constraints. The gauge constraint, ~i, generates local triad rotations. Indeed,
the form of this constraint, together with the commutation relations (2.14) and
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T. Jacobson, L. Smolin / Quantum geometries
(2.17) suggest that we may consider A~, to be a complexified SU(2) Yang-Mills field
and 6 ai to be its conjugate electric field. Making use of this analogy we see that ~i
has the same form as the usual Gauss' law constraint.
This is a useful analogy for understanding the quantization of general relativity in
the self-dual representation. However, it must be employed with care, because
certain things about the formalism are different from the case of Yang-Mills theory.
Most importantly Ai, is valued in the Lie algebra of complexified SU(2) (which may
also be thought of as SL(2, C) with a preferred SU(2) subgroup), while its "electric"
field 6 ~" is restricted to be real.
The diffeomorphism constraint c~ generates a combination of a diffeomorphism,
a gauge transformation, and terms proportional to ~ . This must be expected as the
action of a diffeomorphism does not extend uniquely to objects that are not gauge
singlets.
The remaining constraint, ~, is called the hamiltonian constraint. It is equal to
the old hamiltonian constraint a'(Y when the gauge constraint ffi is satisfied. Thus,
in this formalism the hamiltonian constraint has, in effect, been separated into the
two constraints ~g and ~ *. This is advantageous, because both W and f9 ~ are much
simpler than the old hamiltonian constraint (1.2).
Indeed, the new hamiltonian constraint W manifests an amazing simplification
over its usual form off. Whereas off is nonpolynomial in q~b, cg is quadratic in both
6 ~ and A~. Further, while off is made up of two terms, one involving the p,b, and
the other not, c~ is homogeneous in 6 ~. This very much simplifies the finding of
solutions to the quantum constraint, W'/'[A~] = 0.
2.2. Q U A N T U M T H E O R Y : T H E S E L F - D U A L R E P R E S E N T A T I O N
In this paper we choose to work in the representation in which states are
holomorphic functionals xI'[Ai~] of the connection Aia. Ashtekar [1,2] called this the
self-dual representation, since, as we remarked above, in classical solutions F,' b
captures precisely the self dual piece of the four-dimensional Riemann tensor. In
analogy with the Bargmann representation, +(z), z = q + ip, in quantum mechanics, the holomorphicity of q[A~] ensures that it depends on the correct number of
degrees of freedom. Note that this representation is allowed because the A~ are a
complete set of commuting coordinates on the extended phase space**.
* T h e second spatial derivatives O2q,r, which appear in the v/qR term in ~ are implicit in the
combination of cg and ~ . OA~ appears in cg via F~'h whereas the gauge constraint, fCi = 0 imposes a
relationship between A~ and 06 ~. Thus, F~h is implicitly related to OZqah when f~ = 0. (It might be
thought that the reality condition (2.18) is actually responsible for the relation between A~ and c9~"~.
To see that this is not so, note that in complex relativity there is no such reality condition, and yet
the constraints can still be written in the form (2.21)-(2.23).
* * In the language of geometric quantization [16], we are working with a complex polarization on the
real phase space of general relativity. See ref. [17] for details.
T. Jacobson, L. Smolin / Quantum geometries
307
I n this r e p r e s e n t a t i o n the o p e r a t o r s A;a act b y multiplication. The choice of an
o p e r a t o r to r e p r e s e n t 6 ~i is d e t e r m i n e d b y the c o m m u t a t i o n relations,
[Ai (x),fjb(y)] = - /2h6 6163(x, y),
6 (y) ] =0,
(2.24)
(2.25)
to h a v e the form*
8
Be[A]
8A (x
8A.(x)
(2.26)
w h e r e q) is a n y functional of A ~ * .
A s in o r d i n a r y q u a n t u m mechanics, any choice of • can be c o m p e n s a t e d by
m u l t i p l y i n g all wave functionals b y exp( - q)). In this p a p e r we are only interested in
f i n d i n g wave functionals a n n i h i l a t e d b y the constraints, thus we m a k e the simplest
choice,
oia(x)
8
8A~(x)'
"
(2.27)
This choice c o m p l e t e s the definition of the self-dual representation. Before going
on, however, we need to discuss an a d d i t i o n a l point, which is how the reality of o ";
a n d the c o n d i t i o n (2.18) are to be i m p o s e d in the q u a n t u m theory *'~*. N o w in
q u a n t u m m e c h a n i c s real observables are r e p r e s e n t e d b y o p e r a t o r s which are hermitian with r e s p e c t to the inner product. Thus, reality c o n d i t i o n s in a classical theory
b e c o m e , in the q u a n t u m theory, c o n d i t i o n s involving b o t h the choice of an o p e r a t o r
a n d the choice of an inner product. F o r example, in the case of the h a r m o n i c
oscillator, o n e is free to choose either q = O/Oz or q = O/Oz + f ( z ) , with f ( z ) any
h o l o m o r p h i c function. F o r any f an inner p r o d u c t can be chosen which renders the
real o b s e r v a b l e s q a n d p = ( z - q ) / i hermitian.
I n o u r case, as the c o m m u t a t i o n relations restrict the o p e r a t o r for 6"; to be of the
f o r m (2.26), the o n l y effect of the reality c o n d i t i o n s on the definition of 6 ~; could be
* We are using units in which h = ~/~2** More precisely, 6,a(x) has this form locally in the space of A~'s. When the space of connections mod
gauge transformations and diffeomorphisms has nontrivial topology, the addition of a q~ term can
be useful for studying topological effects. This is presently being studied by Ashtekar, Balachandran
and Mazur [18]. It is interesting in this connection that one possible choice for q) which is gauge and
diffeomorphism invariant is the Chern-Simons integral X - f Tr(A A dA + ~A A A ,'~ A). In this
case the term added to 8[l(x) is proportional to e~mF;~,..
*** It should be mentioned that as neither 6 ai n o r A/ commutes with the constraints they are not
themselves physical observables. It is perhaps sufficient if physical observables which are real by
virtue of the classical reality conditions are represented by hermitian operators.
T. Jacobson, L. Smolin / Quantum geometries
308
in the form of ¢b. However, as we have said, given a state '/" which solves the
constraints with q~ = 0, the state exp(-cb)'/" will solve the same constraints when
the functional gradient of ¢b has been added to 6% Thus, the reality conditions can
have no effect on the problem of finding the solutions to the constraints.*
2.3. THE Q U A N T U M CONSTRAINTS
Following the Dirac quantization procedure [19] for constrained systems, the
constraints (2.21)-(2.23) in the self-dual representation become, with a particular
choice of operator ordering,
=
~2
eJkr (x)
=
8
~o(x)~[A'a] = F ; ~ ( x ) ~ ~ [ A ]
fYi(x)q'[Aio] = N~,
8
8A~(x)
q'[A] = 0.
=0,
0,
(2.28)
(2.29)
(2.30)
This choice of ordering is motivated by the following considerations. The ordering
of the constraint ~i is determined by the requirement that it generate infinitesimal
gauge transformations. With the ordering shown, the condition (2.30) in fact means
that the functional ff'[Aia] is invariant under infinitesimal gauge transformations,
8A i
- ~.A.
_
i
Similarly, the ordering of the diffeomorphism constraint is determined by the
requirement that it generate, on gauge invariant functionals, infinitesimal diffeomorphisms, rather than, for example, infinitesimal diffeomorphisms plus a divergent
constant. With the ordering shown, the diffeomorphism constraint condition (2.29)
means that '/'[A~] is invariant under the infinitesimal transformation 8A~ = NbFl~a
where N b is an arbitrary vector field. This implies that 6Ai~ is then equal to the Lie
derivative of Ai~ in a gauge in which NaA~ = 0. Thus ~b generates what could be
called the "gauge covariant Lie derivative" of the connection with respect to the
arbitrary vector field N b. Thus, with the ordering we have chosen, ~ i x / F ~ 0 and
cgx/, = 0 together imply that g'[A'a] is invariant under the component connected to
the identity of the group of combined gauge transformations and diffeomorphisms.
The ordering of cg is then motivated by the condition that the algebra of
constraints be consistent, and in particular that it have no anomalous c-number
terms. As this is a somewhat complicated subject which involves some issues
* Of course, the physical significance of any given solution will depend on the inner product and
therefore, indirectly, upon the reality conditions.
T. Jacobson, L. Smolin / Quantum geometries"
309
connected with regularization and the consistency of the operator algebra, the
discussion of this choice is postponed to sect. 6.
We should, however, point out that the ordering we are using is different from the
one discussed in Ashtekar's paper [1]. We do not believe that this is a serious
objection to our choice of ordering for reasons that also will be discussed in sect. 6.
Finally, we should stress that the passage from the classical to the quantum
constraints symbolized by eqs. (2.28)-(2.30) has, in any case, been so far only
formal, since these operators require some regularization procedure for their action
on states gt[A~] to be well defined. The regularization of these operators is
discussed in the following two sections.
3. Solutions to the hamiltonian and gauge constraints
In this section we show that there exists, in the context of a well-defined
regularization procedure, a large class of exact solutions to the hamiltonian constraint of general relativity. This will be accomplished by writing down first a large,
and perhaps complete, set of solutions to the gauge constraints fCi expressed in
terms of holonomy elements of the connection Aia. Then we will see that a large
subclass of these functionals, those involving products of traces of holonomy
elements over loops which are differentiable and without intersection, are also
annihilated by the constraint ~, when a suitable regularization procedure has been
imposed. Because ~ and c~ together are equivalent to the usual hamiltonian
constraint ~ , the solutions that we are about to describe are genuinely solutions to
the quantum mechanical hamiltonian constraint of general relativity.
The first thing to note is that the functional
gt0[A/] = 1
(3.1)
is a solution to all of the constraints (2.28)-(2.30), given the ordering that we have
chosen. In the usual, metric, representation, the "potential" term v~-R in the
hamiltonian constraint ~ ' precludes the analogous functional '/'[q,b] = 1.
Now, because we do not at this stage know the physical inner product, the
question of the physical significance of this state, as well as those we are about to
construct, must remain at present unanswered. In particular, we cannot now answer
the question of whether these states will be normalizable with respect to a physically
meaningful inner product.
In this regard, it is interesting to recall that in the usual Bargmann representation
for the harmonic oscillator, the state ff0(z) = 1 is in fact the ground state, and it is
normalized by virtue of the measure factor e ~z in the inner product [11]. It is thus
tempting to attach the label "ground state" to the state (3.1). However, this analogy
cannot be applied too quickly. The reason is that in the representation in which the
310
T. Jacobson, L. Smolin / Quantum geometries
inner product measure has the form e -~z we have ~ = O/Oz, whereas we are working
in a representation for quantum gravity which is more closely analogous to the one
in which q = O/Oz. In this representation the ground state of the harmonic oscillator
is represented by the function ~0 = e z2 and the function + ( z ) = 1 is not even
normalizable ~'.
According to the discussion of subsect. 2.3, the gauge constraint (2.30) means that
q,'[A~] is a gauge invariant functional of the complexified SU(2) connection A~ on
~. Now the holonomy map
hA: ,,~:c ~ SL(2, C),
(3.2)
where La~ is the loop space of S and
hA(y) = Pexp~dsAia(7(s))f~(s),d,
(3.3)
captures precisely the gauge invariant information in Wa. Therefore any gauge
invariant functional '/'[A~] could be regarded equivalently as a functional of hA**
Actually hA(y) is not quite gauge invariant, since it transforms at the basepoint of
the loop. Thus, functionals of the holonomy map h A will not quite be annihilated by
the gauge constraint ~i. We remedy this by considering the functionals of the trace
of the holonomy map. These are gauge invariant and holomorphic and, for holomorphic functionals, involve no loss of generality. We shall now sketch the argument for this.
As explained above, a gauge invariant functional 'P[A] can be regarded as a
functional of the holonomy map h A. Let ~¢ be the space of complexified SU(2)
connections and let ~ c d be the subspace of SU(2) connections. For A E ~ , h A is
a unitary representation of the holonomy group and is therefore determined up to
equivalence by its character (since all unitary representations are so determine by
their character.) Since 'P is gauge invariant, it is therefore actually a functional of
Trh A for A ~..~. Since ~P is holomorphic, its functional form on .~¢ is determined by
analytic continuation from its restriction to ~ , therefore it is functional of Tr h A on
all of d . The restriction to holomorphic functionals is essential to this argument
* In ref. [20] the linearization of the self-dual representation is studied, and it is shown that the
linearized formalism becomes equivalent to the usual linearized formalism when the choice analogous
to z = 3/3z is made. This suggests that the states we are about to discuss will not be normalizable
with respect to the inner product of linearized quantum general relativity. While this has not yet been
rigorously shown, we think that this is in fact likely. Thus, we conclude that if the states we are about
to construct are to be normalizable under the physical inner product of the full nonlinear theory, the
Hilbert space defined by that inner product will not be unitarily equivalent to the Hilbert space of
the linearized theory. (Such inequivalence would be welcome, since it is in any case required if the
inner product is to play a role in softening the contribution from fluctuations at short distances.)
** We are here ignoring all questions of convergence and measures which would be needed to make this
statement rigorously.
T. Jacobson, L. Smolin / Quantumgeometries
311
since, for a general complexified SU(2) connection, T r h a does not d e t e r m i n e A up
to g a u g e equivalence*.
Thus, a n y gauge invariant h o l o m o r p h i c functional of Aia could be r e g a r d e d
e q u i v a l e n t l y as a functional of the trace of the h o l o n o m y m a p of the connection. In
this p a p e r we shall consider a class of such functionals which can be expressed as
l i n e a r c o m b i n a t i o n s of elements of the form
N
]{Y~}) -= 1--I H ( ' y ~ ) ,
~=1
(3.4)
w h e r e H ( ' y ) = T r h A ( ~ ) a n d {'~,}, c~ = 1 . . . . . N is a collection of N loops in Z. W e
then c o n s i d e r states which m a y be expressed as
xtt[Aia] =colO)-~-
M=I
H dYc~ CM[~I ..... ~M]I{~/~)),
a=l
(3.5)
w h e r e the coefficients CM['y1,..., "YM] are densities over (~qz) M. This is the space of
f u n c t i o n a l s we referred to in the i n t r o d u c t i o n as ow(~ez).
W h i l e this is clearly a very large class of h o l o m o r p h i c gauge invariant functionals
of A~, this p a p e r is not the place to investigate whether there is a sense in which the
f u n c t i o n a l s ] { 7 , ) ) form a c o m p l e t e set with respect to a suitable norm. W e may,
however, n o t e that as the loops in the set { ~ , ) can include multiples of one another,
this set c o n t a i n s all p o l y n o m i a l functions of the traces of the h o l o n o m i e s a r o u n d
w h a t e v e r sets of loops are allowed. W e also note that in the lattice regularization of
the t h e o r y [27], these functionals d o i n d e e d form a c o m p l e t e set.
W e n o w go on to the m a i n result of this section, which is that there exists a
r e g u l a r i z a t i o n p r o c e d u r e such that
Cg(N)] { 7, ) ) = 0 ,
(3.6)
w h e n the curves ~, are diffentiable a n d nonintersecting. Here, ~g(N) is the h a m i l t o n i a n c o n s t r a i n t (2.28) smeared with a n y s m o o t h density (of weight - 1 ) ,
•(N)
- fsN(x)CY(x).
(3.7)
* For example, a connection with holonomy entirely in a subgroup of null rotations (i.e. generated by
matrices of the form r l + ir 2) has Trh A = 2 for all loops, but is not gauge equivalent to a flat
connection. Note that this means that a state in quantum gravity cannot distinguish these connections, that is, it must assign them the same amplitude (although it is possible that the measure in the
inner product, not being holomorphic, might distinguish them.) It is amusing to speculate that this is
somehow related to the fact that all known massless particles transform trivially under the null
rotations (which comprise the little group of a null vector).
T. Jaeobson, L. Smolin / Quantum geometries
312
The basic idea behind the construction of these solutions, as well as those involving
intersections described in sect. 5, is to exploit the simple algebraic structure of the
hamiltonian constraint cg. This may be written
82
W(x) = F,'{(x) 8A~(x)SA~(x) '
(3.8)
where F,'L = eijkFfb is antisymmetric separately in both pairs [ab] and [ij]. Thus, if
the second functional derivative of O[A~] is symmetric separately in each of the
pairs, Cd(x)~[A~] will vanish. In a naive, unregulated, calculation, we will see that
the second functional derivative of I()',)) is symmetric separately in each of the
two pairs, if all of the curves 7, are differentiable and nonintersecting. The reason
for this is that there is no source of vector indices other than the tangent to the
curve, so we obtain the symmetric combination ?"9 b. The task then will be to find a
regularization procedure which makes the products of functional derivatives well
defined, and preserves the property that these states are annihilated.
We thus begin by studying the naive unregulated calculation of Cd(N)ha(7),
where 7 is a differentiable loop parametrized by the unit interval [0, 1]. The first
functional derivative yields,
8AJ(x)
hA('Y)6
-_ fo1ds (~3(x, 7(s ))2b(s )U(1, s ) rJU(s, O)
(3.9)
where we employ the notation,
U(t, s) = Pexp ftduAia(7(u))~a(u)r ~
as
(3.10)
and matrix multiplication is implicit. This equation follows from the definition of
the path ordered exponential in (3.3). After a second functional differentiation we
obtain,
Cd(N)hA(y)
=
× [ o ( , - t ) u ( a , s)
-iu(s,
+ o(t- s)u(1,
T. Jacobson, L. Smolin / Quantum geometries
313
where r(i'r j) = - 1/46 ij has been used (recall that the r i's are 1/2i times the usual
Pauli matrices) and J(s) is a jacobian factor.
Now, the integrand in (3.11) vanishes since 9[~(s)~,hl(s)= 0 and 8 tijl = 0. The
difficulty is the overall 82(0), which reflects the 62(0) present in the first functional
derivative (3.9). This in turn reflects the singular, one-dimensional, character of the
functional hA(Y). Our task is then to define a regularization procedure which
replaces the 82(0) with a finite expression, while preserving the algebraic symmetries
that cause the expression to vanish.
To motivate our regularization procedure, let us consider for a moment a slightly
simpler case, namely functionals of an ordinary one-form b,,. Consider functionals
of the special form,
~b[bo] = expfd3xMO(x)bo(x),
(3.12)
where M~(x) is any nonsingular density on N. This functional is infinitely functionally differentiable. In particular, we may take as many functional derivatives as we
like at a single point,
8(n)
8b~(x)...Sba°(X)Cb[b~]=MU~(x)...M""(x)~[ba].
(3.13)
This is in contrast to the situation with functionals of our form hA(y), in which
even the first functional derivative has a 62(0) singularity. Thus the singularities in
~(N)hA(Y ) can be seen as arising from the singular nature of a smearing function
in the exponent with the form, Mb(x) = fds 63(x, y(s))gb(s), rather than from the
coincidence of the two functional derivatives at the same point.
This consideration suggests a regularization procedure involving the definition of
our states, rather than of the operators*. We thus proceed by "smearing" our states
into expressions involving finite "flux tubes" rather than holonomy elements.
We begin by embedding the curve y(s) in a congruence of differentiable nonintersecting curves 7(s; o) parametrized by the two functions o = (o 1, a2). We will
choose the a such that y(s; 0, 0) = y(s). The parameters (s, o i, o 2) thus coordinatize
an open neighborhood of the curve y(s). (See fig. 1.)
The regulated functionals are defined as follows. Consider a two-dimensional
smearing density f ( o ) which might, for example, be chosen to be gaussian. Now,
define
B(s) - f d2of(a)A'~(y(s; a ) ) 9 " ( s ; o ) r ~,
(3.14)
* We stress that, as inner products are not yet involved, we are not claiming that this regularization
procedure will be sufficient to regulate physical quantities. It is, however, sufficient to define eq.
(3.6).
T. Jacobson, L. Smolin / Quantum geometries
314
y(s):y,(so : 0 ~
Fig. 1. Coordinates for the "flux tube" regularization, a stands for the two coordinates o 1 and o 2.
where ~,a(s; o ) = [0`/(s; o)/Os] ~. The regulated holonomy functional is then defined by
hAi(V) -- P e x p f d s B ( s ) .
(3.15)
Note that hf(`/) is not gauge covariant. The gauge covariance will be restored by
considering the limit in which f ( o ) approaches the delta function 62(0). In that
limit, we have
lim h i ( , / ) = h~(,/).
(3.16)
f--.82
We now evaluate the hamiltonian constraint on hf(,/). The first functional
derivative yields,
8a~(x)h~(,/)= f dsUi(1,s) ~ 8 ( s )
UZ(s,O)
= fdsfd2of(o)~g(x,,/(s;o))~h(s;o)Uffl,s),JUqs,O), (3.17)
where, in analogy with (3.10), we have defined
uf(t, s) - Pexp fCduB(u).
(3.18)
It is then convenient to evaluate (3.17) in the coordinate system (s, o), whence,
6
--hf
~A~(x)
( ,/ ) = f ( x )~b(x )Uf(1, x )rJUi( x,O),
(3.19)
where we use the notation x for whichever coordinates of x are relevant, i.e.
2, X3); ,/(X) = , / ( x l ; X 2, X3); U S ( l , x ) = U S ( l , x l ) . Taking another de-
f(x) =f(x
T. Jacobson,L. Smolin / Quantumgeometries
315
rivative, we find
e(x)G(v) =
-
(3.20)
= 0.
The computation of C6(x)h~(~{) is everwhere well defined for any smooth f, and
the result is rigorously zero. Thus, h/A(7) provides an exact solution to the
hamiltonian constraint cg for any smooth f ( o ) and for any differentiable, non-intersecting congruence ~,(s; o). Unfortunately, this has been achieved at the expense of
gauge invariance, as B(s) in (3.14) clearly does not transform appropriately for
h f ( 7 ) to be a gauge covariant quantity*. However, in the limit that rio) approaches a delta function, gauge covariance is recovered (3.16). Therefore, we shall
define the action of C~(N) on h,4(V) by
Cg(N)hA(7) = lim C ~ ( N ) h / A ( 7 ) .
f~62
(3.21)
Note that we are free to define C6(N)hA(7) in this way because it is, to begm
with, not well defined, being proportional to the product of a divergent and a
vanishing quantity. One way of saying this is that hA(Y) is on the boundary of a
domain of functionals of the form h i ( 7 ) on which the action of the operator Cd(N)
is well defined. We are then extending the domain to the boundary by continuity.
Since W(N)hf('f) vanishes for any regular f ( o ) , the right-hand side of (3.21) is
independent of how we take the limit f ( o ) --* 32(o), so this procedure is unambiguous.
With the definition (3.21) we then have
W(N)hA(y) =0
(3.22)
for all differentiable, nonintersecting curves 7(s), and all smearing densities N. It is
trivial to extend the action of C6(N) to sums of products of such functionals, so long
as the curves involved in a given product are differentiable and non-intersecting. In
particular, for the state defined in (3.4) we have
C~(N)l {T~}) = 0.
(3.23)
As for gauge invariance, it is easy to see that
lim ~ ( A ) T r h f ( 7 )
f ~3 2
= ~(A)
lira Trh/A(y) =
fC(A)H(7) =
0,
(3.24)
f--*3 2
where N(A) = fd3x Aif~ i. The non-singular action of the gauge constraints (smeared
* Note that we do not regulate via the gauge invariant integral, f d2of(o)H(3,(o)) because the action
of Cd(N) on it is singular.
316
T. Jacobson, L. Smolin / Quantum geometries
with a smooth A) therefore commutes with the limit which defines the action of
C~(N). We thus see that the regularization procedure described here, while breaking
both coordinate and gauge invariance in the construction of the regulated functionals hf('~), leads in the limit to results which are both gauge invariant and
independent of the coordinates, a, which define the regularization.
4. Point splitting regularization
In this section we repeat the calculation of the previous section using a modified
point splitting regularization rather than the flux tube regularization we used there.
As we indicated there, the flux tube regularization is suggested by the fact that the
divergences which arise in the naive action of the operator C£(N) on the states
I { 7~ }) arise as much from the singular nature of the state as from the coincidence of
operators at a point.
On the other hand, point splitting is a well understood procedure in ordinary
quantum field theory, and we would like to know whether the conclusion that the
states [(~,~}) are annihilated by the hamiltonian constraint could not also be
reached using a point splitting procedure. The answer, as we shall see, is a partial
yes. Some, but not all, of the states which are solutions under the state smearing
regularization are solutions under the point splitting regularization. However, the
point splitting approach applied to this context involves certain difficulties which
are not found in its application to ordinary quantum field theories, or even to
quantum gravity in the usual metric representation.
The reason for these difficulties is that to set up a point splitting procedure for an
operator of the form of (2.28) one needs a metric to tell us how far apart the points
are, and a connection and curve to parallel transport the indices on the functional
derivatives to a common point. The problem is that in the representation in which
we are working the metric and the Christoffel connection are represented by very
complicated operators which involve at least products of second functional derivatives. Thus, these operators are not well defined without a regularization procedure,
and cannot themselves be used in the definition of a regularization procedure. As a
consequence, to set up a point splitting regularization of C~(N) we must introduce
an arbitrary and unphysical metric and connection just to define the point split
operator. Unfortunately, as we shall see, some information about this arbitrary
structure survives in the action of ~ ( N ) on H ( 7 ) in the limit that the two points
are brought together.
We begin by showing exactly why it is not as straightforward to set up a point
splitting procedure in the present context as it is in ordinary quantum field theories.
In the more familiar context, in which there is a fixed background metric, when we
split an operator product A(x)B(x) apart to A(x)B(y) we can parametrize the
regulated point split operator in terms of the physical, geodesic, distance, between x
and y. That this is a physically meaningful thing to do is reflected in the existence
T. Jacobson, L. Smolin / Quantumgeometries
317
of a n o p e r a t o r p r o d u c t expansion,
1
A(x)B(y)
= E [ x _ ylp(gP(x) •
P
I n q u a n t u m gravity, however, we are working on a b a c k g r o u n d manifold, Z o n
which n o metric or c o n n e c t i o n has been defined. The physical distance between two
p o i n t s is a n operator rather than a parameter. I n the usual metric representation this
p r o b l e m m a y n o t be fatal, as the metric acts on the states b y simple multiplication*.
However, in the self-dual representation the metric operator c a n n o t even be simply
w r i t t e n as a n operator in closed form. Its d o u b l y densitised inverse, qab, which is
the simplest operator which measures metrical i n f o r m a t i o n , is a second functional
differential operator,
82
~ a b ( X ) =-- qqab(x) = ½E
i
8Aga(x)6A~(x)
.
(4.1)
Such a n o p e r a t o r could n o t be used in a p o i n t splitting procedure as it itself is not
well d e f i n e d w i t h o u t some regularization procedure.
Thus, in this context we have n o measure of how far apart two n e a r b y points x
a n d y are, as would n o r m a l l y be required in a p o i n t splitting procedure. I n
particular, although we can consider a sequence of points Yi which converge to a
p o i n t x as i ~ oo we c a n n o t give a physical m e a n i n g to the question of how fast
they are a p p r o a c h i n g each other. Thus, we c a n n o t introduce a r e n o r m a l i z a t i o n scale
to p a r a m e t r i z e a n y divergent quantities which might appear when we study the limit
of A ( x ) B ( y i ) as i tends to infinity. T o put it simply, we can regulate, b u t it is n o t
clear that we can renormalize.
Luckily, there are some things that can still be studied in this restricted context.
F o r example, we can study a limit of the form,
limA(x)B(y)l~).
y~x
I n particular, we will be able to show that there do exist states which have the
p r o p e r t y that they are annihilated by a suitably p o i n t split version of W ( N ) in the
* It is an interesting, and to our knowledge an open, question whether or not an operator product
expansion exists in this case. Even here, however, there may be difficulties not present in the usual
case. For example, we might consider the action of an operator D(x, y) which measures the geodesic
distance between two nearby points x and y on a state which is itself superposition of two states,
I1)+ 12). It could happen that a sequence of points y, could be written down such that
(liD(x, Yi)l 1) converges to zero, while (2ID (x, yi)12) does not converge, or converges to some
finite distance. If this can be the case then we would have trouble using D(x, y) to define a
physically meaningful operator product expansion.
318
T. Jaeobson, L. Smolin / Quantum geometries
limit that the two points approach each other. We will be able to do this because,
for those states which are annihilated by the point split operator in the limit, no
divergent quantities appear as the points approach each other. However, if we study
the action of the point split Cg(N) on other states which are not annihilated in the
limit, we will see that the limit depends on arbitrary structures which must be
introduced to define the point split operators, and thus do not have an unambiguous physical meaning.
Let us now turn to the problem of constructing a point split version of the
hamiltonian constraint operator Cg(x), given by (2.28). For the purposes of this
computation, we return to the notation employing explicit spinor indices, A, B, C . . . .
instead of the frame indices, i, j, k . . . . . Now we might define the point split
operator by just displacing one of the functional derivatives to a point y:
9
8
8
C(x, y) " eBCSONFaAN(x) 8A~B(X ) 8ACD(y ) .
(4.2)
However, this operator is neither gauge invariant nor diffeomorphism invariant,
since we are contracting indices at different points. In order to restore gauge
invariance we can introduce a curve ~ connecting the points x and y, and then
parallel transport the "gauge" (i.e. spinor) indices at y back to the point x. This
amounts to the replacement
e~c8~ ---, u~Cu~
(4.3)
in (4.2), where U, is the spinor propagator along ~/defined in (3.10). Since we are in
the self-dual representation, in which the operator A2 B acts by simple multiplication, this replacement is straightforward.
But what are we to do about diffeomorphism invariance? Not only can't we use
the metric to determine the curve 71 as, say, the geodesic joining x and y, but we
have no way to transport the vector index b. The spin connection A AB does not tell
us how to transport spatial indices, and it is of no help to try to construct an
operator for the usual Christoffel connection, as this contains terms of the form
o b Oco ~ which involve products of operators at points, and are thus not themselves
defined in the absence of a regularization procedure.
In the absence of any way to use the physical metric in the definition of
the regularization procedure, we must introduce an unphysical, arbitrary curve
and connection F~c to tell us how to carry the indices of 8/8AC°(y) back to the
point x. Of course, this does not solve the problem of breaking diffeomorphism
invariance in the construction of the regularized operator. The reason is that from
the point of view of the operators of the quantum field theory, diffeomorphisms are
generated by the action of the constraint cg~. This constraint will have no action on
the arbitrary connection Eft, and curve 7- Thus, the commutation relation
T. Jacobson, L. Smolin / Quantum geometries
319
[ ~ ( x ) , ~a(Y)] = 3~33(x, y)rg(x), which tells us that ~ ( x ) transforms under diffeomorphisms like a scalar density, will not become anything simple when C~(x) is
replaced by its point split regulated version.
Finally, in addition to the arbitrary curve and connection we would like to
introduce some measure e of the distance between x and y, to parametrize the
splitting of the points. Although it is certainly possible to introduce all of these
structures independently, it is simplest just to choose a coordinate chart 0 and use
the (flat) metric, connection and geodesics determined by those coordinates, and
this is what we shall do. Hatted indices will denote components in the chart 4~.
Now, the breaking of a symmetry such as diffeomorphism invariance in the
construction of a regularization procedure may not be fatal in the case that
the symmetry can be recovered in the unregulated limit. We will see that this is not
the case here. In particular, we will see that the point splitting regularization
procedure introduces an extra term into ~'(N)[{,& }) which in general diverges in
the limit x ---,y. This term is proportional to ( l / e ) f d s Fa~'~a~;~, where e is a measure
of the coordinate distance between x and y. As a consequence, when the action
~ ( N ) [ { y ~ } ) is defined through our point splitting procedure the state is annihilated by ~ ( N ) only if it satisfies the additional condition that each y~ is a geodesic
with respect to the arbitrary coordinate chart q5.
This is an unsatisfactory situation, which bodes ill for any possibility of finding a
subspace of the space of states annihilated by q~(N) which are also annihilated by
the generators of diffeomorphism. In order to proceed, some way must be found to
remove this diffeomorphism noninvariant condition on the space of solutions to the
hamiltonian constraint. We will return to this question at the end of this section.
For the moment, we proceed to give the details of the calculations we have been
describing.
The point split hamiltonian constraint ~ * ( x , y) is defined with respect to two
points, x and y, and it takes the form
8
cg+(x,
y) - ufcU~NF~ff(x) 8A~8(x)
8
8ACD(y) '
(4.4)
where Us is the spinor propagator along the coordinate geodesic (" straight" line in
the coordinate system ~)joining x and y.
We are interested in computing the action of the point split constraint on the
functional U(1, 0) = hA(y),
~ * ( M ~ ) U ( 1 , 0 ) --
Here the biscalar
M~(x, y)
fd3xd3yM~(x, y)~'(x,
y)U(1,0).
(4.5)
is a smearing function, the choice of which completes the
T. Jacobson, L. Smolin / Quantum geometries"
320
definition of the point splitting regularization. We take for this smearing function
3
M,(x, y ) - ~ e 3 0 ( e - I x - y l ) N ( x ) ,
where N(x) is the usual smearing density and I x - y [
between x and y. We then have
lim
M~(x,
y ) = 63(x,
(4.6)
is the coordinate distance
y)N(x),
(4.7)
e~O
so that in the limit we recover the usual constraint ~'(N). However, we stress that
the rate at which this limit is approached is a coordinate dependent notion, and can
have no physical meaning.
We are now ready to compute. In order to simplify many of the formulae which
arise in this and the following section we employ a modified index notation for the
spinor indices on parallel propagators. First, the index associated with the argument
t of U(t, s) is written on the left of U, and that associated with s remains on the
right. Thus, instead of U(t, s)A ~ we write AU(t, s)B. When we do this, only indices
that live in the spin space at the same point end up sitting next to each other.
Second, indices associated with free ends of curves and which are not relevant for
the calculation at hand are suppressed. Using this notation we can write the basic
formula (3.9) for the functional derivative of a parallel propagator as
8U(1,0)
1
-- [ d s
3
8 (x,
7(s));t~(s)U(1, s)(AB)U(s,O)
(4.8)
,o
Here the indices at 0 and 1 are suppressed. Note that the symmetrized pair (AB)
stands in the place of the ~J matrix in (3.9). This makes sense because the "ri are,
after all, nothing but a basis for the three-dimensional space of symmetric two index
spinors at a point.
We find
~ * ( M ~ ) U(1,O)
= folds foldtM~(T(s), ~[(t))F:/V('d(s))'?a(s)~b(t)
× u~Cu~N[O(s -
t)U(1,
S)(AB)U(s, t)(cD)V(t,O) + (S, A, B ~ t, C, D ) ] .
(4.9)
Now note that the first part of the right-hand side of this expression is of order
l / e , since fdtM~ is of order 1/e 2 and we must expand "~b(t) to at least first order
around s if the antisymmetry of Fab is not tO make the whole thing vanish. Thus in
1". Jacobson, L. Smolin / Quantum geometries
321
the rest of the expression we can neglect terms of order e 2. This simplifies things
greatly, since we have
U, = Uv + O(~2),
(4.10)
where the O ( e 2) t e r m is the curvature integrated over the area b o u n d e d by the
curves y and the ~-coordinate geodesic. Using (4.10) in (4.9), the U(s, t) is " z i p p e d
u p " and we get
c~*(M~)U(1,0) =
folds rfU( y( s ) )'~a( s )U(1, S ) ANU( s, O)
x fldtM~(v(s), v ( t ) ) ~ ( t ) [ O ( s -
t) - O ( t -
ao
s)]
+o(~).
Now suppose we expand
~,b(t)
(4.11)
around s,
,;,g(t) = Z
1
A
n=o ~-. ~°~"(,)(t-,)
~.
(4.12)
T a k i n g the definition (4.6) of M e into account, we see that the n t h term of (4.12)
contributes to (4.11) at most to O(en-2), so we need only consider n = 0,1, 2. The
n = 0 term vanishes by antisymmetry of Fab, i.e. for the same reason the unregulated
expression vanishes. The n = 2 term is actually O(e) rather than O(1), due to the
a n t i s y m m e t r i c combination of 0-functions. This is because MF(~,(s), y(t)) effectively
sets the range of t integration to be Is - e3 , s + e6+], where 6 and 8+ are O(1)
and 8 + - 8 = O(e)*. Thus the t-integral for the n = 2 term becomes
fsS2
,t ol,
= o(~).
There remains, however, the n = 1 term, which is O(e-1). Thus, the result is,
c~*(M~)U(1 0) = 1 3 ( 8 2 +
82) fldsU(y(s))F~U(y(s))~a(s):~b(s )
,~
8q7"
× U(1,
S)ANU(s,O) + O(e).
Jo
(4.13)
* We are assuming here and above that, for sufficiently small e, the e-ball around any point on the
curve "r intersects ~/ in a single curve segment. That is, we assume that y is not pathologically wiggly
with respect to the coordinate chart ~.
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T. Jacobson, L. Smolin / Quantum geometries
Therefore cg*(M,)U(1,O) vanishes as e ~ 0 only if the integrand vanishes for all
Aa. As we mentioned above, this happens only if the curve 7(s) is geodesic with
respect to the coordinates q~, since then
aa(s) ~a(s)=f(s)~a(s)
(4.14)
for some function f(s), and the antisymmetry of F,b again takes its toll.
This situation is clearly unsatisfactory. As we argued above, it is not acceptable to
have a regularization procedure which picks out a class of solutions to the hamiltonian constraint in a way which breaks diffeomorphism invariance. We need to find
some way to further modify the point splitting procedure so that the space of
solutions to the regulated hamiltonian constraint does not break diffeomorphism
invariance. If we cannot do this then, given that we know of another regularization
procedure which is satisfactory in this regard, we should give up the point splitting
technique as unsuitable for this context.
We can imagine several ways to improve the point splitting regularization. As we
do not yet know if any of them will work out satisfactorily, we will simply list them
here, suppressing details.
(i) We can introduce an arbitrary affine connection F~,. and use it rather than the
coordinate to transport the spatial index at y back to x in W(x, y). Then in
the expression (4.13) will appear the acceleration with respect to Fff~ instead of the
"coordinate" acceleration. We can then append a last step to the regularization in
which we introduce an average over all connections F/,~e. This is motivated by an
analogy with a problem that comes up in geodesic point splitting, which is that there
can appear subdivergent terms which depend on the direction in which the points
are split. This problem is remedied by introducing an average over directions, which
leaves only covariant terms.
While it is straightforward to introduce a formal integration over connections
which eliminates terms linear in the acceleration aa(s) of a curve, it is not clear that
these formal expressions can be given a more precise meaning. In particular, it
appears likely that any convergent definition of the functional integral involved,
which eliminates these terms, will have to depend on the curve. Thus, the definition
of the regulated operator will depend on the state on which it acts.
(ii) We can imagine making the connection F~c into an operator which makes a
measurement of the curve y(s) and then adjusts itself so that y(s) is geodesic with
respect to it. The problem is how to construct such an operator in a way that will
not, in itself, require regularization.
(iii) One may notice that the expression (4.13) has the same form as the action of
the diffeomorphism constraint on our states. It is easy to show that
1
cd*(M~) U(1,O)
8"/r
fd3xN'%U(1,O)
+ O(e)
(4.15)
T. Jacobson, L. Smolin / Quantum geometries
323
where N " is any smooth vector field which agrees with N ( , / ( s ) ) ~ a on the curve "f. It
is thus conceivable that in a procedure which regulated both the hamiltonian and
d i f f e o m o r p h i s m constraints, and in which simultaneous solutions to both were
sought, such a coordinate dependent term would not survive.
5. Intersections
W e now turn to the study of the action of the hamiltonian constraint on states
involving intersecting curves. We shall see that there are solutions which involve
intersections, and that acting on such solutions, the role of the hamiltonian
constraints is to restrict the ways in which the group indices are traced through the
points of intersection.
In this section we will describe a particular solution, which is indicated by fig. 2.
By putting this in a gauge invariant state, such as the one shown in fig. 3, we see
that the effect of the hamiltonian constraint is to mix states with different topology,
where we include in the meaning of the topology of a state the n u m b e r of traces on
closed loops comprising that state. While the role of time in q u a n t u m gravity is not
entirely clear, the fact that the solutions of the hamiltonian constraint involve
superpositions of states with different topology certainly suggests that in circumstances in which evolution via a hamiltonian can be defined, that hamiltonian
m a y induce transitions between states of different topology. This feature, we might
add, is also seen in the lattice formulation of the theory [27].
= ~ Fig. 2. A schematic drawing of the routings of an intersection state which is annihilated by cg.
Fig. 3. The intersection of fig. 2 imbedded in a gauge mvarimat state, drawn again schematically to
emphasize the routings. One can see that the solution contains a superposition of loop states with
different numbers of loops.
324
T. Jacobson, L. Smolin / Quantum geometries
Fig. 4. Some types of intersections for which no linear combination of the available routings are
annihilated by (~. Thus, we see that the only way a four point intersection can be a solution is if it
contains two pairs of lines, each sharing a common tangent vector at the intersection point.
Fig. 4 shows several configurations for which no linear combinations of routings
yields a solution to the hamiltonian constraint. Thus, the solution indicated in fig. 2
is to some extent unusual. While we have found no other solutions, we have not yet
made a systematic enough study to determine whether this solution is unique.
As with the solutions we described in the previous sections, the existence of a
solution can be conjectured from a naive unregulated calculation, but can only be
demonstrated in the context of a suitable regularization procedure. We will thus
begin with a "naive" discussion of the unregulated states, and then go on to show
that the states exist in the context of the "flux tube" regularization procedure we
discussed previously.
5.1. U N R E G U L A T E D INTERSECTIONS
We consider any state of the form [ { ~,~}) in which an intersection of two smooth
curves, 7(s) and 8(t) takes place at a point p = Y(s0) = 8(t0). The point p could as
well be a self-intersection point of a single curve ~,(s). Let us consider a factor in the
state consisting of just a segment from each of the two curves which includes p. This
segment will be denoted × and written
× = 7(1,0) ® 8 ( 1 , 0 ) ,
(5.1)
Here we use the notation y(1, 0) -= U~(1, 0) and similarly for 8, where the propagator
U is as defined in (3.10). As in the previous section, we are suppressing indices at
the free endpoints of the curves. The symbol ® is needed to distinguish the tensor
product from implicit index summation, which will be employed in later formulae.
Thus, adorned with all of its indices, the state x reads,
X g--By(1,0)A ° 3 ( 1 , 0 ) c .
(5.2)
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T. Jacobson, L. Smolin / Quantum geometries
We shall also be interested in states where the parallel transport is routed
differently through the intersection point. There are two additional routings, namely
(5.3)
> < = ~,(1, So)d(to,O ) ® a(1, to)~,(So,0)
v/k = y(1, s o ) 3 ( t o , 1) ® y(0, s0)6(to,0)
(5.4)
In these and following expressions, the composition of parallel propagators is
written with summation over indices suppressed, e.g. Thus,
(5.5)
3;(1, S o ) 6 ( t o, O) = y(1, So)MMS(to,O ) .
The three states ×, > < and
v
A
are not independent, however. Because of the
identity
(5.6)
together with the fact that the propagators have unit determinant, we have the
relation
X- > <-
V
=0.
A
(5.7)
We are interested in computing the action of the constraint W(N) on these states.
Using the spinor notation, this constraint takes the form
~2
¢g(N) =
f dxN(x)(8EBcg)FN(x)~ A ~ B ( x ) a A C D ( x )
.
(5 8)
This makes explicit the tracing operator, for which we shall need the identity
~(AeB)(C,~D )
lt, A ¢,D B C - - lEAD B C
(M ~
~N) = 20(M ON)E
-~"
~(M~N) "
(5.9)
We first examine the action of ~ ( N ) on the state ×. As "y(s) and 6 ( t ) are
assumed to be differentiable curves, the results of sect. 3 show that, at least
formally, the only non-vanishing contributions occur when one of the functional
derivatives acts on y(s) while the other acts on 6(t). We find
Cg(N) × = ~)o1ds
X2y(1, s)aB~,(s,0 ) ® 6(1, t ) c D 8 ( t , 0 ) .
(5.10)
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T. Jacobson,L. Smolin / Quantumgeometries
Now using the identity (5.9) to contract with the tracing operator the propagator
part of (5.10) becomes
~/(1, s) MNS(t,0) ® 3(1, t)'/(s,O) -- y(1, s)8(t,O) ® 3(1, t)MN'y(s,O)
(5.11)
(the M and N are symmetrized by contraction with F~N.) While this does not
vanish, we shall now show that by forming a linear combination with the state > <
we can find a state annihilated by C~(N).
Acting with ~ ( N ) on the state > < there are now contributions only when both
derivatives act on either > or < , since otherwise the symmetric combination of
vectors ('~"+ ~a)(.~b + ~b) arises and dies when contracted with F,b. We find the
same first part of the integrand as in (5.10), while the propagator part yields
½[Y(x, So),, Bc D8(,o,O) ® 8(1, to) ,(so,O)
s0)8(t0,0) ® 8(1, t0)A .c re(So,0)],
(5.12)
where the minus sign in front of the second term arises because we have switched
the vector indices contracted with Fab in order to agree with the first part of the
integrand in (5.10). The factor of ½ arises because
fs idS fOtOdt83(~[(S),8(t)) ~ ¼f01dsf01dt83(~(S), 8(t)).
(5.13)
Using the delta function 33(-f(s), 3 ( 0 ) to make the replacements s o ~ s and t o ~ t,
and contracting with the tracing operator using the identity (5.9) again, (5.12)
becomes
½[Y(1,S)MN3(t,0 ) ® 6 ( 1 , t ) Y ( s , O ) - Y ( 1 , s ) 3 ( t , O ) ® 6 ( 1 , t ) M u ' / ( s , O ) ] .
(5.14)
N o w this expression is precisely ½ what we found for the state ×. We have thus
shown that, formally,
~(N)(×
- 2 > < ) = 0,
(5.15)
or equivalently, using (5.7),
Cg(N)( V - > / x < ) = 0 .
(5.16)
We say these are only "formally" solutions because the expression (5.10) is
actually singular. In the factor
f01 ds foldt 33(y(s ), 3 ( t ) ) ,
T. Jacobson,L. Smolin / Quantumgeometries
pt~~~
327
[) ~,(s)
Fig. 5. Coordinates of the "flux tube" regularization of the intersection states. P is the point (So, to, Uo).
the three-dimensional delta function is only matched by two dimensions of integration. We can
handle this factor as follows. We introduce a coordinate
system (s, t, u) as shown in fig. 5, such that the curve 7(s) is given by the conditions
t = to, u = u 0, while
is given by s = s 0, u = u 0. The point of intersection is at
p = (s 0, t 0, u0). Then we have, for any function
the formal evaluation
formally
3(0
G(s, t),
foadSfoldt33(y(s),3(t))G(s,t)
=foldSfoldt~(S--So)~(t--to)8(blO--blo)G(s,t)
8(0)
-
J
G(so,
(5.17)
to) ,
where J is the jacobian factor arising from transforming to the coordinates (s, t, u).
We m a y note that this expression is less singular than those encountered in the
naive evaluation of the action of ff at differentiable, nonintersecting points, as it
involves only a one dimensional delta function. This observation will be relevant for
the regularization treated in the following subsection.
Thus, in place of (5.15), it is better to write
Cg(N)(X - 2 >
<)=
3(0)
J
X0.
(5.18)
To summarize, there is a two-dimensional vector space of states associated with
different superpositions of routings at each intersection point. The requirement that
the state be annihilated by the hamiltonian constraint is only satisfied by a single
state which, however, may be written in different ways using the identity (5.7).
5.2. R E G U L A T I O N OF INTERSECTIONS
We now repeat the calculation of the previous subsection using the flux tube
regularization developed in sect. 3. We will see that the intersection states that we
have been discussing are still solutions, provided that the limits involved in
shrinking the flux tubes down to lines are taken in a certain way. As we just
328
T. Jaeobson, L. Smolin / Quantum geometries
mentioned, the action of Cg(N) on the intersecting states is less singular at the
intersection point than at the other points along the curves. In fact, this divergence
can be regularized by smearing the states along a single direction which is linearly
i n d e p e n d e n t from the two tangent directions at the intersection. Of course this will
not eliminate the singularities at the other points which, as we saw in sect. 3, require
smearing in two dimensions transverse to the curve. Our strategy is thus the
following.
Let us work in the coordinate system (s, t, u) introduce earlier (see fig. 5). The
curve y ( s ) is given by t = t o , u = u 0 while 3(0 is given by s = s o , u = u 0. N o w
consider that ~,(s) is imbedded in a family of curves "/(s; t, u) - (s, t, u), one for
each value of t and u. Likewise 3(0 is imbedded in a family 3(t; s, u). We assume
that (s, t, u) is a nonsingular coordinate system in an open neighborhood of the
intersection point p = (s 0, t 0, u0) so that in this neighborhood each of these two
families constitutes a smooth congruence of curves. We will define, as we did in sect.
3, smeared states,
yf(1,0) =
PexP fotdSB/(s ) ,
8gO,O) = PexPfoa
dtBg(t),
(5.19)
(5.20)
where
BY(s) = f dtduf(t, u)A~('y(s; t, u))'?O(s; t, u),
(5.21)
Bg(t) = fdsdug(s, u)A~(8(t; s, u))Sa(t; s, u),
(5.22)
and
with "~ = ( 3y/Os) a, 3~ - ( O3/Ot) ~.
W e m a y n o w define the smeared intersection states,
X f ' s = "t/(1,0) ® 3 g ( 1 , 0 ) ,
> < f'g ~ y/(1,
So)3g(to,O ) ® ag(1,
(5.23)
to)Yf(so,O ) .
(5.24)
N o w let us characterize the spread of f and g in the t and s directions by the
p a r a m e t e r e, and the spread in the u direction by 7- For example, we can take f and
g to be square step functions,
1
f(t,u)~- ~--~e O(e- [t-tol)O(~ 1- [U-Uo]),
g ( s , u) =
1
70(e-
IS-Sol)O(rl- l u - uol).
(5.25)
T. Jacobson, L. Smolin / Quantum geometries
329
At finite e and ~/the expression
<~( N)( X f . g - 2 > < f,g)
(5.26)
is regular but non-zero.
N o w suppose we let e ~ 0 while keeping B fixed. At any point off the line
(So, to, u), e is eventually small enough that the state depends on at most one curve
through that point (rather than one from each of the two congruences). We have
already seen in sect. 3 that in this situation C~(N) annihilates the smeared state, thus
if we first take the limit e --* 0 we may confine our attention to the intersection line
(So, to, u). N o w the contribution to (5.26) from this line vanishes (even at finite e)
for precisely the same reason as in the formal calculation of the previous subsection.
The only difference is that in place of the infinite coefficient 8(0) in (5.18) we will
have something of the form
1
f duj(So, to, u) G(So, 'o, u),
where G(so, to, u) = 0.
In conclusion, we have shown that
lim C~(N)( x / ' g - 2 > < f'g) = 0.
(5.27)
e-~0
Gauge invariance however is recovered only if we also take the limit 71 ~ 0, in which
f and g become delta functions and the state becomes the unsmeared, manifestly
gauge invariant state.
N o t e that it is essential that we take the limits in this order, first e---, 0, then
~/---, 0. If they are taken in the opposite order the result is divergent.
6. Consistency and operator ordering
In the previous three sections we have exhibited a large class of wave functionals
which are annihilated by the hamiltonian constraint, but are not diffeomorphism
invariant. The primary purpose of this section is to raise, and answer the following
question. H o w could the existence of any of these solutions be consistent with the
algebra of constraints, which includes the relation [5]
[Cg(x), Cg(y)] = 3~8'(x --y)~l"b(X)Cgb(X) -- (x ~ y ) ?
(6.1)
The question arises from the following considerations. It follows from the algebra
that if a state Iq~) satisfies
c~(U)14, ) = 0
(6.2)
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T. Jacobson, L. Smolin / Quantum geometries
for all N, then it also satisfies
f O3xMb l+> = 0
(6.3)
for all one-form fields, M~, where ~gb is the operator
(6.4)
b(x) =
The quotes are meant to indicate that the operator is to be ordered in a manner
determined by the ordering of ¢g(N).
This statement is the content of a theorem by Moncrief and Tietelboim [24]. More
precisely, the meaning of the theorem is the following: Given a choice of operator
ordering and a regularization prescription for ~ such that, for some t~b), and for all
N, Cg(N)]~) = 0 in the limit that the regularization is removed, then there exists a
choice of ordering and regularization for the operator Wb, which is a consequence of
those made in the definition of cg, such that (6.3) holds in the limit that the
regularization is taken away.
Now from this theorem one might try to deduce the further conclusion that, since
the metric qab is in general invertible, ~ ( N ) l ~ b ) = 0 for all N implies that the
diffeomorphism constraint must necessarily also be satisfied, i.e.,
~(N~)l~b) --
f d3xNa
l+> = 0
(6.5)
for all vector fields N ~. This conclusion is incorrect, however. Indeed, it is in
contradiction with the existence of the non diffeomorphism invariant solutions of
the hamiltonian constraint described in the previous sections.
The resolution of this apparent contradiction is the following. For the choice of
ordering we used, given by (2.28), we may formally compute the commutator
[Cg(N), ~ ( M ) ] . The ordering this implies, through (6.1), that ~ b ( x ) is given by
1
k
= Cga(X)qab(x),
8
8A(x)]
(6.6)
where in the last line cga is ordered as in (2.29). Note that this is the way Wa is
required to be ordered if it is to generate diffeomorphisms on gauge invariant
functionals of A k (cf. subsect. 2.3).
Now we know that since our states are annihilated by W(N), it follows as an
identity that they are also annihilated by cgb(x) in (6.6). We shall now show
explicitly just how this occurs.
T. Jacobson, L. Smolin / Quantum geometries
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As gauge invariance is not at issue, we shall work here with the smeared
functionals U((1, 0) (defined in (3.19)) of the "flux tube" regularization discussed in
sect. 3. Now, in view of (6.6) we must first determine how q"b(x) acts on our
solutions. We have
1
q~6(x)Uf(l'O)
~
2 ~Ak~(x) 6A~(x) U ( ( I ' 0 )
= (-3/4)[f(x)]2~a(x)'~b(x)Uf(1,O)
(6.7)
in the coordinate system (s; o), with the notation of (3.19). Acting next with cgb we
obtain
Wb( x )q~bU((1, 0) = _ 3 [ f ( x )]3Fkc( X )'~c( x )-~b( x ) ~ a ( x ) U f ( 1 ' X )'rkUf( x, O)
=0.
(6.8)
A similar calculation shows that the operator (6.6) also annihilates the intersection states described in sect. 5. Thus, we see that the existence of the solutions
described in the earlier sections is in fact consistent with the algebra of constraints,
given the ordering and regularization prescriptions we have used. Furthermore, (6.8)
does not imply that U((1,0) is annihilated by Wb(X), since (i) ~ab acts first and (ii)
even if it acted second, we see from (6.7) that it is not invertible on states of the
form U((1, 0).
This resolution raises a disturbing question, however. It is not true that, since the
diffeomorphism constraint ~ga does not act first in (6.6), there are additional,
unwanted constraints imposed on the states? Suppose we had a state [~b) which was
annihilated by all of the constraints. Then (6.6) would imply that, in addition, it
must satisfy
[cga(x), qab(x)][~ ) = 0 .
(6.9)
This is certainly not one of the original constraints. On the other hand, it is
ill-defined, as it involves delta functions and derivatives of delta functions evaluated
at coincident points. The question of whether or not this is a serious problem thus
can be resolved only in the context of a regularization procedure in which meaningless expressions such as (6.9) do not arise.
A reasonable response to this situation is to seek an ordering for the original
constraints in which the above problem does not occur [19]. As was first found by
Ashtekar [1], there is in fact a choice of ordering for cg such that cgb ends up to the
right of ~ab in a formal calculation of the commutator. This ordering, which will be
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T. Jacobson, L. Smolin / Quantum geometries
denoted by a bar, is given by
8
8
ug= e i j k _ _
Fib(x) "
(6.10)
With this choice, the right-hand side of (6.1) is given by
(6.11)
with
8
- 8A~(x) F~c(x)"
(6.12)
Unfortunately, this latter ordering for the diffeomorphism constraints does not
generate infintesimal diffeomorphisms. Instead we have, formally,
~a(X) = ~a(X) + [6bk(X), F~h(X)] = W~(X) + 3v~-h 0,63(x, x).
(6.13)
Thus, given Ashtekar's choice of ordering, the operators on the right-hand side of
(6.1) end up formally in the correct order, but with the diffeomorphism constraint
itself ordered in such a way that it differs from the generator of infintesimal
diffeomorphisms by a divergent constant.
In fact there is no ordering for the constraints written in the self-dual representation that achieves consistency in such a formal calculation. This does not mean that
consistency cannot be achieved in the context of the self-dual representation, but it
does mean that the question can be resolved only in the context of a regularization
procedure which gives meaning to the divergent expressions we have encountered*.
Such a regularization procedure must not only make the action of the constraints on
states well defined, it must eliminate the divergent constants that come from
reorderings of the operators inside of the constraints.
In conclusion, the issues of the consistency of the quantum constraints, and of the
existence of a sufficiently large space of states which are annihilated by all of the
constraints of quantum gravity, cannot be settled here. In future work we will want
to keep two questions in mind. (i) Is there a regulated version of Dirac consistency
which can be achieved in the context of a complete regularization of the theory? (ii)
If not, is there a weaker consistency condition, which still permits the existence of a
sufficiently large space of physical states for the gravitational field?
* The need to regulate the factor-ordering problem has been clearly argued in a recent paper of Tsamis
and W o o d a r d [22]. This paper also contains a useful historical review and critique of attempts to
solve the problem.
T. Jacobson, L. Smolin / Quantum geometries
333
7. Diffeomorphism invariant states
In this section we discuss how to make states in the self-dual representation which
are annihilated by the diffeomorphism as well as the hamiltonian and gauge
constraints. We begin by giving an example of a large class of states which are
annihilated by all of the constraints. These states are formal, in the sense that their
definition involves functional integration. If the construction can be regulated in a
m a n n e r consistent with the constraint algebra, then the functionals we are about to
describe are genuine physical states of the gravitational field.
There is an additional class of states which are, in the asymptotically fiat case,
annihilated by all of the constraints. These are states which are evaluated in the
asymptotic region of the three manifold, and manage to be annihilated by the
integrated constraints because of the fall off conditions necessary in the asymptotically flat case. These "asymptotic states" were originally introduced by Ashtekar,
and are described below.
Following this, we describe several additional ways to construct states in the
self-dual representation which are diffeomorphism and gauge invariant, but are not
necessarily annihilated by the hamiltonian constraint.
7.1. SOME PHYSICAL STATES OF THE GRAVITATIONAL FIELD
We begin by noting that a large class of diffeomorphism invariant states in the
self-dual representation can be constructed by functional transform from diffeomorphism invariant functionals in the metric representation. In the context of the metric
representation it is easy to write lots of diffeomorphism invariant functionals. For
example, we can write any function of an integral of a scalar density constructed
from the three metric, qab, and its curvature tensor, Rabcd.
Let us consider one such state, which we will call XP[qab]. We can, at least
formally, obtain from this a functional of Aia by writing a functional transform such
as
• [A~] =
f[dq]'P[q]x[A,q],
(7.1)
where [dq] is a diffeomorphism invariant measure and x[A, q] is any holomorphic,
gauge-invariant functional of A which is diffeomorphism invariant when both A
and q are transformed. ~[A~] is then holomorphic, and gauge and diffeomorphism
invariant. As an example, one could take x[A, q] = a(fzv/qTrF, bF~aq"Cqbd) with a
any holomorphic function.
Now, in this class of functionals there are some which are also annihilated by the
hamiltonian constraint. To see this, note that
C#(N)•[Ai,] : f[dq]q'[q]~(N)x[A, q].
(7.2)
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334
The state will be annihilated if Cg(N)x[A, q] = 0 for all qab" One way to satisfy this
is if
x [ A , q ] = H(Tq),
(7.3)
where yq is a differentiable loop in ~ determined in a covariant fashion by the
metric q~b"
For example, in the asymptotically flat context, 7q might be specified by its
coordinates ~qa in the harmonic coordinate system xa[q] of q~b" The coordinates
x~[q] (d = 1, 2, 3) are determined by the metric via the covariant Laplace equation,
A qX a = 0, subject to some boundary conditions at infinity. Under a diffeomorphism
of q~b, the harmonic coordinates are transformed and therefore so is the curve. As
another example, one might let yq be determined by the geodesic equation of qab,
subject to some boundary conditions.
To be more explicit, we need to define the measure [dq] and the original
functional "l"[qab]. We are completely free to choose these, as long as they are
diffeomorphism invariant, and define a convergent functional integral. For example,
we might take
1
[dq] ~[ qab] = -~ 1-I dqab( X ) e- stq°~] ,
x
(7.4)
where S[q] is any diffeomorphism invariant functional, and Z is an infinite
normalisation factor. More specifically, we might take S to be of the form
S[ql = fd3x V~[X + aR(q)
+ bR~hRab]
(7.5)
and
f
z=
d q q b ( X ) e -S[q°b] .
(7.6)
Putting these together we have one explicit example of a physical state of the
gravitational field,
•
1
= = ( ] - I dqob(X)e-Slq°~]H(yq) •
ZJ
x
(7.7)
It is interesting to express these states directly in terms of a quantum gravity
theory in three dimensions. Any suitable S defines a functional integration formulation of a theory of quantum gravity in three euclidean dimensions. In this theory,
the vacuum expectation value of any operator O[q] is given by
fl-I~ dqob(X )O[ q ]e - slqob]
(0) =
fFi~dq~b(x)e_S[q~ 1
(7.8)
T. Jacobson, L. Smolin / Quantum geometries
335
We may note that, as the functional integral is not directly defining a quantum
thoery, unitarity is not a concern, and we are free to choose S as we need to so that
the functional integrals are convergent. We then may write physical states of the
gravitational field as
~[Ai~] = ( x [ A , q ] ) ,
(7.9)
with x[A, q], chosen to be (7.3), or any other expression which is annihilated by
C~(N) and fgi for all q~b.
7.2. ASYMPTOTIC PHYSICAL STATES
When asymptotically flat boundary conditions are imposed, there is another class
of physical states of the gravitational field, which exploit the fall off conditions on
the gauge and diffeomorphism constraints which are required by functional differentiability of the integrated constraints. These states have been proposed by
Ashtekar [23, 25], but as they are potentially very useful, we include them here for
completeness. One class of such asymptotic states is the following. Define a two
sphere Sr of asymptotic radius r in the neighborhood of infinity, with ~2 the
coordinates on the sphere. Consider any linear functional of A~,, restricted to the two
sphere and then taken to infinity, as follows,
kO[A] = lim fs d I 2 f ( I 2 ) a i A ( r , ~ ) i a .
r
~
(7.10)
r
For any f(12) ~i this defines a functional of Ai,. We may first note that as the
canonical fall off conditions [2] require that Ai~ fall like 1/r 2 this functional will
exist. Furthermore, it is annihilated by the integrated gauge and diffeomorphism
constraints, fA'f~ ~ and fNac~a, because A ~ and N ~ are required to fall off at least as
1/r. Finally, g'[A] is annihilated by the Hamiltonian constraint because it is a
linear functional of A/.
Another example, which makes use of the loop construction of this paper is the
following. Consider a differentiable, nonintersecting, loop 6(s) = (O(s), CO(s)) on the
coordinate two-sphere. Let us then construct a loop in ~ at constant radial
coordinate, r, by y(s) = (r, O(s), CO(s)). Then, let us consider the functional
ebb[A] =
TrPexpr~dsA~(y(s))'~(s).
(7.11)
This is annihilated by the hamiltonian constraint, as for each r it is based on a
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differentiable loop. We may then define
• ~ [ A ] - = lim ~ [ A ] .
(7.12)
r~oo
This is well defined as A, = O(1/r=) while ~,a increases linearly with r. The limit is
obviously still annihilated by Cg(N). Furthermore, we may note that since F,b =
O ( 1 / r 3) we have
~(Na)~[A]=~dsNa(7(s))Tr[Fab(7(s))Uy(s + 1,s)]'~(s)r
= o(a/r)
(7.13)
even for asymptotic translations such that N" goes to a constant at infinity. Thus,
we have
C~(NO) q)~ [A ] = 0 .
Similarly, for gauge transformations that fall off as
the integrated gauge constraint.
(7.14)
1/r,
this state is annihilated by
7.3. O T H E R D I F F E O M O R P H I S M I N V A R I A N T STATES
We continue our discussion by giving several additional ways in which states
which are diffeomorphism invariant may be constructed in the self-dual representation.
(i) First of all, we might try to construct superpositions of the form
• [A] = f d ~ [ v ] a [ v ] l v ) .
(7.15)
Here, d/~[y] is a measure on loop space, a['~] is a complex function on loop space
which may be thought of as assigning an amplitude to different loops, and 17) is the
state H(~,).
At the formal level, if ~[A] is to be diffeomorphism invariant the loop space
measure d/~[7]a[l'] must be invariant under the action of Diff(X), the group of
diffeomorphisms of IJ. Thus, the basic problem connected with this approach is the
construction of a measure on the loop space of S which is invariant under the
action of Diff(I;). As the latter is an infinite dimensional group this is a nontrivial
problem, especially as there is no natural metric on S itself. At present it is not
known whether any such measure can be constructed at all. However, if one
assumes the existence of such a measure then a rather interesting conclusion follows.
T. Jacobson, L. Smolin / Quantum geometries
337
In this case the physical states of the gravitational field which are of the form (7.15)
will be distinguished by the functions a[7] which are constant under the action of
Diff(N). Now, interestingly enough, the loop space of a three dimensional manifold
is decomposed by the action of Diff(Z) into an infinite number of components,
which are exactly the knot classes of the manifold. Thus, diffeomorphism invariance
implies
a[7] = a[K(7)],
(7.16)
where K ( 7 ) is the knot class of y [26]. Consequently, on the assumption that a
diffeomorphism invariant measure for the loop space exists, we see that there is a
space of physical quantum states of the gravitational field on a three manifold
which is spanned by a basis in one to one correspondence with the knot classes
of 2.
It is not hard to generalize this result to many loop states, with the result that
under the same assumptions, the physical states of the gravitational field are
classified by the link classes of Z. Unfortunately, at the present time it is not at all
clear that measures on the loop space with the required properties exist.
(ii) It is interesting to note that there is a structure on the space of A~'s which
can, for some purposes, play the role of a metric. To see what this is, let us recall
that, from the point of view in which we treat A~ as a Yang-Mills connection, the
usual metric is constructed from the conjugate "electric" field, 6% These three
vector densities make up a (densitised) orthonormal frame field for the metric qobSince, in the self-dual representation, we are interested in structures which depend
on Aio and not o n •ai, we might instead try to define a metric in terms of the
"magnetic" fields of Aio [25].
The magnetic fields are given by
~oi = eObCF/,c,
(7.17)
where eob~. is the Levi-Civita density. We may define a "magnetic metric" hob for
which these/}~i play the role of a (densitized) orthonormal frame field,
h °b - B°iBbi/det B ,
(7.18)
as long as the three magnetic field densities /}ai form a nondegenerate matrix.
For general SL(2, C) connections, hob is complex, whereas when the connection is
restricted to SU(2), hab is positive semi-definite. In any case, integrals of local
scalar densities formed from hob and its curvature tensor provide diffeomorphism
invariant functionals of A~, just as do those of qob in the metric representation.
These functionals are also gauge invariant, since the magnetic metric is itself gauge
invariant. If they are to qualify as wave functionals in the self-dual representation
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T. Jacobson, L. Smolin / Quantum geometries
they must in addition be holomorphic. Although hab and its curvature are holomorphic functions of Ai~, the only way to form a density of weight one is to take the
square root (det/~)1/2 which, unfortunately, is not holomorphic. Thus it seems that
another ingredient is required, for example a background scalar density which is
then functionally integrated over.
One difficulty with all of this is that there will be points in the space of A~'s
where d e t / ~ ' = 0 and therefore the metric hab is singular. This is perhaps not a
problem if the set of A~'s at which hab is singular is of zero measure in the physical
inner product.
(iii) Recall that the state 17) fails to be invariant under diffeomorphisms of A~
because the loop 7 is not a function of the dynamical variable Aia and therefore
remains fixed. If instead y were determined by A~ in some covariant manner, it
would transform along with A~ and the state would be diffeomorphism invariant.
For example, one might define 7[A] as the loop that extremises 17) with respect
to variations of 7 for each A~*. Then the functional
g'[A] = Tr Pexp ~dsAa(7[A])~(s
)
(7.19)
is diffeomorphism invariant. Unfortunately, the resulting functional is now no
longer annihilated by if, due to the additional dependence of 7[A] on Aia . Whether
functionals which are annihilated by all the constraints can be constructed in this
way is thus an open, and nontrivial, problem.
(iv) A fourth approach to the problem of diffeomorphisms is to impose coordinate fixing conditions on the theory before quantization. There are then only the
hamiltonian and gauge constraints to solve. However, these are considerably complicated by the imposition of the coordinate fixing conditions. The challenge of this
approach is to choose a coordinate fixing condition on the phase space (6, A) such
that the resulting forms of fCi and ff can be solved.
8. Discussion
It is by now completely clear that what the sages of the quantum gravity have
been telling us for many years is true: if there is to be a quantum theory of gravity
* Of course such an extremising loop may not exist, or if it does exist it may not be unique. We use it
here just to illustrate the idea. It is interesting to note that the condition, 8/3ya(s)ly) = 0, for y[A]
to be a stationary point of I'Y) --- H ( y ) is equivalent to the equation "~b(s)Tr F~(~,(s))Uy(s + 1, s) = 0
(where Uv(s + 1, s) is the parallel transport around y beginning and ending at the point s). The
question of the existence of a closed curve that satisfies this equation at every s is somewhat like that
of the existence of closed geodesics on a riemannian manifold. Note that a simple class of solutions is
given by loops of unit holonomy, since Tr Fuh --- 0.
T. Jacobson, L. Smolin / Quantum geometries
339
based on the dynamics of general relativity then nonperturbative effects must
dominate the behavior of the theory at short distances, softening the contributions
from fluctuations at Planck scales and shorter. Several interesting proposals as to
how this might occur have been offered [4], but to date none have been supported
by reliable computations.
It has also been clear for many years that the hamiltonian method offers in
principle the best tool for investigating the behavior of quantum general relativity
nonperturbatively, as it does not depend on a semiclassical perturbation expansion.
As we emphasized in the introduction, the fact that the inner product of the exact
theory will depend on the details of the dynamics offers one direct way for
nonperturbative effects to influence the counting of physical states with structure at
scales shorter than the Planck scale*. Unfortunately, in spite of the great promise of
this approach, technical difficulties with the usual, metric, representation have so far
prevented sufficient progress from being made with the hamiltonian approach to
make it possible to test the conjecture that general relativity has, after all, a sensible
quantum theory.
We have shown in this paper that some of these obstacles are lifted in the
self-dual representation based on Ashtekar's new variables. We have exhibited a
large class of states which are annihilated by the hamiltonian constraint, and have
also given several different techniques whereby diffeomorphism invariant states may
be constructed in this representation. We have even found it possible to give an
explicit, although at this stage formal, construction of a large class of physical states,
i.e. states annihilated by all of the constraints.
In order to put this work in perspective, we may recall the four steps of canonical
quantization outlined in the introduction and, with these in mind, ask how much
remains to be done and what should be the plan of attack in the near future.
(i) Choice of the representation space. The discussion of sect. 2 establishes the
self-dual representation as a satisfactory representation space for quantum gravity.
In particular, the discussion leading to eq. (2.27) determines the form of the
operator for 6 ai up to terms that can, at least locally in the space of connections, be
removed by multiplying all wavefunctionals by a common phase factor. Thus, the
only ambiguities which remain in the definition of the self-dual representation
concern the possibility that the representation space might be topologically nontrivial. It is in fact possible that such effects may be present, and important, and they
are currently under study [18].
Reality conditions of the classical theory are to become hermiticy conditions with
respect to the quantum inner product, and as such they have not yet been imposed
(see item 4 below).
* This could happen only if the perturbative and nonperturbative Hilbert spaces are not unitarily
equivalent.
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T. Jacobson, L. Smolin / Quantum geometries
(ii) Regularization and operator ordering of the constraints. In this paper we have
studied a particular ordering of the hamiltonian constraint, which is the simplest
from the point of view of regularization, as the functional derivatives do not act on
the F j / ( x ) in the constraint itself. We described two different regularization
methods which are sufficient for defining the action of the hamiltonian constraint,
giving this ordering, on some states in the self-dual representation.
We showed in sect. 6 that although our states are consistent with the algebra of
the constraints in our form, this algebra does not satisfy the Dirac consistency
condtion [19]. It fails to close by some singular operators which are not well defined
in the context of our regularization procedure. In order to make sense of this
situation, a more complete regularization procedure is required.
One proposal for a complete regularization of the self-dual representation is to
put the theory on a lattice [27] ~'. On the lattice there are operators which represent
regularized versions of the constraints of quantum gravity. Their algebra is presently
under investigation, and results will be reported in the near future [28]. In this
connection it is encouraging to note that a somewhat similar problem, a lattice
transcription of the Virasoro algebra, has recently been solved [29].
(iii) The physical state space. We have exhibited a large space of solutions to the
hamiltonian constraint and, if the functional integrals involved in subsect. 7.1 can
be suitably defined, a large subspace of these solutions which are physical states.
The simple loop states 13') are not diffeomorphism invariant because the loop 3' is
a fixed structure. It is interesting to note however that, by virtue of reparametrization invariance of the holonomy functional, IT) is invariant under the subgroup of
diffeomorphisms that map the image of T into itself.
Of course full diffeomorphism invariance is what is required. In sect. 7 we
described several diffeomorphism covariant structures which exist in the self-dual
representation and which might be useful in constructing physical states.
In this paper the spatial diffeomorphisms have been treated only at a formal level.
If canonical quantum gravity is to be studied at the nonperturbative level we need
to know a great deal about the representation theory of the three dimensional
diffeomorphism group. A measure of how much remains to be learned about this
subject, and how valuable it could be for physics, can be obtained by recalling the
important role that the representation theory of the diffeomorphism group of the
circle plays in string theory. Indeed, it is not much of an exaggeration to say that
much of what were considered "miracles" and "surprises" in the theory of strings
have been found to be consequences of the highly nontrivial structure of the
representations of the diffeomorphism group of the circle, and its central extensions
[30]. There is no reason to expect the representation theory of the three dimensional
diffeomorphism group to be any less intricate than that of the circle. Indeed, the
~' It was in fact several results of the lattice formulation which initially suggested the investigation of
states based on holonomy elements.
T. Jacobson, L. Smolin / Quantum geometries
341
results of this paper could be taken as a hint that this representation theory will
involve both the loop space and the knot invariants in a fundamental way.
(iv) The physical inner product. We have not touched the problem of the inner
product in this paper. It is essential for imposing the reality conditions inherited
from the classical theory and for the probability interpretation of the quantum
theory.
It is instructive to consider the way the inner product is handled in the old
covariant quantization of the relativistic string [31]. The constraint algebra there is
the Virasoro algebra. Rather than waiting until the space of physical states is
obtained, and then imposing an inner product on the physical subspace, one starts
by constructing a Fock space of string oscillators. An inner product on this space is
implicitly defined in the following way. One defines the norm of the ground state to
be unity, and one assumes that x~(o) and p~(o), whose classical counterparts are
real, are hermitian operators. The commutation relations then determine the inner
products of all Fock space states. This implicitly defined inner product is not
positive definite, however the Virasoro constraints then limit one to a physical
subspace of positive norm states (together with spurious zero norm states).
Now we cannot just do something similar in the self-dual representation of full,
non-linear general relativity because the variables AZ~ and A~a do not have oscillator
commutation relations. One can, of course, always construct a conjugate pair of
oscillator variables, however the constraints will be terribly complicated operators in
terms of these, so it will become hopeless to try and find the physical subspace. A
more concrete approach therefore seems to be needed. That is, given the form of our
operators in the self-dual representation (subsect. 2.2) one should look for an inner
product which enforces the classical reality conditions.
Even if such an inner product is found, it will be necessary somehow to divide out
by the volume of the gauge and diffeomorphism groups in order to obtain finite
norms for physical states. This would be particularly simple in the lattice regularization of the theory. Another possibility is to gauge fix. This means solving the
constraints for non-dynamical variables and substituting in the hamiltonian constraint. Although this would be proceeding in an entirely different direction than we
have pursued in this paper, it is certainly an important avenue to investigate, and
would remain closer to conventional techniques in field theory.
Given the fact that we are at this stage lacking essential elements required for a
complete physical interpretation of the formalism so far developed, it would be
useful to invent heuristic or model calcualtions, which might capture some of the
structure of quantum gravity in the self-dual representation. Such models might also
lead to systematic approximation techniques. We will end by indicating some
possibilities for this kind of work.
One very promising development of this kind is the study of the strong-coupling
limit in terms of the new variables recently carried out by several people [32].
Because of the simplicity of the formalism, there is also some hope that a strong-
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coupling expansion could also be developed [33]. This would allow us for the first
time to systematically study the nature of the vacuum state of quantum gravity at
sub-Planck scales.
Another approach is to attempt to impose, either implicitly or explicitly, an inner
product on some "minisuperspace", i.e. on a limited class of wave functionals.
Intuition about the interpretation of the formalism might then be developed along
these lines.
Yet another idea involves an attempt to give the solutions we have described in
this paper a semiclassical interpretation. At first sight this seems a surprising thing
to do, as the solutions are one dimensional they are singular in terms of any smooth
three dimensional structure. However, it is interesting, and perhaps important, to
note that Ashtekar's formulation of general relativity allows such singular structures
at the classical level. An example of such structures is in fact given by some of the
loop states which, as we will now show, may also be interpreted as Hamilton
principal functions and, as such, lead to an infinite class of initial data which exactly
solve the hamiltonian (but not the diffeomorphism) constraint.
In the new variables, the Hamilton-Jacobi equations take the form
8S
8S
eijkr~b(X) 8Ai (x ) 8AJb(X)
O,
(8.1)
8S
Fib(x) 8A~(x)
O,
(8.2)
8S
2aSA"x'~t) - 0 '
(8.3)
corresponding respectively to the hamiltonian, diffeomorphism and gauge constraints. Here S = S[Aia] is the Hamilton principal functional.
It is easy to verify that the regulated function
S = Y'~Hf(7.)
(8.4)
solves the first Hamilton-Jacobi equation, as long as the congruences of loops 7~ are
non-intersecting.
Such an S represents an infinite family of complex classical geometries. For every
A / there is a 5 ~i which has the form
6"i(x) =SS/SAi~(x) = ~f(x)'~"(x)Tr.c'Uv[,(s + 1,s).
(8.5)
Together the A~ and this 5 ai solve the classical hamiltonian constraint. Of course,
T. Jacobson, L. Smolin / Quantum geometries
343
they solve neither the three classical diffeomorphism constraints, nor the gauge
constraints, and so do not provide good initial data for the Einstein equations. (The
gauge invariance can, of course, be restored by using for S instead the gauge
invariant holonomy elements that arise in the limit f ( o ) ~ 62(0), although the 6 ai
associated with this S is then singular.
Perhaps the beginnings of the quantum interpretation of the geometry of the loop
states follows from the fact that, as shown by equation (6.7), the loop states without
intersections are eigenstates of the operator ~ b ( x ) , with degenerate eigenvalues
proportional ?~(x)gb(x), or zero if ~, does not pass through x. It would be very
interesting to see if the infinite superpositions of loop states necessary to construct
diffeomorphism invariant functionals could cure the problem of the degeneracy and
singular nature of the eigenvalues of the metric.
A general question raised by these thoughts is whether the classical non-degeneracy of 8 ai should necessarily be enforced in the quantum theory. Since Ashtekar's
formalism makes perfect sense with degenerate 6 ai it is actually an extension of
general relativity [25]. The question is, might this extension be physically significant?
In summary, quantum general relativity may or may not be a sensible theory. A
great deal remains to be done to decide this question. We believe that we have
shown, at least, that the self-dual representation of the theory reveals that the space
of solutions to the constraints has nonperturbative structure which was missed by
analyses based both on perturbation theory and on the usual metric representation
of the canonical theory. This structure suggests that quantum geometry at Planck
scales might be much simpler when explored in terms of the parallel transport of
left-handed spinors than when explored in terms of the three metric. Whether this
structure is in fact significant for the question of the existence of a nontrivial
quantization of general relativity is a question that will only be decided by further
work.
We would like, first of all, to thank Abhay Ashtekar for the encouragement, help
and criticism he has given us over the last year and a half. Paul Renteln has
followed every stage of this work, and made several valuable suggestions. We are, in
addition, grateful to David Boulware, Louis Crane, Carlos Kazomeh, Karel KuchaE
Jorge de Lyra, Vincent Moncrief, Ted Newman, Andrew Strominger and Richard
Woodard for comments, criticisms and discussions which have much improved our
understanding and presentation of this work.
We would also like to thank Professor R. Schrieffer and the organizers of the
workshop "Approaches to quantum gravity" at the Institute for Theoretical Physics,
in the stimulating atmosphere of which this work was begun. In addition this work
was supported by NSF grants PHY 82-01094 at Brandeis and PHY 85-03072 at
Yale, PHY 85-06686 and PHY 82-17853, supplemented by funds from NASA, to
UC Santa Barbara and a grant from the Research Corporation to LS.
344
T. Jacobson, L. Smolin / Quantum geometries
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