The Erwin Schr odinger International Boltzmanngasse
ESI Institute for Mathematical Physics A Wien Austria
Quantum Theory of Geometry I
Area Op erators
Abhay Ashtekar
Jerzy Lewandowski
Vienna Preprint ESI August
Supp orted by Federal Ministry of Science and Research Austria
Available via http www esi ac at
Quantum Theory of Geometry I Area Op erators
Abhay Ashtekar and Jerzy Lewandowski
August
Center for Gravitational Physics and Geometry
Physics Department Penn State University Park PA U S A
Institute of Theoretical Physics
Warsaw University ul Hoza Warsaw Poland
Max Planck Institut fur Gravitationphysik
Schlaatzweg Potsdam Germany
Abstract
A new functional calculus develop ed recently for a fully non p erturbative
treatment of quantum gravity is used to b egin a systematic construction of
a quantum theory of geometry Regulated op erators corresp onding to areas
of surfaces are intro duced and shown to b e self adjoint on the underlying
kinematical Hilb ert space of states It is shown that their sp ectra are purely
discrete indicating that the underlying quantum geometry is far from what
the continuum picture might suggest Indeed the fundamental excitations
of quantum geometry are dimensional rather like p olymers and the
dimensional continuum geometry emerges only on coarse graining The full
Hilb ert space admits an orthonormal decomp osition into nite dimensional
sub spaces which can b e interpreted as the spaces of states of spin systems
Using this prop erty the complete sp ectrum of the area op erators is evaluated
The general framework constructed here will b e used in a subsequent pap er
to discuss dimensional geometric op erators e g the ones corresp onding to
volumes of regions
It is a pleasure to dedicate this article to Professor Andrzej Trautman who was one of the
rst to recognize the deep relation b etween geometry and the physics of gauge elds which
lies at the heart of this investigation
Intro duction
In his celebrated inaugural address Riemann suggested that geometry of space
may b e more than just a ducial mathematical entity serving as a passive stage
for physical phenomena and may in fact have a direct physical meaning in its own
right General relativity proved this vision to b e correct Einstein s equations put
geometry on the same fo oting as matter Now the physics of this century has
shown us that matter has constituents and the dimensional ob jects we p erceive
as solids in fact have a discrete underlying structure The continuum description
of matter is an approximation which succeeds brilliantly in the macroscopic regime
but fails hop elessly at the atomic scale It is therefore natural to ask if the same
is true of geometry Do es geometry also have constituents at the Planck scale
What are its atoms Its elementary excitations Is the space time continuum only a
coarse grained approximation If so what is the nature of the underlying quantum
geometry
To prob e such issues it is natural to lo ok for hints in the pro cedures that have
b een successful in describing matter Let us b egin by asking what we mean by
quantization of physical quantities Let us take a simple example the hydrogen
atom In this case the answer is clear while the basic observables energy and
angular momentum take on a continuous range of values classically in quantum
mechanics their sp ectra are discrete So we can ask if the same is true of geometry
Classical geometrical observables such as areas of surfaces and volumes of regions can
take on continuous values on the phase space of general relativity Are the sp ectra
of corresp onding quantum op erators discrete If so we would say that geometry is
quantized
Thus it is rather easy to p ose the basic questions in a precise fashion Indeed
they could have b een formulated so on after the advent of quantum mechanics An
swering them on the other hand has proved to b e surprisingly di cult The main
reason it seems is the inadequacy of the standard techniques More precisely the
traditional approach to quantum eld theory has b een p erturbative where one begins
with a continuum background geometry It is then di cult to see how discreteness
would arise in the sp ectra of geometric op erators To analyze such issues one needs
a fully non p erturbative approach geometric op erators have to b e constructed ab
initio without assuming any background geometry To prob e the nature of quantum
geometry we can not b egin by assuming the validity of the continuum picture We
must let quantum gravity itself decide whether this picture is adequate at the Planck
scale the theory itself should lead us to the correct microscopic picture of geometry
In this pap er we will use the non p erturbative canonical approach to quantum
gravity based on connections to prob e these issues Over the past three years this
approach has b een put on a rm mathematical fo oting through the development
of a new functional calculus on the space of gauge equivalent connections
This calculus do es not use any background elds such as a metric and is therefore
well suited to a fully non p erturbative treatment The purp ose of this pap er is to
use this framework to explore the nature of quantum geometry
In section we recall the relevant results from the new functional calculus and
outline the general strategy In section we present a regularization of the area
op erator Its prop erties are discussed in section in particular we exhibit its
entire sp ectrum Our analysis is carried out in the connection representation and
the discussion is self contained However at a non technical level there is a close
similarity b etween the basic ideas used here and those used in discussions based
on the lo op representation Indeed the development of the functional
calculus which underlies this analysis itself was motivated in a large measure by
the pioneering work on lo op representation by Rovelli and Smolin The relation
b etween various approaches will discussed in section
The main result of this pap er should have rami cations on the statistical me
chanical origin of the entropy of black holes along the lines of This issue is
b eing investigated
Preliminaries
This section is divided into three parts In the rst we will recall the basic
structure of the quantum con guration space and in the second that of the Hilb ert
space of kinematic quantum states The overall strategy will b e summarized
in the third part
Quantum con guration space
In general relativity one can regard the space A G of SU connections mo dulo
gauge transformations on a spatial manifold as the classical con guration
space For systems with only a nite numb er of degrees of freedom
the classical con guration space also serves as the domain space of quantum wave
functions i e as the quantum con guration space For systems with an in nite
numb er of degrees of freedom on the other hand this is not true generically the
quantum con guration space is an enlargement of the classical In free eld theory in
Minkowski space as well as exactly solvable mo dels in low space time dimensions
for example while the classical con guration space can b e built from suitably smo oth
elds the quantum con guration space includes all temp ered distributions This
is an imp ortant p oint b ecause typically the classical con guration spaces are of zero
measure wave functions with supp ort only on smo oth con gurations have zero norm
The overall situation is the same in general relativity The quantum con guration
A G is a certain completion of A G space
A G inherits the quotient structure of A G i e A G is the quotient The space
of the space A of generalized connections by the space G of generalized gauge trans
formations To see the nature of the generalization involved recall rst that each
R
A smo oth connection de nes a holonomy along paths in h A P exp
p
p
Generalized connections capture this notion That is each A in A can b e de ned
as a map which assigns to each oriented path p in an element A p of SU
such that i A p A p and ii A p p A p A p where p is
obtained from p by simply reversing the orientation p p denotes the comp osition
of the two paths obtained by connecting the end of p with the b eginning of p and
A p A p is the comp osition in SU A generalized gauge transformation is a
map g which assigns to each p oint v of an SU element g x in an arbitrary
p ossibly discontinuous fashion It acts on A in the exp ected manner at the end
p oints of paths A p g v A p g v where v and v are resp ectively the
b eginning and the end p oint of p If A happ ens to b e a smo oth connections say A
we have A p h A However in general A p can not b e expressed as a path
p
ordered exp onential of a smo oth form with values in the Lie algebra of SU
Similarly in general a generalized gauge transformation can not b e represented by
a smo oth group valued function on
A G and A G seem to o large to b e mathematically At rst sight the spaces
controllable However they admit three characterizations which enables one to in
tro duce di erential and integral calculus on them We will conclude this
sub section by summarizing the characterization as suitable limits of the corre
sp onding spaces in lattice gauge theory which will b e most useful for the main
b o dy of this pap er
We b egin with some de nitions
An edge is an oriented dimensional sub manifold of with two b oundary
p oints called vertices which is analytic everywhere including the vertices A graph
in is a collection of edges such that if two distinct edges meet they do so only at
vertices
In the physics terminology one can think of a graph as a oating lattice i e
a lattice whose edges are not required to b e rectangular Indeed they may even b e
non trivially knotted Using the standard ideas from lattice gauge theory we can
construct the con guration space asso ciated with the graph Thus we have the
space A each element A of which assigns to every edge in an element of SU
and the space G each element g of which assigns to each vertex in an element
of SU Thus if N is the numb er of edges in and V the numb er of vertices
N V
A is isomorphic with SU and G with SU G has the obvious action
on A A e g v A e g v The gauge invariant con guration space
A G and A G can asso ciated with the oating lattice is just A G The spaces
b e obtained as well de ned pro jective limits of the spaces A G and A G
Note however that this limit is not the usual continuum limit of a lattice gauge
For technical reasons we will assume that all paths are analytic An extension of the framework
to allow for smo oth paths is b eing carried out The general exp ectation is that the main results
will admit natural generalizations to the smo oth category In this article A has the physical
dimensions of a connection length and is thus related to the con guration variable A in the
old
literature by A GA where G is Newton s constant
old
theory in which one lets the edge length go to zero Here we are already in the
continuum and have available to us all p ossible oating lattices from the b eginning
We are just expressing the quantum con guration space of the continuum theory
as a suitable limit of the con guration spaces of theories asso ciated with all these
lattices
A G is a sp eci c extension of To summarize the quantum con guration space
the classical con guration space A G Quantum states can b e expressed as complex
A G or equivalently as G invariant square valued square integrable functions on
integrable functions on A As in Minkowskian eld theories while A G is dense
A G top ologically measure theoretically it is generally sparse typically A G is in
contained in a subset set of zero measure of A G Consequently what matters is
the value of wave functions on genuinely generalized connections In contrast with
the usual Minkowskian situation however A G and A G are all compact spaces in
their natural Gel fand top ologies This fact simpli es a numb er of technical
issues
Our construction can b e compared with the general framework of second quanti
zation prop osed by Kijowski already twenty years ago He intro duced the space
of states for a eld theory by using the pro jective limit of spaces of states asso ciated
to a family of nite dimensional theories He also suggested as an example the
lattice approach The common element with the present approach is that in our case
A is also the pro jective limit of the spaces of measures the space of measures on
de ned on nite dimensional spaces A
Hilb ert space
Since A G is compact it admits regular Borel normalized measures and for every
such measure we can construct a Hilb ert space of square integrable functions Thus
to construct the Hilb ert space of quantum states we need to select a sp eci c measure
A G on
o
A admits a measure that is preferred by b oth mathemat It turns out that
o
ical and physical considerations Mathematically the measure is natural
b ecause its de nition do es not involve intro duction of any additional structure it
is induced on A by the Haar measure on SU More precisely since A is iso
N o
morphic to SU the Haar measure on SU induces on it a measure in
the obvious fashion As we vary we obtain a family of measures which turn out
o
A to b e compatible in an appropriate sense and therefore induce a measure on
This measure has the following attractive prop erties i it is faithful i e for any
R
o
A d f equality holding if and only continuous non negative function f on
if f is identically zero and ii it is invariant under the induced action of Di
o o
the di eomorphism group of Finally induces a natural measure on A G
o o
A to A G is simply the push forward of under the pro jection map that sends
o
Physically the measure is selected by the so called reality conditions More
precisely the classical phase space admits an over complete set of naturally de ned
con guration and momentum variables which are real and the requirement that the
corresp onding op erators on the quantum Hilb ert space b e self adjoint selects for us
o
the measure
o o
A G d as our Hilb ert space Elements Thus it is natural to use H L
o
of H are the kinematic states we are yet to imp ose quantum constraints Thus
o
H is the classical analog of the ful l phase space of quantum gravity prior to the
intro duction of the constraint sub manifold Note that these quantum states can
A In fact since the spaces un b e regarded also as gauge invariant functions on
o
der consideration are compact and measures normalized we can regard H as the
o o
gauge invariant sub space of the Hilb ert space H L A d of square integrable
functions on A In what fol lows we we wil l often do so
What do typical quantum states lo ok like To provide an intuitive picture we
can pro ceed as follows Fix a graph with N edges and consider functions of
generalized connections of the form A A e A e for some smo oth
N
N
function on SU where e e are the edges of the graph Thus the
N
functions know ab out what the generalized connections do only to those paths
which constitute the edges of the graph they are precisely the quantum states
of the gauge theory asso ciated with the oating lattice This space of states
although in nite dimensional is quite small in the sense that it corresp onds to
the Hilb ert space asso ciated with a system with only a nite numb er of degrees
of freedom However if we vary through all p ossible graphs the collection of
all states that results is very large Indeed one can show that it is dense in the
o
Hilb ert space H If we restrict ourselves to which are gauge invariant we
o
obtain a dense sub space in H Since each of these states dep ends only on a nite
numb er of variables b orrowing the terminology from the quantum theory of free
elds in Minkowski space they are called cylindrical functions and denoted by Cyl
Gauge invariant cylindrical functions represent the typical kinematic states In
R of smo oth functions of compact many ways Cyl is analogous to the space C
o
supp ort on R which is dense in the Hilb ert space L R d x of quantum mechanics
Just as one often de nes quantum op erators e g the p osition the momentum and
rst and then extends them to an appropriately larger the Hamiltonians on C
o
domain in the Hilb ert space L R d x we will de ne our op erators rst on Cyl
and then extend them appropriately
Cylindrical functions provide considerable intuition ab out the nature of quan
tum states we are led to consider These states represent dimensional p olymer
like excitations of geometry gravity rather than dimensional wavy undulations
on at space Just as a p olymer although intrinsically dimensional exhibits
dimensional prop erties in su ciently complex and densely packed con gurations
the fundamental dimensional excitations of geometry can b e packed appropriately
to provide a geometry which when coarse grained on scales much larger than the
Planck length lead us to continuum geometries Thus in this descrip
tion gravitons can arise only as approximate notions in the low energy regime
o
At the basic level states in H are fundamentally di erent from the Fo ck states
of Minkowskian quantum eld theories The main reason is the underlying di eo
morphism invariance In absence of a background geometry it is not p ossible to
intro duce the familiar Gaussian measures and asso ciated Fo ck spaces
Statement of the problem
We can now outline the general strategy that will b e followed in sections and
i
on Recall that the classical con guration variable is an SU connection A
a
a manifold where i is the su internal index with resp ect to a basis Its
i
b
conjugate momentum E has the geometrical interpretation of an orthonormal triad
j
with density weight one the precise Poisson brackets b eing
i b b i
fA x E y g G x y
a j a j
where G is Newton s constant Recall from fo otnote that the eld A used here
is related to A used in the literature via A GA
old old
Therefore geometrical observables functionals of the metric can b e expressed
a
in terms of this eld E Fix within the manifold any analytic nite surface
i
S without b oundary such that the closure of S in is a compact The area A of
S
S is a well de ned real valued function on the ful l phase space of general relativity
a
which happ ens to dep end only on E It is easy to verify that these kinematical
i
observables can b e expressed as
Z
1
i
2
E dx dx E A
S
i
S
where for simplicity we have used adapted co ordinates such that S is given by
x and x x parameterize S and where the internal index i is raised by a the
inner pro duct we use on su k Tr
i j i j
Our task is to nd the corresp onding op erators on the kinematical Hilb ert space
o
H and investigate their prop erties
There are several factors that make this task di cult Intuitively one would ex
a i
p ect that E x to b e replaced by the op erator valued distribution ihG A x
i a
Unfortunately the classical expression of A involves square roots of products of E s
S
and hence the formal expression of the corresp onding op erator is badly divergent
One must intro duce a suitable regularization scheme Unfortunately we do not
have at our disp osal the usual machinery of Minkowskian eld theories and even the
precise rules that are to underlie such a regularization are not apriori clear
There are however certain basic exp ectations that we can use as guidelines i
o
the resulting op erators should b e well de ned on a dense sub space of H ii their
We assume that the underlying manifold is orientable Hence principal SU bundles over
are all top ologically trivial Therefore we can represent the SU connections on the bundle
by a su valued form on The matrices are anti Hermitian given e g by i times
i
the Pauli matrices
nal expressions should b e di eomorphism covariant and hence in particular inde
p endent of any background elds that may b e used in the intermediate steps of the
regularization pro cedure and iii since the classical observables are real valued the
op erators should b e self adjoint These exp ectations seem to b e formidable at rst
Indeed these demands are rarely met even in Minkowskian eld theories in presence
of interactions it is extremely di cult to establish rigorously that physically inter
esting op erators are well de ned and self adjoint As we will see the reason why
one can succeed in the present case is two folds First the requirement of di eo
morphism covariance is a p owerful restriction that severely limits the p ossibilities
Second the background indep endent functional calculus is extremely well suited for
the problem and enables one to circumvent the various road blo cks in subtle ways
Our general strategy will b e following We will de ne the regulated versions of
area and volume op erators on the dense sub space Cyl of cylindrical functions and
show that they are essentially self adjoint i e admit unique self adjoint extensions
o
to H This task is further simpli ed b ecause the op erators leave each sub space H
spanned by cylindrical functions asso ciated with any one graph invariant This in
e ect reduces the eld theory problem i e one with an in nite numb er of degrees of
freedom to a quantum mechanics problem in which there are only a nite numb er
of degrees of freedom Finally we will nd that the op erators in fact leave invariant
o
certain nite dimensional sub space of H asso ciated with extended spin networks
intro duced in Sec This p owerful simpli cation further reduces the task of
investigating the prop erties of these op erators in e ect the quantum mechanical
problem in which the Hilb ert space is still in nite dimensional is further simpli ed
to a problem involving spin systems where the Hilb ert space is nite dimensional
It is b ecause of these simpli cations that a complete analysis is p ossible
Regularization
Our task is to construct a well de ned op erator A starting from the classical ex
S
pression As is usual in quantum eld theory we will b egin with the formal
expression obtained by replacing E in by the corresp onding op erator valued
i
distribution E and then regulate it to obtain the required A For an early dis
S
i
cussion of non p erturbative regularization see in particular Our discussion
will b e divided in to two parts In the rst we intro duce the basic to ols and in the
second we apply them to obtain a well de ned op erator A
S
To simplify the presentation let us rst assume that S is covered by a single
chart of adapted co ordinates Extension to the general case is straightforward
one mimics the pro cedure used to de ne the integral of a di erential form over a
manifold That is one takes advantage of the co ordinates invariance of the the
resulting lo cal op erator and uses a partition of unity
To ols
The regularization pro cedure involves two main ingredients We will b egin by sum
marizing them
The rst involves smearing of the op erator analog of E x and p oint splitting
i
of the integrand in Since in this integrand the p oint x lies on the surface S
let us try to use a dimensional smearing function Let f x y b e a parameter
family of elds on S which tend to the x y as tends to zero i e such that
Z
lim d y f x x y y g y y g x x
S
for all smo oth densities g of weight and of compact supp ort on S Thus f x y
x will is a density of weight in x and a function in y The smeared version of E
i
b e de ned to b e
Z
x E y d y f x y E
f
i i
S
so that as tends to zero E tends to E x The p oint splitting strategy now
f
i i
provides a regularized expression of area
Z Z Z
1
i
2
d z f x z E z d y f x y E y d x A
S f
i
S S S
Z
1
i
2
x E x d x E
f f
i
S
which will serve as the p oint of departure in the next subsection To simplify techni
calities we will assume that the smearing eld f x y has the following additional
prop erties for su ciently small i for any given y f x y has compact sup
p ort in x which shrinks uniformly to y and ii f x y is non negative These
conditions are very mild and we are thus left with a large class of regulators
We now intro duce the second ingredient To go over to the quantum theory
i
However it is not apriori iGh A in by E we want to replace E
i i
is a well de ned op erator b ecause i our wave clear that even after smearing E
f
i
functions are functionals of generalized connections A whence it is not obvious
what the functional derivative means and ii we have smeared the op erator only
along two dimensions Let us discuss these p oints one by one
A First let us x a graph and consider a cylindrical function on
A A e A e
N
where as b efore N is the total numb er of edges of and where is a smo oth
N
function on SU Now a key fact ab out generalized connections is that for
For example f x y can b e constructed as follows Take any non negative function f of
R
compact supp ort on S such that d xf x and set f x y f x y Here we
have implicitly used the given chart to give f x y a density weight in x
any given graph each A is equivalent to some smo oth connection A Given
any A there exists an A such that
Z
A A e h A P exp
k k
e
k
for all k N For any given A the smo oth connection A is of course not
unique However this ambiguity do es not a ect the considerations that follow
A and Hence there is a corresp ondence b etween the cylindrical function on
function h A h A on the space A of smo oth connections and we can apply
E
the op erator E to the latter The result is
f
i
Z
N
X
h
I
3
d y f x y E x A iGh j A
f
y
i
i
y A h
S
I
a
I
Z Z
N
X
dt e t y e t y e t e t i d y f x y
I
I I I P
S
I
A
i
A h t h t
I I
B
A
h
I
B
p
Gh is the Planck length the index I lab els the edges in the graph where
P
R
0
t
A e s t e t is any parameterization of an edge e h t t P exp
a I I I I
t
a
s ds is the holonomy of the connection A along the edge e from parameter value e
I
I
t to t Thus the functional derivative has a well de ned action on cylindrical func
tions the rst of the two problems mentioned ab ove has b een overcome
However b ecause of the presence of the delta distributions it is still not clear
is a genuine op erator rather than a distribution valued op erator To that E
f
i
explicitly see that it is we need to sp ecify some further details Given a graph we
can just sub divide some of its its edges and thus obtain a graph which o ccupies
the same p oints in as but has trivially more vertices and edges Every function
which is cylindrical with resp ect to the smaller graph is obviously cylindrical
with resp ect to the larger graph as well The idea is to use this freedom to
simplify the discussion by imp osing some conditions on our graph We will assume
that i if an edge e contains a segment which lies in S then it lies entirely in the
I
closure of S ii each isolated intersection of with the surface S is a vertex of
and iii each edge e of intersects S at most once
I
The overlapping edges are often called edges tangential to S they should not
b e confused with edges which cross S but whose tangent vector at the intersection
p oint is tangent to S If the given graph do es not satisfy one or more of these
conditions we can obtain one which do es simply by sub dividing of some of the
edges Thus these conditions are not restrictive They are intro duced to simplify
the b o ok keeping in calculations
Let us now return to If an edge e has no p oint in common with S it do es
I
not contribute to the sum If it is contained in S e vanishes identically whence its
I
contribution also vanishes For a subtlety see the remark b elow Eq We are
thus left with edges which intersect S in isolated p oints Let us rst consider only
those edges which are outgoing at the intersection Then at the intersection p oint
is p ositive negative the value of the parameter t is zero and for a given edge e e
I I
if e is directed upwards along increasing x downwards along decreasing x
I
Hence b ecomes
Z
N
h
X
i
A
i P
d y f x y y e y e h E x
I I f
I I i
B
A
h
S
I
B
I
N
X
i
P
i
f x e L A e A e
I I N
I
I
where the constant asso ciated with the edge e is given by
I I
if e is tangential to S or do es not intersect S
I
if e has an isolated intersection with S and lies ab ove S
I I
if e has an isolated intersection with S and lies b elow S
I
i
is the left invariant vector eld in the i th internal direction on the and where L
I
copy of SU corresp onding to the I th edge
i A i
A e A e A e L
N I
B I
A
A e
I
B
If some of the edges are incoming at the intersection p oint then the nal expression
a
of E x can b e written as
f
i
N
h i
X
i
P
i
E x f x v X A e A e
f I N
i I
I
I
i
is an op erator assigned to a vertex v and an edge e intersecting v by the where X
I
I
following formula
i A
when e is outgoing A e
A I I
B
A e
I i
B
A e A e X
N
I
A i
when e is incoming A e
A I I
B
A e
I
B
Remark Let us brie y return to the edges which are tangential to S In this case
although e vanishes we also have a singular term in the x direction in
I
Hence to recover an unambiguous answer for these edges we need to smear
0
also in the third direction using an additional regulator say g x y When this
is done one nds that the contribution of the tangential edges vanishes even b efore
removing the regulator as stated earlier the tangential edges do not contribute
We did not intro duce the smearing in the third direction right in the b eginning to
R
1
In the rst step we have used the regularization dz g z z g which follows if the
z is obtained in the standard fashion as a limit of functions which are symmetric ab out
emphasize the p oint that this step is unnecessary for the edges whose contributions
survive in the end
The right side again de nes a cylindrical function based on the same graph
o o
Denote by H the Hilb ert space L A d of square integrable cylindrical
o
is the induced Haar measure on functions asso ciated with a xed graph Since
A and since the op erator is just a sum of right left invariant vector elds standard
results in analysis imply that with domain Cyl of all C cylindrical functions
o
based on it is an essentially self adjoint on H Now it is straightforward to
o
verify that the op erators on H obtained by varying are all compatible in the
appropriate sense Hence it follows from the general results in that E x with
f
i
domain Cyl the space of all C cylindrical functions is an essentially self adjoint
o
op erator on H For notational simplicity we will denote its self adjoint extension
also by E x The context should make it clear whether we are referring to the
f
i
essentially self adjoint op erator or its extension
The fact that this op erator is well de ned may seem surprising at rst sight since
we have used only a dimensional smearing Recall however that in free eld theory
in Minkowski space the action of the momentum op erator on cylindrical functions
is well de ned in the same sense without any smearing at all In our case a
dimensional smearing is needed b ecause our states contain one rather than three
dimensional excitations
Area op erators
Let us now turn to the integrand of the smeared area op erator corresp onding to
Denoting the determinant of the intrinsic metric on S by g we have
S
i
x E x g x E
f f S f
i
X
P
i i
X X f x v I J f x v
J I J I
I J
where the summation go es over all the oriented pairs I J v and v are the
I J
vertices at which edges e and e intersect S I J equals if either of
I J I J
the two edges e and e fails to intersect S or lies entirely in S if they lie on the
I J
same side of S and if they lie on the opp osite sides For notational simplicity
from now on we shall not keep track of the p osition of the internal indices i as
noted in Sec they are contracted using the invariant metric on the Lie algebra
su The next step is to consider vertices v at which intersects S and simply
rewrite the ab ove sum by re grouping terms by vertices The result simpli es if we
0 0 0
Given two graphs and we say that if and only if every edge of can b e written
0
as a comp osition of edges of Given two such graphs there is a pro jection map from A to A
o o
0
into H A family of op erators of H which via pull back provides an unitary emb edding U
0
0 0 0 0 0
D D and U O U O O on the Hilb ert spaces H o is said to b e compatible if U
0
for all g g
cho ose su ciently small so that f x v f x v is zero unless v v We
I J I J
then have
X X
P
i i
I J X X f x v g x
S f
I J
I J
where the index lab els the vertices on S and I and J lab el the edges at the
vertex
The next step is to take the square ro ot of this expression The same reasoning
that established the self adjointness of E x now implies that g x is a non
f S f
i
negative self adjoint op erator and hence has a well de ned square ro ot which is also
a p ositive de nite self adjoint op erator Since we have chosen to b e su ciently
small for any given p oint x in S f x v is non zero for at most one vertex v
We can therefore take the sum over outside the square ro ot One then obtains
X X
1
1
P
i i
2
2
I J X X f x v g x
S f
I J
I J
Note that the op erator is neatly split the x dep endence all resides in f and the
op erator within the square ro ot is internal in the sense that it acts only on copies
of SU
Finally we can remove the regulator i e take the limit as tends to zero By
integrating b oth sides against test functions on S and then taking the limit we
conclude that the following equality holds in the distributional sense
X X
1
p
P
i i
d
2
I J X X x v g x
S
I J
I J
Hence the regularized area op erator is given by
X X
1
P
i i
2
A I J X X
S
I J
I J
Here as b efore lab els the vertices at which intersects S and I lab els the edges
of at the vertex v With Cyl as its domain A is essentially self adjoint on the
S
o
Hilb ert space H
Let us now remove the assumption that the surface is covered by a single chart
of adapted co ordinates If such a global chart do es not exist we can cover with a
family U of neighb orho o ds such that for each U U there exists a lo cal co ordinates
a
system x adapted to Let b e a partition of unity asso ciated to U
U U U
We just rep eat the ab ove regularization for a slightly mo di ed classical surface area
functional namely for
Z
1
i
2
dx dx E E A
U S U
i
S
which has supp ort within a domain U of an adapted chart Thus we obtain the
op erator A Then we just de ne
SU
X
A A
S S U
U U
The result is given again by the formula The reason why the functions
U
disapp ear from the result is that the op erator obtained for a single domain of an
adapted chart is insensitive on changes of this chart This concludes our technical
discussion
The classical expression A of is a rather complicated It is therefore
S
somewhat surprising that the corresp onding quantum op erators can b e constructed
rigorously and have quite manageable expressions The essential reason is the un
derlying di eomorphism invariance which severely restricts the p ossible op erators
Given a surface and a graph the only di eomorphism invariant entities are the
intersection vertices Thus a di eomorphism covariant op erator can only involve
structure at these vertices In our case it just acts on the copies of SU asso ciated
with various edges at these vertices
We have presented this derivation in considerable detail to sp ell out all the
assumptions to bring out the generality of the pro cedure and to illustrate how reg
ularization can b e carried out in a fully non p erturbative treatment While one is
free to intro duce auxiliary structures such as preferred charts or background elds
in the intermediate steps the nal result must resp ect the underlying di eomor
phism invariance of the theory These basic ideas will b e used rep eatedly for other
geometric op erators in the sequel to this pap er
General prop erties of op erators
Discreteness of the spectrum By insp ection it follows that the total area op erator
A leaves the sub space of Cyl which is asso ciated with any one graph invariant
S
o o
of H corresp onding to Next and is a self adjoint op erator on the sub space H
o o
recall that H L A d where A is a compact manifold isomorphic with
N
SU where N is the total numb er of edges in As we explained b elow the
o
is given by certain commuting elliptic di erential op erators restriction of A to H
S
on this compact manifold Therefore all its eigenvalues are discrete Now supp ose
o
that the complete sp ectrum of A on H has a continuous part Denote by P the
S c
o o
asso ciated pro jector Then given any in H P is orthogonal to H for any
c
graph and hence to the space Cyl of cylindrical functions Now since Cyl is
o o
dense in H P must vanish for all in H Hence the sp ectrum of A has no
c S
continuous part
Note that this metho d is rather general It can b e used to show that any self
o o
adjoint op erator on H which maps the intersection of its domain with H to
o o
H and whose action on H is given by elliptic di erential op erators has a purely
o
discrete sp ectrum on H Geometrical op erators constructed purely from the triad
eld tend to satisfy these prop erties
Area element Note that not only is the total area op erator well de ned
p
d
g which is an op erator valued but in fact it arises from a lo cal area element
S
distribution in the usual sense Thus if we integrate it against test functions the
o
op erator is densely de ned on H with C cylindrical functions as domain and the
matrix elements
p
d
0
g x i h
S
are dimensional distributions on S Furthermore since we did not have to renor
malize the regularized op erator b efore removing the regulator there are no
free renormalization constants involved The lo cal op erator is completely unambigu
ous
g versus its square root Although the regulated op erator g is well
S f s f
de ned if we let to go zero the resulting op erator is in fact divergent roughly it
would lead to the square of the dimensional distribution Thus the determinant
of the metric is not a well de ned in the quantum theory As we saw however the
square ro ot of the determinant is wel l de ned We have to rst take the square ro ot
of the regulated expression and then remove the regulator This in e ect is the
essence of the regularization pro cedure
To get around this divergence ofg as is common in Minkowskian eld theories
S
we could have rst rescaled g by an appropriate factor and then taken the limit
S f
Then result can b e a well de ned op erator but it will dep end on the choice of
the regulator i e additional structure intro duced in the pro cedure Indeed if the
resulting op erator is to have the same density character as its classical analog g x
S
which is a scalar density of weight two then the op erator can not resp ect the
underlying di eomorphism invariance For there is no metric chart indep endent
distribution on S of density weight two Hence such a renormalized op erator is not
useful to a fully non p erturbative approach For the square ro ot on the other hand
we need a lo cal density of weight one And the dimensional Dirac distribution
provides this now is no apriori obstruction for a satisfactory op erator corresp onding
to the area element to exist This is an illustration of what app ears to b e typical in
non p erturbative approaches to quantum gravity Either the limit of the op erator
exists as the regulator is removed without the need of renormalization or it inherits
back ground dep endent renormalization elds rather than constants
Vertex operators As noted already in the nal expressions of the area element
and area op erators there is a clean separation b etween the x dep endent and the
internal parts Given a graph the internal part is a sum of square ro ots of the
If on the other hand for some reason we are willing to allow the limiting op erator to have a
di erent density character than its classical analog one can renormalize g x in such a way as
f
0
to obtain a background indep endent limit For instance we may use f jx x j
and rescale g by b efore taking the limit Then the limit is a well de ned di eomorphism
f
covariant op erator but it is a scalar density of weight one rather than two
op erators
X
i i
I J X X
S v
I J
I J
asso ciated with the surface S and the vertex v on it It is straightforward to check
that op erators corresp onding to di erent vertices commute Therefore to analyze
the prop erties of area op erators we can fo cus just on one vertex op erator at a time
Furthermore given the surface S and a p oint v on it we can de ne an op erator
o
on the dense sub space Cyl on H as follows
S v
P
i i
I J X X if intersects S in v
I J
I J
S v
Otherwise
where I and J lab el the edges of which have v as a vertex Recall that every
cylindrical function is asso ciated with some graph As b efore if intersects S at
v but v is not a vertex of one can extend just by adding a new vertex v and
orienting the edges at v to outgoing It is straightforward to verify that this de
nition is unambiguous if a cylindrical function can b e represented in two di erent
0 0
are two representations of then and ways say as and
S v S v
A There is a precise sense in which can b e regarded the same function on
S v
o
as a Laplacian op erator on H The area op erator is a sum over all the p oints v of
S of square ro ots of Laplacians
q
X
P
A
S v S
v S
Here the sum is well de ned b ecause for any cylindrical function it contains only
a nite numb er of non zero terms corresp onding to the isolated intersection p oints
of the asso ciated graph with S We will see in the next subsection that this fact is
re ected in its sp ectrum
p
Gauge invariance The classical area element g is invariant under the inter
S
a
nal rotations of triads E its Poisson bracket with the Gauss constraint functional
i
vanishes This symmetry is preserved in the quantum theory the quantum op erator
p p
o
d d
g commutes with the induced action of G on the Hilb ert space H Thus g
S S
and the total area op erator A map the space of gauge invariant states to itself
S
o
they pro ject down to the Hilb ert space H of kinematic states
Note however that the regulated triad op erators E are not gauge invariant
f
i
o
they are de ned only on H Nonetheless they are useful they feature in an im
p ortant way in our regularization scheme In the lo op representation by contrast
one can only intro duce gauge invariant op erators and hence the regulated triad op
erators do not exist Furthermore even in the de nition of the regularized
area element one must use holonomies to transp ort indices b etween the two p oints
y and z While this manifest gauge invariance is conceptually pleasing in practice
it often makes the calculations in the lo op representation cumb ersome one has to
keep track of these holonomy insertions in the intermediate steps although they do
not contribute to the nal result
Overal l Factors The overall numerical factors in the expressions of various
op erators considered ab ove dep end on two conventions The rst is the convention
noted in fo otnote used in the regularization pro cedure Could we not have used a
R R
di erent convention setting dz g z z cg and dz g z z c g
for some constant c The answer is in the negative For in this case the
constant would take values
I
if e is tangential to S or do es not intersect S
I
c if e has an isolated intersection with S and lies ab ove S
I
I
q if e has an isolated intersection with S and lies b elow S
I
It then follows that unless c the action of the area op erator A on a given
S
cylindrical function would change if we simply reverse the orientation on S keeping
the orientation on the same Since this is physically inadmissible we must have
c there is really no freedom in this part of the regularization pro cedure
The second convention has to do with the overall numerical factor in the action
which dictates the numerical co e cients in the symplectic structure Here we have
adopted the convention of see chapter which makes the Poisson bracket
i a i b b i
x x as iGh A x y enabling us to express E y g G x E fA
a i j a j a
Had we rescaled the action by as is sometimes done in our expressions
Newton s constant G would b e replaced by G
Eigenvalues and Eigenvectors
This section is divided into three parts In the rst we derive the complete sp ectrum
of the area op erators in the second we extend the notion of spin networks and in
the third we use this extension to discuss eigenvectors
The complete sp ectrum
We are now ready to calculate the complete sp ectrum of A Since A is a sum
S S
of square ro ots of vertex op erators which all commute with one another the task
reduces to that of nding the sp ectrum of each vertex op erator Furthermore since
vertex op erators map C cylindrical functions asso ciated with any one graph to
C cylindrical functions asso ciated with the same graph we can b egin with an
arbitrary but xed graph Consider then a vertex op erator and fo cus on the
S v
edges of which intersect S at v Let us divide the edges in to three categories
let e e lie b elow S down e e lie ab ove S up and let e e
d d u u t
b e tangential to S As b efore the lab els down and up do not have an invariant
signi cance the orientation of S and of enable us to divide the non tangential
edges in to two parts and we just lab el one as down and the other as up Let us
set
i i
d u
i i
i i
J i X X J i X X
S v d S v d u
i i i i
u t d u d
i i
J J i X X J J
S v S v u t S v S v
i
where X is the op erator de ned in assigned to the p oint v and an edge e
I
I
at v This notation is suggestive We can asso ciate with each edge e a particle with
i
only a spin degree of freedom Then the op erators iX can b e thought of as the
e
ith comp onent of angular momentum op erators asso ciated with that particle and
i i i
t u d
as the total down up and tangential angular momentum and J J J
S v S v S v
op erators at the vertex v
By varying the graph we thus obtain a family of op erators It is easy to check
i i i
u d t
that they satisfy the compatibility conditions and thus de ne op erators J J J
S v S v S v
i
d u
and J on Cyl It is also easy to verify that they all commute with one another
S v
Hence one can express the vertex op erator simply as
S v
i i i i
u d u d
J J J J
S v
S v S v S v S v
b ecause of the factor I J in the edges which are tangential do not feature
in this expression
The evaluation of p ossible eigenvalues is now straightforward It is simplest to
express as
S v
d u d u
J J J
S v
S v S v S v
and as in elementary text b o oks go to the representation in which the op erators
d u d u
J J and J are diagonal If we restrict now the op erators to Cyl
S v S v S v
asso ciated to a xed graph it is obvious that the p ossible eigenvalues of are
S v
given by
d d u u d u d u
j j j j j j
S v
d u d u
where j j and j are half integers sub ject to the usual condition
d u d u d u d u
j fjj j j jj j j j j g
Returning to the total area op erator we note that the vertex op erators asso ciated
with distinct vertices commute Although the sum is not nite restricted to
any graph and Cyl it b ecomes nite Therefore the eigenvalues a of A are
S S
given by
1
i h
X
2
P
d u d u u u d d
a j j j j j j
S
where lab els a nite set of p oints in S and the non negative half integers assigned
to each are sub ject to the inequality The question now is if all these
eigenvalues are actually attained i e if given any a of the form there are
S
o
eigenvectors in H with that eigenvalue In Sec will show that the full sp ectrum
o
is indeed realized on H
o o
The area op erators map the subspace H of gauge invariant elements of H to
o
itself Hence we can ask for their sp ectrum on H We will see in Sec that
further restrictions can now arise dep ending on the top ology of the surface S There
are three cases
i The case when S is an op en surface whose closure is contained in An
example is provided by the disk z x y r in R In this case there
o
is no additional condition all a of sub ject to are realized
S
ii The case when the surface S is closed S and divides into disjoint
op en sets and i e S with An example
is given by R and S S In this case there is an a condition on the
d u
half integers j and j that app ear in in addition to
X X
d u
j N and j N
for some integers N and N
iii The case when S is closed but not of typ e ii An example is given by
S S S and S S S In this case the additional condition is
milder
X
d u
j j N
for some interger N
Next let us note some prop erties of this sp ectrum of A By insp ection it is clear
S
that the smallest eigenvalue is and that the sp ectrum is unb ounded from ab ove
One can ask for the area gap i e the value of the smallest non zero eigenvalue On
o
the full Hilb ert space H it is given by
p
o
a
P S
This is a sp ecial case of the situation when there is only one term in the sum in
d u d u
with j j j j Then
q
P
j j a
S
o o
On the Hilb ert space H of we obtain the eigenvalue a and if we cho ose j
S
gauge invariant states on the other hand b ecause of the constraints on the sp ectrum
discussed ab ove the area gap is sensitive to the top ology of S
p
o
if S is of typ e i a
P S
p
o
a if S is of typ e ii
S
o
if S is of typ e iii a
P S
Another imp ortant feature of the sp ectrum is its b ehavior for large a As noted
S
ab ove the sp ectrum is discrete However an interesting question is if it approaches
continuum and if so in what manner We will now show that as a the
S
di erence a b etween a and its closest eigenvalue satis es the inequality
S S
p
a a O a
S P S S
P P P
and hence tends to zero irresp ective of the top ology of S Sp eci cally given o dd
p
N we will obtain an eigenvalue a integers M and N satisfying M
S N M
of A such that for su ciently large N the b ound is explicitly realized
S
Let us lab el representations of SU by their dimension n j Let n
P
M N
j Then n N and jn M b e o dd integers such that
M
for each M we have from an eigenvalue a
S N M
M
q
X
P
j a j
S N M a
M
X
P
n O
n N
M k M
P
N O
N N N
p
N a for some integer k M As M varies b etween and varies
S N M
p
N Hence N N k N N and b etween
P P P
given a su ciently large a there exist integers N M satisfying the conditions given
S
ab ove such that a ja a j satis es the inequality
S S n m
We will conclude this discussion of the sp ectrum by providing an alternative
form of the expression which holds for gauge invariant states This form will
b e useful in comparing our result with those obtained in the lo op representation
where from the b eginning one restricts oneself to gauge invariant states Let
A Then the Gauss constraint implies b e a gauge invariant cylindrical function on
that at every vertex v of the following condition must hold
X
i
X
I
I
i
where I lab els the edges of at the vertex v and X is assigned to the p oint v and
I
vertex e see Therefore
I
i i i
d u t
J J J
S v S v S v
This calculation was motivated by the results of Bekenstein and Mukhanov and our esti
mate has an interesting implication on whether the Hawking sp ectrum is signi cantly altered due
to quantum gravity e ects Because the level spacing a go es to zero as a go es to in nity the
S S
considerations of do not apply to large black holes in our approach and there is no reason to
exp ect deviations from Hawking s semi classical results On the other hand for small black holes
i e the nal stages of evap oration the estimate do es not apply and one exp ects transitions
b etween area eigenstates to show signi cant deviations
Hence one can now express the op erator in an alternate form
d u t
J J J
S v
S v S v S v
Furthermore if it happ ens that has no edges which are tangential to S at v
implies
d u
J J
S v
S v S v
q
P
j j whence the corresp onding restricted eigenvalues of A are given by
S
P
where j are half integers
Extended spin networks
As a prelude to the discussion on eigenvectors in this sub section we will generalize
the constructions and results obtained in on spin networks and spin
network states The previous work showed that the spin network states provide us
o
with a natural orthogonal decomp osition of the Hilb ert space H of gauge invariant
states in to nite dimensional sub spaces Here we will extend those results to the
o
space H
We b egin by xing some terminology Given N irreducible representations
m m
k +1 N
of SU an asso ciated invariant tensor c is a multi linear
N m m
1
k
N N
k N
map from to such that
I I
I I k
n n n m m m n n
1
k +1 N k k +1 N N k +1
c g g c g g
n n N m m N k
1 1
k m n k m m
1
k N k +1
for arbitrary g SU where g is the matrix representing g in the represen
I
m m
k +1 N
tation An invariant tensor c is also called an intertwining tensor
I m m
1
k
from the representations into All the invariant tensors are
k k N
given by the standard Clebsch Gordon theory
An extended spin network is an quintuplet c M consisting of
i A graph
ii A lab eling of the edges e e of that graph with irre
N N
ducible and non trivial representations of SU
iii A lab eling of the vertices v v of with irreducible repre
V V
sentations of SU the constraint b eing that for every vertex v the represen
tation emerges in the decomp osition of the tensor pro duct of representations
assigned by to the edges intersecting v
iv A lab eling c c c of the vertices v v of with certain invariant
V V
tensors namely assigned to a vertex v is an intertwining tensor c from the
representations assigned to the edges coming to v and to the representa
tions assigned to the outgoing edges at v and
v A lab eling M M M M of the vertices v v of
V V V
which assigns to every vertex v a vector M in the representation
It should b e emphasized that every is necessarily non trivial whereas may
I
b e trivial i e dimensional In the gauge invariant context are all
trivial whence the items iii and v are unnecessary The details of these conditions
may seem somewhat complicated but they are necessary to achieve the orthogonal
decomp osition
o
From spin networks we can construct states in H An extended spin network
A constructed from an extended state N is simply a C cylindrical function on
c M
spin network c M
V N
O O
V
M c A e N A
I I
c M
I
for all A A where as b efore A e is an element of G asso ciated with an edge
I
e and stands for contracting at each vertex v of the upp er indices of the
I
matrices corresp onding to all the incoming edges the lower indices of the matrices
assigned to all the outgoing edges and the upp er index of the vector M with all
the corresp onding indices of c We skip and in the symb ol for the extended
spin network function b ecause the intertwiners c contain this information Thus
for example in the simple case when the network has only two vertices and all
edges originate at the rst vertex and end at the second N can b e written out
c M
explicitly as
0 0
m m
m m n n
1
N
1 2 1
N
0
c M M c A e N A e
0
n n m N N
1 2 m m
m N
c M
1
1 N
where indices m n range over j and m ranges over j Given
I I I
A which is square integrable with any spin network provides a function on
o
A the resp ect to the measure Given an extended spin network function on
the range R of the asso ciated graph is completely determined Thus two spin
networks can de ne the same function on A if one can b e obtained from the other
by sub dividing edges and changing arbitrarily the orientations
It turns out that the spin network states provide a decomp osition of the full
o
Hilb ert space H into nite dimensional orthogonal sub spaces compare with
Given a triplet de ned by i iii ab ove consider the vector space
H spanned by the spin network functions N given by all the p ossible choices
c M
for c M compatible with xed lab elings Note that according to the repre
sentation theory of compact groups every H is a nite dimensional irreducible
representation of G in Cyl The group acts there via
g v M A M Ag N N g
0
c M c M
Mo dulo the obvious completions we have the following orthogonal decomp osition
M
o
H H
R
where given a graph the lab elings and range over all the data de ned ab ove by
i iii whereas for in the sum we take exactly one representative from every range
of an analytic graph in When is trivial we skip in H On H the action
G is trivial and we have the following orthogonal of the gauge transformations group
decomp osition of the Hilb ert space of gauge invariant cylindrical functions
M
o
H H
R
where we used the same conventions as in Thus we recover the result on
spin network states obtained in
We conclude this sub section with a general comment on spin network states
Consider tri valent graphs i e graphs each vertex of which has three or less
edges In this case the standard Clebsch Gordon theory implies that the the num
b er of asso ciated gauge invariant spin network functions is severely limited the
o
corresp onding sub space of H is one dimensional Hence on the sub space Cyl
o
of H corresp onding only to tri valent graphs the normalized spin network states
provide a natural orthonormal basis What is remarkable is that these spin networks
were rst intro duced by Penrose already twenty ve years ago to prob e the mi
croscopic structure of geometry although in a di erent context Because of the
simplicity and other attractive prop erties of these Penrose spin network states it
is tempting to hop e that they might su ce also in the present approach to quantum
gravity Indeed there were conjectures that the higher valent graphs are physically
redundant However it turns out that detailed physical considerations rule out this
p ossibility quantum gravity seems to need graphs with unlimited complexity
Eigenvectors
We are now ready to exhibit eigenvectors of the op erators and A for any of
S v S
the p otential eigenvalues found in section We will b egin with the full non gauge
o o
invariant Hilb ert space H and consider an arbitrary surface S Since H serves as
the gravitational part of the kinematical Hilb ert space in theories in which gravity
is coupled to spinor elds our construction is relevant to that case In the second
o
part of this sub section we will turn to the gauge invariant Hilb ert space H and
exhibit eigenvectors for the restricted range of eigenvalues presented in section
Fix a p oint v in the surface S We will investigate the action of the op erators
d u d u
J J J and on extended spin network states Without loss of
S v
S v S v S v
generality we can restrict ourselves to graphs which are adapted to S and contain
v as a vertex say v v Given a graph and lab eling and of its edges and
vertices by representations of SU we shall denote by C the linear space of the
v
intertwining tensors which are compatible with and at v in the sense of section
Let c M b e an extended spin network and N b e the corresp onding
c M
state As one can see from Eqs each of the four op erators ab ove is given
by a linear combination with constant co e cients of gauge invariant terms of the
i
i
1 E
form b X where b is a constant tensor and all the X s are asso ciated X
i i i i
1 1
E E I I
1
E
with the p oint v and the edges which meet there On N the action of any
c M
op erator of this typ e reduces to a linear op erator o acting in C More precisely if
v v
O is any of the ab ove op erators we have
ON N
0
c M c M
where N is again an extended spin network state and the network c M
0
c M
di ers from the rst one only in one entry of the lab eling c corresp onding to the
vertex v c c for all the vertices v v and c o c Consequently the
v
problem of diagonalizing these op erators reduces to that of diagonalizing a nite
symmetric matrix of o Note that a constant vector M assigned to v do es not play
v
any role in this action and hence will just make eigenvectors degenerate
d u d u
In the case of op erators J J and J the simultaneous eigen
S v S v S v
states are given by the group representation theory We can now sp ell out the
general construction
Let us x a graph and arrange the edges that meet at v into three classes
as b efore e e e e e e Let us also x a lab eling of
d d u u t t
these edges by irreducible non trivial representations of SU and an irreducible
p ossibly trivial representation which emerges in the decomp osition of
t
Consider now the following ingredients
a Irreducible representations and
d u d u
00
0
m m m
u
d+1
m m m m
1
d
0 00
c b Invariant tensors c and c asso ciated resp ec
d u d m m
u
tively to the representations and to and nally
d d u
d u
to and
d u d u
m m m
t
u+1
asso ciated to c Invariant tensor c
u t
d u t
n
From this structure construct the following invariant tensor
0 00
m m m m m m m m m n m m n
t u t
u+1 1 1
d d+1
0 00
c c c c c
t n d u d u m m
asso ciated with the representations To obtain a non trivial result in
t
the end we need all the tensors to b e non zero The existence of such tensors is
equivalent to the following two conditions on the data a c
d The representations and emerge resp ectively in and
d
d u
and
d u
e the representation emerges b oth in and
u t
d u d u
Finally intro duce an extended spin network c M such that
c c c c
t N V V
the remaining entries b eing arbitrary Then the corresp onding state N is
c M
d u d u
an eigenvector of the op erators J J and J with the eigenvalues
S v S v S v
d d u u d u d u
j j j j and j j resp ectively where the half integers
d u d u
j j and j corresp ond to the representations and Hence
d u d u
this N is also an eigenvector of with the eigenvalue It is obvious
S v
c M
that for any triple of representations and satisfying the constraint
d u d u
there exists an extended spin network
This construction provides al l eigenvectors of The key reason b ehind this
S v
completeness is that given any choice of and as ab ove the
d u t
invariant tensors which can b e written in the form with any and
d u
span the entire space C of invariant tensors at v compatible with that data
v
d u
Since the de ning formula for a spin network function is linear with resp ect
to every comp onent of c given any spin network c M it su ces to decom
p ose the comp onent c of c at v v into invariant tensors of the form in
any manner to obtain a decomp osition of the corresp onding spin network function
into a linear combination of extended spin network functions given by
o
The desired result now follows from the orthogonal decomp osition of H in to the
extended spin network subspaces
Let us now turn to the op erator A A basis of eigen vectors can b e obtained in
S
the following way Since the area op erator can b e expressed in terms of and com
d u u d
at any p oint v in S we can simultaneously and J J mutes with J
S v S v S v
diagonalize all these op erators Because for every graph the area op erator preserves
the subspace of spin network states asso ciated with that graph and for two di erent
graphs the spin network spaces spaces are orthogonal it is enough to lo ok for eigen
vectors for an arbitrary graph Given a graph lab elings and M as in
section at every vertex v contained in the surface S cho ose a basis in the space
C consisting of invariant tensors of the form The set of the spin network
v
functions constructed by varying M and picking at each vertex v an
o
element of the basis in C constitutes a basis in H If we restrict the lab elings to
v
consisting only of the trivial representations then the resulting set of spin network
o
states provide a basis for the space H of gauge invariant functions Each of such
states is automatically an eigenvector of A with eigenvalue
S
We conclude the rst part of this sub section with a simple example of an eigen
vector of the area op erator with eigenvalue a where a is any real numb er satisfying
S S
d u u d
W is a nite set of triples j j Example Supp ose j
of half integers which for every satisfy Rather than rep eating the
construction a e ab ove step by step we will sp ecify only the simplest of
the resulting extended spin networks In S cho ose W distinct p oints v
W To every p oint v assign two nite analytic curves e and e
d u
starting at v not intersecting S otherwise and going in opp osite directions of
S For a graph take the graph fe e e e g the vertices b eing the
d u d W u W
intersection p oints v and the ends of the edges e and e the curves b eing
d u
chosen such that the p oints v are the only intersections Lab el each edge e
d
d
with the irreducible representation corresp onding to a given j and every
d
u
edge e with the irreducible representation de ned by j That de nes
u u
a lab eling of The absence of edges e is equivalent to intro ducing these
t
edges in any manner and assigning to them the trivial representations To
de ne a lab eling at the vertices v assign to every vertex v a representation
d u
de ned by a given j Next to each vertex v assign an invariant tensor
m m m
u
d
c asso ciated to the triple of representations intro duced
d u
ab ove The construction of a spin network is completed by i lab eling that
end p oint of each e and resp ectively of e which is not contained in
d u
S with the representation and resp ectively ii
d d u u
lab eling of these ends of the edges with the unique invariants corresp onding to
the representations or resp ectively to iii de ning
d d u u
a lab eling M of vertices which can b e chosen arbitrarily provided at a vertex
v the asso ciated vector M b elongs to the representation and at an
d u
endp oint of either of the edges e the asso ciated M b elongs to
d u d u d u
o
As we noted in section the Hilb ert space H is the quantum analog of the full
phase space Now in the classical theory the imp osition of the Gauss constraint on
the phase space do es not restrict the allowed values of the functional A of
S
It is therefore of interest to see if this feature p ersists in the quantum theory Is
o
the sp ectrum of A on the full H the same as that on its gauge invariant sub
S
o
space H As was indicated in Sec the answer is in the a rmative only if the
surface is op en If S is closed there are restrictions on the sp ectrum which dep end
on top ological prop erties of S emb edded in The second part of this section is
devoted to this issue As indicated in Sec we need to consider three separate
cases
Case i S and S
We will mo dify the spin network of the ab ove Example in such a way as to
obtain a gauge invariant eigenstate without changing the eigen value of the area
op erator Let and the lab eling b e the ones de ned in the Example To each
vertex v assign one more edge e b eginning in v and contained in S Lab el
t
d u
it by the representation corresp onding to a given j at that p oint The
t
lab eling is now taken to b e trivial To every p oint v assign as in the Example
an invariant tensor c asso ciated now to the representations Every
d u t
extension of this data to a spin network will de ne a spin network state which is
gauge invariant at each of the p oints v Now we need to de ne a closed spin
network which contains all the edges e e e and provides an extension for
d u t
the lab elings already intro duced For this we use a key prop erty of the area op erator
asso ciated to a surface with b oundary vertices which lie on S do not contribute to
the action of the op erator Therefore we can simply extend every edge e within S
t
to the b oundary of S Denote the intersection p oint with S by v Next for every
t
we extend in a piecewise analytic way the edges e and e such that they end
d u
at v The extended edges form a graph fe e e e e e g
t d u t d W u W t W
Let us lab el each primed edge by the irreducible representation assigned b efore to
the edge it is an extension of This de nes a lab eling of Finally assign
to each new vertex v the non zero invariant tensor c which is unique
t t m m m
u t
d
up to rescaling asso ciated to the triplet of representations This
d u t
completes the construction of a gauge invariant extension of a spin network state
constructed in the Example Thus for an op en surface the sp ectrum of the area
o o
op erator A on H is the same as that on H
S
Case ii S and S splits in to two open sets
In this case we can not rep eat the ab ove construction Since S has no b oundary
if additional vertices are needed to close the op en spin network they must now lie
in S and can make unwanted contributions to the action of the area op erator Con
sequently there are further restrictions on the p ossible eigenvalues of the op erators
d u d u
J J and J To see this explicitly consider arbitrary spin network
S v S v S v
state c given by the construction a e of Sec Let fv v g b e a set of
W
the vertices of contained in the surface S Graph can b e split into three graphs
which is contained in S which is contained in a one side of S in and
t u d
contained in the other side of S in The only intersection b etween the two parts
is the set fv v g of vertices of which are contained in S Let b e one of the
W r
parts of i e r d or r u or r t According to the construction a e the
lab elings and c de ne naturally on an extended spin network The lab eling of
r
the edges of by irreducible representations is de ned just by the restriction of to
r
The lab eling of the vertices by irreducible representations and invariant tensors
r
is de ned in the following way For the vertices of which are not contained in S
r
the lab elings are taken to b e again the restriction of which are all trivial and
c To a vertex v contained in S we assign the representation corresp onding to a
given j and the invariant tensor c de ned in b for r d u and c for r t
r r
of the construction a e Finally we complete it by arbitrary nonzero lab eling M
of the vertices with vectors in appropriate representations The construction a
e guarantees that a resulting extended spin network state is not zero Now for an
extended spin network c M we have the following fermion conservation
law
X
0
j N
v
v
for some integer N where v runs through the vertices of a graph and each j is
v
an half integer corresp onding to an representation assigned to v by In our case
we therefore obtain the restriction
X
j N
r r
for r d u d u which gives the conditions listed in Sec In fact either
two of the ab ove conditions imply the third one
The conditions are also su cient for an eigen vector to exist Supp ose we
are given a set of half integers as in the Example ab ove which satisfy the restriction
A statement converse to the fermion conservation law is that for any set
fv v g of p oints in S and any assignment v j where j are non negative
W
half integer satisfying there exists an extended spin network c M
such that every v is its index j corresp onds to the representation assigned to v
by and for every vertex v v W of the representation assigned
by is trivial From extended spin networks provided by the ab ove statement it is
easy to construct an eigen vector of the corresp onding eigen values
Case iii S but S does not split
The only di erence b etween this case and the previous one is that now a graph
representing an eigen vector is cut by S into two comp onents contained in S
t
and which corresp onds to the rest of Since can b e now connected by
d u d u
the same arguments as ab ove we prove that a necessary and su cient condition for
d u
an eigen vector to exists is imp osed only on the half integers j
Discussion
In section we b egan by formulating what we mean by quantization of geometry
Are there geometrical observables which assume continuous values on the classical
phase space but whose quantum analogs have discrete sp ectra In the last two
sections we answered this question in the a rmative in the case of area op era
tors In the next pap er in this series we will show that the same is true of other
dimensional op erators The discreteness came ab out b ecause at the micro
scopic level geometry has a distributional character with dimensional excitations
This is the case even in semi classical states which approximate classical geometries
macroscopically
We will conclude this pap er by examining our results on the area op erators from
various angles
Inputs The picture of quantum geometry that has emerged here is strikingly
di erent from the one in p erturbative Fo ck quantization Let us b egin by recalling
the essential ingredients that led us to the new picture
This task is made simpler by the fact that the new functional calculus provides
the degree of control necessary to distill the key assumptions There are only two
essential inputs The rst assumption is that the Wilson lo op variables T
R
A should serve as the con guration variables of the theory i e that the Tr P exp
Hilb ert space of kinematic quantum states should carry a representation of the
C algebra generated by the Wilson lo op functionals on the classical con guration
o
space A G The second assumption singles out the measure In essence if we
a i
assume that E b e represented by ih A the reality conditions lead us to
i a
o
the measure Both these assumptions seem natural from a mathematical
physics p ersp ective However a deep er understanding of their physical meaning is
still needed for a b etter understanding of the overall situation
Compactness of SU plays a key role in all our considerations Let us therefore
brie y recall how this group arose As explained in one can b egin with
a i
the ADM phase space in the triad formulation i e with the elds E K on
i a
as the canonical variables and then make a canonical transformation to a new
i i i a i i
pair A K E where K is the extrinsic curvature and the spin
a a a i a a
a i
connection of E Then A is an SU connection the con guration variable
i a
with which we b egan our discussion in section It is true that in the Lorentzian
signature it is not straightforward to express the Hamiltonian constraint in these
variables one has to intro duce an additional step e g a generalized Wick transform
However this p oint is not directly relevant in the discussion of geometric
op erators which arise at the kinematical level See however b elow Finally we
could have followed the well known strategy of simplifying constraints by using
C i i i i
A iK in place of the real A The internal group a complex connection
a a a a
would then have b een complexi ed SU However for real Lorentzian general
C i
relativity the kinematic states would then have b een holomorphic functionals of A
a
To construct this representation rigorously certain technical issues still need to b e
overcome However as argued in in broad terms it is clear that the results will
b e equivalent to the ones obtained here with real connections
Kinematics versus Dynamics As was emphasized in the main text in the
classical theory geometrical observables are de ned as functionals on the ful l phase
space these are kinematical quantities whose de nitions are quite insensitive to the
precise nature of dynamics presence of matter elds etc Thus in the connection
dynamics description all one needs is the presence of a canonically conjugate pair
consisting of a connection and a density weighted triad Therefore one would ex
p ect the result on the area op erator presented here to b e quite robust In particular
they should continue to hold if we bring in matter elds or extend the theory to
sup er gravity
There is however a subtle caveat In eld theory one can not completely sep
arate kinematics and dynamics For instance in Minkowskian eld theories the
kinematic eld algebra typically admits an in nite numb er of inequivalent represen
tations and a given Hamiltonian may not b e meaningful on a given representation
Therefore whether the kinematical results obtained in any one representation actu
ally hold in the physical theory dep ends on whether that representation supp orts the
Hamiltonian of the mo del In the present case therefore a key question is whether
In particular in the standard spin Fo ck representation one uses quite a di erent algebra of
con guration variables and uses the at background metric to represent it It then turns out that
the Wilson lo ops are not represented by well de ned op erators our rst assumption is violated
One can argue that in a fully non p erturbative context one can not mimic the Fo ck space strategy
Further work is needed however to make this argument water tight
o
the quantum constraints of the theory can b e imp osed meaningfully on H Results
to date indicate but do not yet conclusively prove that this is likely to b e the case
for general relativity The general exp ectation is that this would b e the case also
for a class of theories such as sup er gravity which are near general relativity The
results obtained here would continue to b e applicable for this class of theories
Dirac Observable Note that A has b een de ned for any surface S Therefore
S
these op erators will not commute with constraints they are not Dirac observables
To obtain a Dirac observable one would have to sp ecify S intrinsical ly using for
example matter elds In view of the Hamiltonian constraint the problem of pro
viding an explicit sp eci cation is extremely di cult However this is true already
in the classical theory In spite of this in practice we do manage to sp ecify surfaces
and furthermore compute their areas using the standard formula from Riemannian
geometry which is quite insensitive to the details of how the surface was actually de
ned Similarly in the quantum theory if we could sp ecify a surface S intrinsically
we could compute the sp ectrum of A using results obtained in this pap er
S
Comparison Let us compare our metho ds and results with those available in
the literature Area op erators were rst examined in the lo op representation The
rst attempt was largely exploratory Thus although the key ideas were recog
nized the very simplest of lo op states were considered and the simplest eigenvalues
were lo oked at there was no claim of completeness In the present language this
corresp onds to restricting oneself to bi valent graphs In this case apart from an
overall numerical factor which do es however have some conceptual signi cance
our results reduce to that of
A more complete treatment also in the framework of the lo op representation
was given in It may app ear that our results are in contradiction with those in
on two p oints First the nal result there was that the sp ectrum of the area
q
P
j j where j are half integers rather than by op erator is given by
l l l
P
However the reason b ehind this discrepancy is rather simple the p ossibility that
some of the edges at any given vertex can b e tangential to the surface was ignored
in It follows from our remark at the end of section that given a surface S
if one restricts oneself to only to graphs in which none of the edges are tangential
our result reduces to that of Thus the eigenvalues rep orted in do o ccur in
our sp ectrum It is just that the sp ectrum rep orted in is incomplete Second
it is suggested in that as a direct consequence of the di eomorphism covariance
of the theory lo cal op erators corresp onding to volume and by implication area
elements would b e necessarily ill de ned which makes it necessary to by pass the
intro duction of volume and area elements in the regularization pro cedure This
p
d
g is a well assertion app ears to contradict our nding that the area element
S
de ned op erator valued distribution which can b e used to construct the total area
Note that this issue arises in any representation once a su cient degree of precision is reached
In geometro dynamics this issue is not discussed simply b ecause generally the discussion is rather
formal
op erator A in the obvious fashion We understand however that the intention
S
of the remark in was only to emphasize that the volume and area elements
are genuine op erator valued distributions thus there is no real contradiction
The di erence in the metho dology is p erhaps deep er First as far as we can
tell in only states corresp onding to trivalent graphs are considered in actual
calculations Thus even the nal expression Eq in of the area op erator
after the removal of the regulator is given only on tri valent graphs similarly
their observation that every spin network is an eigenvector of the area op erator
holds only in the tri valent case Second for the limiting pro cedure which removes
the regulator to b e well de ned there is an implicit assumption on the continuity
prop erties of lo op states sp elled out in detail in A careful examination shows
that this assumption is not satis ed by the states of interest and hence an alternative
limiting pro cedure analogous to that discussed in section is needed Work is
now progress to ll this gap Finally not only is the level of precision achieved
in the present pap er signi cantly higher but the approach adopted is also more
systematic In particular in contrast to in the present approach the Hilb ert
space structure is known prior to the intro duction of op erators Hence we can b e
con dent that we did not just omit the continuous part of the sp ectrum by excising
by at the corresp onding sub space of the Hilb ert space
Finally the main steps in the derivation presented in this pap er were sketched
in the App endix D of The present discussion is more detailed and complete
Manifold versus Geometry In this pap er we b egan with an orientable ana
lytic manifold and this structure survives in the nal description As noted in
fo otnote we b elieve that the assumption of analyticity can b e weakened without
changing the qualitative results Nonetheless a smo othness structure of the under
lying manifold will p ersist What is quantized is geometry and not smo othness
Now in dimensions using the lo op representation one can recast the nal de
scription in a purely combinatorial fashion at least in the so called time like sector
of the theory In this description at a fundamental level one can avoid all refer
ences to the underlying manifold and work with certain abstract groups which later
on turn out to b e the homotopy groups of the reconstructed derived manifold
see e g section in One might imagine that if and when our understanding
of knot theory b ecomes su ciently mature one would also b e able to get rid of the
underlying manifold in the theory and intro duce it later as a secondary derived
concept At present however we are quite far from achieving this
In the context of geometry however a detailed combinatorial picture is emerg
ing Geometrical quantities are b eing computed by counting integrals for areas and
volumes are b eing reduced to genuine sums However the sums are not the ob
vious ones often used in approaches that begin by p ostulating underlying discrete
structures In the computation of area for example one do es not just count the
numb er of intersections there are precise and rather intricate algebraic factors that
dep end on the representations of SU asso ciated with the edges at each intersec
tion It is striking to note that in the same address in which Riemann rst
raised the p ossibility that geometry of space may b e a physical entity he also intro
duced ideas on discrete geometry The current program comes surprisingly close to
providing us with a concrete realization of these ideas
Acknowledgments
Discussions with John Baez Bernd Bruegman Don Marolf Jose Mourao Roger
Picken Thomas Thiemann Lee Smolin and esp ecially Carlo Rovelli are gratefully
acknowledged Additional thanks are due to Baez and Marolf for imp ortant informa
tion they communicated to JL on symmetric tensors in the representation theory
This work was supp orted in part by the NSF Grants PHY and PHY
the KBN grant P and by the Eb erly fund of the Pennsylvania
State University JL thanks the memb ers of the Max Planck Institute for their
hospitality Both authors acknowledge supp ort from the Erwin Schro dinger Inter
national Institute for Mathematical Sciences where the nal version of this pap er
was prepared
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