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Intro duction

The lo op representation is a formulation of quantum eld theory suit

able when the degrees of freedom of the theory are given by a gauge eld

or a connection This formulation has b een used in the context of contin

uum and lattice gauge theory and it has found a particularly eective

application in quantum gravity b ecause it allows a description of the

dieomorphism invariant quantum states in terms of knot theory and

at the same time b ecause it partially diagonalizes the quantum dynamics of

the theory leading to the discovery of solutions of the dynamical constraints

Recent results in quantum gravity based on the lo op representation

include the construction of a nite physical Hamiltonian op erator for pure

gravity and fermions the computation of the physical sp ectra of area

and volume and the develop ement of a p erturbation scheme that

may allow transition amplitudes to b e explicitely computed A

mathematically rigorous formulation of quantum eld theories whose cong

uration space is a space of connections inspired by the lo op representation

has b een recently develop ed and the kinematics of the theory is now

on a level of rigor comparable to that of constructive quantum eld theory

This approach has also pro duced interesting mathematical spino s

such as the construction of dieomorphism invariant generalised measures

on spaces of connections and could b e relevant for a constructive eld

theory approach to nonab elian YangMills theories

Applications of the lo op representation however have b een burdened by

complications arising from two technical nuisances The rst is given by the

Mandelstam identities b ecause of which the lo op states are not indep endent

and form an overcomplete basis The second is the presence of a certain sign

n

factor in the denition of the fundamental lo op op erators T for n This

sign dep ends on the global connectivity of the lo ops on which the op erator

acts and obstructs a simple lo cal graphical description of the op erators

action In this work we describ e an elegant way to overcome b oth of these

complications This comes from using a particular basis which we denote

as spin network basis since it is related to the spin networks of Penrose

The spin network basis has the following prop erties

i It solves the Mandelstam identities

n

ii It allows a simple and entirely lo cal graphical calculus for the T

op erators

iii It diagonalises the area and volume op erators

The spin network basis states b eing eigenstates of op erators that corresp ond

to measurement of the physical geometry provide a physical picture of the

three dimensional quantum geometry of space at the Planckscale level

The main idea b ehind this construction long advo cated by R Loll

is to identify a basis of indep endent lo op states in which the Mandelstam

identities are completely reduced We achieve such a result by exploiting the

fact that all irreducible representations of SU are built by symmetrized

p owers of the fundamental representation We will show that in the lo op

representation this translates into the fact that we can suitably antisym

metrize all lo ops overlapping each other without lo osing generality More

precisely the suitably antisymmetrized lo op states span but do not over

span the kinematical state space of quantum gravity

The indep endent basis states constructed in this way turn out to b e

lab elled by Penroses spin networks and by a direct generalization of

these A spin network is a graph whose links are colored by integers sat

isfying simple relations at the intersections Roger Penrose intro duced spin

networks in a context unrelated to the present one remarkably however

his aim was to explore a quantum mechanical description of the geometry

of space which is the same ambition that underlies the lo op representation

construction

The idea of using a spin network basis has app eared in other contexts in

which holonomy of a connection plays a role including lattice gauge theory

and top ological quantum eld theory The

use of this basis in quantum gravity has b een suggested previously but

its precise implementation had to await resolution of the sign diculties

mentioned ab ove Here these diculties are solved by altering a sign in

the relation b etween the graphical notation of a lo op and the corresp onding

quantum state This mo died graphical notation for the lo op states allows

us to reduce the lo op states to the indep endent ones by simply antisym

metrizing overlapping lo ops

The spinnetwork construction has already suggested several directions

of investigation which are b eing pursued at the present time The fact that

it diagonalizes the op erator that measures the volume of a spatial slice

gives us a physical picture of a discrete quantum geometry and also makes

the spin network basis useful for p erturbation expansions of the dynamics of

general relativity as describ ed in It has also played a role in the

mathematically rigorous investigations of refs Another intriguing

suggestion is the p ossibility of considering q deformed spinnetworks on

which we will comment in the conclusion

The details of the application of the spin network basis to the diagonal

ization of the volume and area op erators have b een describ ed in an earlier

pap er The primary aim of this pap er is to give an intro duction to

the spin network basis and to its use in nonp erturbative quantum gravity

We emphasize the details of its construction at a level of detail and rigor

that we hop e will b e useful for practical calculations in quantum gravity

No claims are made of mathematical rigor for that we p oint the reader to

the recent works by Baez and Thiemann where the spin network

basis is put in a rigorous mathematical context Finally we note that in this

pap er we work with SL C or SU spinors which are relevant for the

application to quantum gravity but a spin network basis such as the one we

describ e exists for all compact gauge groups

This pap er is organized as follows In the next section we briey explain

the two problems that motivate the use of the spin network basis This leads

to section in which we provide the denition of spin network states in the

lo op representation In section we describ e the spin network states as

they app ear in the connection representation The pro of that the spin

network states do form a basis of indep endent states may then b e given in

section Following this in section we review the general structure of the

transformation theory in the sense of Dirac b etween the lo op representa

tion and the connection representation The use of the spin network basis

considerably simplies the transformation theory as we show here Simi

larly old results on the existence of solutions to the hamiltonian constraint

and exact physical states of quantum gravity may b e expressed in a simpler

way in terms of the spin network basis Its use makes it unnecessary to

explicitly compute the extensions of characteristic states of nonintersecting

1

knots to intersecting lo ops as describ ed in

Finally an imp ortant side result of the analysis ab ove is that it indicates

how to mo dify the graphical calculus in lo op space in order to get rid of the

annoying nonlo cality due to the dep endence on global ro oting The new

notation that allows a fully lo cal calculus is dened in section The pap er

closes with a brief summary of the results in section and with a short

1

These characteristic states were previously dened to b e equal to one on the knot

class of a nonintersecting lo op zero on all other nonintersecting lo ops with an extension

to the classes of intersecting lo ops dened by solving the Mandelstam identities

Now they may b e succinctly describ ed as b eing equal to one on one element of the spin

network basis and zero on all the others

app endix in which we discuss the details of the construction of higher than

trivalent vertices

Denition of the problem

The lo op representation is dened by the choice of a basis of bra states hj

on the state space of the quantum eld theory These states are lab elled by

lo ops By a lo op we mean here a set of a nite numb er of single lo ops

1

by a single lo op we mean a piecewise smo oth map from the circle S into

the space manifold The lo op basis is characterised dened by the action

on this basis of a complete algebra of observables A quantum states j i

is represented in this basis by the lo op space function

hj i

For detailed intro ductions and notation we refer to As shown in

the functions that represent states of the system must satisfy a

set of linear relations which we denote as the Mandelstam relations These

co de among other things the structure group of the Ashtekars connection

R

A b e the parallel propa A of the classical theory Let U A exp

gator matrix or holonomy of the connection A along the curve and let

T A T r U A b e its trace Then the Mandelstam relations are dened

for the present purp oses as follows For every set of lo ops and

1 N

complex numb ers c c such that

1 N

X

c T A

k k

k

holds for all smo oth connections A the states must satisfy

X

c

k k

k

It follows that the states of the bra basis hj are not indep endent as linear

functionals on the ket state space but satisfy the identities

X

c h j

k k

k

The basis hj is therefore overcomplete Let us from now on concentrate

on the SL C case There are two cases of the relation that are

particularly interesting The rst one yields the spinor identitity or prop er

SL C Mandelstam identy Given any two SL C matrices A and B

the following holds b etween the traces we can construct in terms of them

1

T r AT r B T r AB T r AB

1

Let and b e two lo ops that intersect in a p oint p Let and b e

the two lo ops that are obtained by starting at p going around and then

1 1

U U Then around either or so that U U U and U

1

implies that for every A

1

T A T A T A

Therefore we have

1

h j h j h j

The second example which is easily seen deriving from is the retracing

identity

1

h j hj

where is an op en segment with an end p oint on the lo op a tail In

earlier work it has b een assumed that all identities can b e derived

from the two relations and We are not aware of any complete pro of

or of a counterexample of this conjecture

The redundancy intro duced in the lo op representation by the Mandel

stam identities is cumb ersome It is not the spinor identity by itself nor the

retracing identity by itself that create much complications since the rst

could b e solved by simply cho osing a list of indep endent intersections and

the second by discarding all lo ops with tails from the theory It is the

combination of the two relations which makes it dicult to isolate a set of

indep endent lo op functionals To see this consider two lo ops and that

do not intersect At rst sight one would say that these are not aected by

the spinor relation but they are To see how consider an op en segment

with one end on and one end on Combining and we have

1 1

h j h j h j

So that even nonintersecting multiple lo op state enters the Mandelstam

identities Equation can b e represented graphically as

- - = 0

The problem that we consider in this pap er is to nd a basis of states

hsj that are fully indep endent so that no linear combination of them can b e

set to zero using the identities Such a basis will b e dened in section

and the pro of of indip endence given in section The rest of this section

describ es the motivations underlying the denitions in section

The sign diculty

There is a natural strategy for getting rid of the redundancy expressed in

eq which we are now going to describ e This strategy however is

obstructed by a sign diculty which previously prevented its complete im

plementation The natural strategy is to get rid of the degeneracy by anti

symmetrizing all lines running parallel to each other For instance out the

tree lo op states involved in relation we may pick the two indep endent states

i)

ii) = -

where we have indicated antisymmetrization by a wiggly line Since the

1 1

symmetric combination h j h j is equal to h j

by equation the two states ab ove exhaust all p ossible indep endent lo op

states that can b e constructed out of three original states This pro cedure

should b e combined with some suitable restriction to the indep endent inter

sections

Let us intro duce some terminology We denote a set of lo op segments

that fully overlap as a rop e and we call the numb er of lo ops that form

1

it without regard to orientation the order of the rop e Thus and

form a rop e of order two in the second and third states in ii ab ove Given

an intersection p oint p of the lo op a p oint on the supp ort of the lo op where

this supp ort fails to b e a submanifold of we denote the numb er of rop es

that emerge from p as the order of the intersection and we say that a lo op

is nvalent if it has intersections of order at most n To b egin with we

shall only consider trivalent lo ops For instance in the example ab ove the

1

intersection b etween and in the lo op is trivalent b ecause and

1

form a single set of overlapping lo op segments a single rop e emerging

from the intersection We will deal with nontrivalent intersections in the

App endix

We may hop e to reduce the degeneracy by replacing every overlapping

segment with a suitable antisymmetrized combination plus tails that can

b e got rid of by means of the retracing identity In the example considered

1

ab ove for instance we can reduce the state h j to a linear

combination of the two states dened in as follows

= 1/2 { - } + 1/2 { + } =

= 1/2 + 1/2 =

= 1/2 + 1/2

So we may hop e that any time we have two parallel lines we could use the

spinor identity as follows

= 1/2 { - } + 1/2 { + } =

= 1/2 + 1/2 =

= 1/2 + 1/2

Unfortunately this do es not work To understand why consider a lo op

and an op en segment that starts and ends in two dierent p oints of

Denote by and the two segments in which the two intersections with

1 2

partition Then we have due to the spinor and retracing identities

1

hj h j h j

1 1 1 1 namely

- + = 0

If we want to pick two indep endent linear combinations we have to chose the

1

symmetric combination hj h j and not the antisymmetric

1 1

one as b efore Namely we have to cho ose

i)

ii) +

Thus to pick the indep endent combination of lo op states we have to anti

symmetrize the rop e in one case but we have to symmetrize it in the other

case In general the choice b etween symmetrization and antisymmetriza

tion can only b e worked out by writing out explicitly the full pattern of

ro otings in the multiple lo op In other words equation is in general

wrong if taken as a calculation rule that can b e used in dealing with any lo op

state More precisely at every intersection the spinor identity provides a

linear relation b etween the three multiple lo ops obtained by replacing the

intersection with the three p ossible ro otings through the lo op

-+ -+ = 0

but the sign in front of each term dep ends on the global ro oting of the lo ops

There is a simple way out of this diculty which do es allow us to get

rid of the spinor identities among trivalent lo ops simply by antisymmetriza

tion In order to determine the correct signs of the various terms in eq

we have to take the global ro oting into account There are only three p ossibilities

+ - = 0

-- = 0

- + = 0

The signs derive immediately from eq Now the way out of the sign

diculty is provided by the following observation If we multiply each term

of the linear combinations in eq by the signfactor

n( )

where n is the numb er of single lo ops in the term we obtain the

following relations

ε + ε + ε = 0

ε + ε + ε = 0

ε + ε + ε = 0

n( )

Thus if we multiply all terms by we can use the algebra of eq

and therefore reduce every overlapping lo op to fully antisymmetrized

terms plus terms where two overlapping disapp ear by means of the retracing

identity In other words the indip endent states must b e constructed by fully

antisimmetrizing the segments along the rop es and multiplying the resulting

n( )

terms by

Let us study the set of states determined in this way It is easy to

convince oneself that if the three rop es adjacent to a trivalent no de are com

pletely symmetrized then the ro otings of the single lo opsegments through

the intersection are uniquely determined It follows that the trivalent

states that we have obtained by antisymmetrizing the rop es are fully deter

mined solely by their supp ort the order of each rop e and an overall sign

Equivalently they are determined by a trivalent graph the supp ort with

integers assigned to each link the order of the rop e plus an orientation of

the graph Furthermore the orders of the three rop es adjacent to a given

no de are constrained to satisfy some relations among themselves First we

can assume that no lo op through the no de can go back to the rop e it comes

from otherwise we can retrace it away Thus there are three sets of lo ops

that run through a trivalent intersection the ones ro oted from the rst to

the second rop e lets say we have a of them the ones ro oted from the

second to the third rop e b of them and the ones ro oted from the third to

the rst rop e c of them It follows that the order of the three rop es are

resp ectively

p c a q a b r b c

The three numb ers a b and c are arbitrary p ositive integers but not so the

orders p q and r of the adjacent rop es It follows immediately from

that they satisfy two relations

i Their sum is even

ii None is larger than the sum of the other two

and that these two conditions on p q and r are sucient for the existence

of a b and c We conclude that our states are lab elled by oriented trivalent

graphs with integers p asso ciated to each links l such that at every no de

l

the relations i and ii are satised By denitions these are Penroses spin

networks Thus a linear combination of trivalent lo ops with the same

supp ort in which every rop e is fully antisymmetrized is uniquely determined

by an an imb edded oriented trivalent spin network We shall denote these

fully antisymmetrized states as spinnetwork states From the discussion we

have just had we can see that they comprise an indep endent basis

Using the ab ove discussion as motivation in the next section we provide

a complete denition of spin networks and spin network quantum states

Spinnetwork states in the lo op representation

In this section we dene the spin network states and their corresp onding

dieomorphism invariant knot states As dened by Penrose a spin network

is a trivalent graph in which the links l are labled by p ositive integers

p denoted the color of the link such that the sum of the colors of three

l

links adjacent to a no de is even and none of them is larger than the sum of

the other two To each spin network we may asso ciate an orientation

or determined by assigning a cyclic ordering to the three lines emerging

from each no de In particular an orientation is determined by a planar

representation of the graph by the clo ckwise ordering of the lines and

gets reversed by redrawing one of the intersection with two lines emerging

in inverted order Here we consider imb edded oriented spin networks

and we denote such ob jects by the capitalized latin letters S T R An

imb edded spin network is a spin network plus an immersion of its graph

in the threedimensional manifold Later in discussing the solution of

the dieomorphism constraint we will consider equivalence classes of these

spin networks under dieomorphisms these will b e called sknots and

indicated by lower case latin letters s t r

Given a trivalent imb edded oriented spin network from now on just spin

network S we can construct a quantum state of the lo op representation as

follows First we replace every link l of the spin network by a rop e of degree

p where p is the color of the link l Then at every intersection we join the

segments that form the rop e pairwise in such a way that each segment is

joined with one of the segments of a dierent rop e As illustrated in the

previous section the constraints on the coloring turn out to b e precisely

the necessary and sucient conditions for the matching to b e p ossible The

S

Then we matching pro duces a multiple lo op which we denote as

1

Q

S

p lo ops m M that can b e obtained from consider the M

l

m

l

S

by p ermutations of the lo ops along each rop e Each rop e of color p

l

1

c(m)

pro duces p terms We assign to each of these lo ops a sign factor

l

S

which is p ositivenegative for eveno dd p ermutations of the lo ops in

1

Equivalently one can identify cm with the numb er of crossings along

rop es in a planar representation of the lo ops Finally we dene the state

X

c(m) n(m) S

h j hS j

m

m

S

as nm for short we recall that here and in he following we write n

m

n is the numb er of single lo ops forming the multiple lo op We denote

the state hS j dened by equation as spin network state or the quantum

state asso ciated with the spin network S

Notice that up to the overall sign the linear combinaton that denes hS j

is indep endent on the particular ro oting through the intersections chosen in

S

b ecause every other ro oting is pro duced by the p ermuta constructing

1

tions The overall sign is xed by the orientation of the spin network For

concreteness let us assign an orientation to the spin network by pro jecting

c(m) S

among the it on a plane and assign to the unique lo op

1

S

that can b e drawn without crossings the segments c along the

m

rop es and in the no des We will show in section that the states hS j we

have dened form a basis of indep endent states for the trivalent quantum

states

We represent spin network states simply by drawing their graphs and

lab eling the edges with the corresp onding colors and if necessary with

the name of the lo op or segment they corresp ond to As an example and

in order to illustrate how the signs are taken into account by the ab ove

denitions consider the spin network

1 β

2 γ

1 α

This is expanded in lo ops as

1

2 = - +

1

b ecause for the rst lo op we have c n and for the second we have

c n therefore the spin network represents the state

1 β

γ 2 = α ∗ γ ∗ β-1 ∗ γ-1 - α ∗ γ ∗ β ∗ γ -1

1 α

On the other hand the spin network

1 2 1

is expanded in lo ops as

1 2 1 = +

b ecause we have c n for the rst lo op and c n for the

second therefore the spin network represents the state

α γ -1 α γ α γ α γ α γ α = 1 **2 + * * *

1 1 2 1 2 1 2

Notice the plus sign contrary to the minus sign of the previous example

The construction ab ove can b e easilly extended to lo ops with intesections

of valence higher than This is done by means of a simple generalization

of the spin networks obtained by considering nontrivalent graphs colored

on the vertices al well as on the links Or equivalently by trivalent spin

networks in which sets of no des are lo cated in the same spacial p oint This

is worked out in detail in the App endix

Now since the spin network states hS j span the lo op state space it

follows that any ket state j i is uniquely determined by the values of the

hS j functionals on it Namely it is uniquely determined by the quantities

S hS j i

Furthermore since as we shall prove later the bra states hS j are linearly

indep endent any assignement of quantities S corresp onds to some ket

j i Therefore quantum states in the lo op representation can b e represented

by spin network functionals S By doing so we can forget the diculties

due to the Mandelstam identities which the lo op states must satisfy

In particular we can consider spin networks characteristic states S

T

dened by S We will later see that the AshtekarLewandowski

T T S

measure induces a scalar pro duct in the lo op representation under which the

spin network states are orthonormal then we can identify the characteristic

states as the Hilb ert duals of the spin network bra states On the other

hand this identication dep ends on the scalar pro duct and thus in general

one should not confuse the spin network characteristic states kets with the

spin network states bras

It is easy to see that the calculations of the action of the Hamiltonian

constraint C presented in imply immediately that if S vanishes on

all spin networks S which are not regular formed by smo oth and non self

intersecting lo ops then C S Notice that this follows from the

combination of two results the rst is that hS jC for all regular S the

second is that C S if S is not regular b oth these results are dis

cussed in Thus states S with supp ort on regular spin networks solve

the Hamiltonian constraint and at the same time satisfy the Mandelstam

identities Indeed they are precisely the extensions of the lo op states with

supp ort on regular lo ops dened implicitely in and discussed in detail

in The spin network basis allows these solutions to b e exhibited in a

much more direct form

The same conclusion may b e reached using the form of the hamiltonian

constraint describ ed in in which we consider the classically equivalent

p

R

f form of the constraint C where f are smo oth functions on



Dieomorphism invariance and spin networks in knot

space

One of the main reasons of interest of the lo op representation of quantum

gravity is the p ossibility of computing explicitely with dieomorphism in

variant states These are given by the knot states A knot K is an equivalent

class of lo ops under dieomorphisms We recall from that a knots state

which we denote as or simply as jK i in Dirac notation is a state of

K

the quantum gravitational eld with supp ort on all the lo ops that are in the

equivalence class K

if K

hjK i

K

otherwise

Clearly the same idea works for the spin network states Let us consider

the equivalent classes of emb edded oriented spin networks under dieomor

phisms Such equivalence classes are entirely identied by the knotting

prop erties of the emb edded graph forming the spin network and by its col

oring We call these equivalence classes knotted spin networks or sknots

for short and indicate them with a lower case Latin letter as s t r An

sknot s can therefore b e thought of as an abstract top ological ob ject inde

p endent of a particular imb edding in space in the same fashion as knots

Then for every knotted spin network s we can dene a quantum state jsi

a ket of the gravitational eld by

if S s

S hS jsi

s

otherwise

Notice that in general a knot state jK i do es not satisfy the Mandelstam

relations Dieomorphism invariant lo op functionals representing physical

states should b e constructed by suitable linear combination of the elemen

tary knot states jK i This must b e done by nding suitable extensions of

the states from their values on regular knots to intersecting knots using the

Mandelstam identities as describ ed in However these constructions

are rather cumb ersome in spite of the fact that the rigorous results based

on the use of dieomorphism invariant measures on lo op space ensure

us of the existence of such states The spin network construction provides

a way to circumvent this diculty Indeed the sknot state form a com

plete set of solutions of the dieomorphism constraint and the sknot states

corresp onding to regular spin networks are solutions of all the constraints

combined

The space of the trivalent sknots is numerable for the same reason for

which the set of the knots without intersections is numerable However we

recall that dieomorphism classes of graphs with intersections of order higher

than ve are continuous To construct a separable basis for dieomorphism

invariant states including spin networks of all valences a sep erable basis

must b e selected for functions on each of these mo duli spaces As these

spaces are nite dimensional this can b e accomplished For a classication

of the resulting mo duli spaces of higher intersections see

This concludes the construction of the spin network states and of the

sknot states in the lo op representation To set the stage for the demon

stration of their indep endence we rst dene the spin network states in the

connection representation

The connection representation

We recall that in the connection representation one may consider a lo op state

2

or ji dened by the trace of the holonomy of the Ashtekar SL C

connection A along

A hAji T A T r U

Consider a spin network S We can mimic the construction of the lo op

representation and dene the quantum state

X

c(m)+n(m) S

T A A hAjS i

S

m

m

where we recall n is the numb er of single lo ops and c counts the terms

of the symmetrization Let us analyse this state in some detail For every

every link l with color p there are p parallel propagators U A along the

l l

link l each one in the spin representation that enter the denition

2

Note that here we do not use the factor of = that has b een conventional since the

work of Ashtekar and Isham but return to the original convention of ref This

choice is substantally more convenient for the present formalism b ecause otherwise we

have to keep track of a factor of = for every trace and these factors come into the

relations b etween the spin networks and the lo op states

of A Let us indicate tensors indices explicitely we intro duce spinor

S

B

indices A B with value The connection A has comp onents A

A

which form an sl C matrix and its parallel propagator along a link l is a

B

matrix U in the SL C group Since SL C is the group of matrices

l A

with unit determinant we have

CD A B

U U det U

AB

l

l C l D

where is the totally antisimmetric two dimensional ob ject dened by

AB

01

01

B

One can write A explicitely in terms of the parallel propagarors U

S

l A

AB B

the ob jects and and the Kro eneker delta Thus any spin network

AB

A

state can b e expressed by means of a certain tensor expression formed by

sl C tensors and ob jects

Penrose has describ ed in a graphical notation for tensor expressions

of this kind This notation is going to play a role in what follows so we

b egin by recalling its main ingredients We indicate twoindices tensors

with thick lines with the indices at the op en ends of the line resp ecting

the distinction b etween upp er indices indicated by lines p ointing up and

lower indices corresp onding to lines p ointing down The sum over rep eated

indices is indicated by tying two lines together More precisely we indicate

B

the matrix of the parallel propagator U of an op en or closed curve as

A

a vertical b old line as in

B

B α Uα A

A

where the lab el is understo o d unless needed for clarity we indicate the

antisymmetric tensors as in

AB εAB

ε AB

AB

and the Kroneker delta as in

A δ A B

B

Finally we indicate the sum over rep eated indices by connecting the op en

ends of the lines where the indices are We then have for instance

Tr =

β α β = α

α-1 = - α

B B - A =

A

Also

= - ABAB

B A = B A

α α =

The most interesting relation is the identity

BD B D D B

AC

C A C A

which b ecomes

-- = 0

which is of course related to the lo op representation spinor identity Because

of this last relation in the Penrose diagram of any lo op state we can use the

graphical relation

= 1/2 + 1/2

where the bar indicates simmetrization on any true intersection or over

lapping lo op

Now consider a generic multiple lo op state in the connection repre

sentation this is given as a pro duct of terms each of which is the trace

of a pro duct of matrices We can represent these traces in terms of the

corresp onding graphical tensor diagram which will result as a set of closed

lines We adopt the additional convention of drawing lines that form a rop e

as nearby parallel lines and of repro ducing the intersections of the original

lo ops as intersections in the Penrose diagram true intersections represent

ing intersections of the lo op states should b e distinguished from accidental

intersection forced by the planar nature of the Penrose diagram In this

way every multiple lo op state is represented as a closed diagram Let us

denote this diagram as G for Graphical Tensor notation

Notice that the diagram G repro duces the top ological features of

the original lo op it can b e naively thought as a simple twodimensional

drawing of the lo op itself But the corresp ondence is not immediate as is

clear from the fact that the sign relations ab ove imply that the same lo op

may corresp ond to either G or to G dep ending on the way the lo op

is drawn

α ; α - α

In order to distinguish b etween the drawing of the lo op and the graphical

tensor notation G of the trace of the corresp onding holonomy from now

on we denote the drawing of the lo op as D G cannot b e immediately

identied with D However the relation b etween the graphical tensor

diagram G of the tensor A and the planar representation D of

the lo op is not to o dicult to work out In fact we have

m()+c()+n()

G D

where m is the numb er of minima in the diagram D c is the num

b er of crossings and n is as b efore the numb er of single lo op comp onents

of This is an imp ortant formula since it allows one to translate rigor

ously b etween graphical relations of the lo op pictures and tensor relations of

the corresp onding holonomies In a sense this formula renders explicit an

intuition that underlies the entire construction of the lo op representation

Let us work out the main consequence of this formula in the spin net

work context The denition of the spin network states b ecomes in tensor

graphical notation

X

c(m) n(m)+1 s

G Gs

m

m

Therefore expressing the right hand side in D notation we have

X

c(m)+1 n(m)+1 m(m)+c(m)+n(m) s

D Gs

m

m

or simplifying the even exp onents noticing that the numb er of minima

do es not dep end on the p ermutations and absorbing an overall sign in the

orientation

X

s

D Gs

m

m

Thus we obtain the crucial conclusion that the tensor representing the spin

network is obtained by writing one of the lo op states and consider all the

p ermutations with no sign factor namely by considering all symmetrizations

of the lines along each rop e The resulting linear combination of graphical

tensors gives directly the tensor representing the spin network state up

to an overall sign that we can absorb in the orientation Thus we can

conclude that the antisymmetrization that denes the spin network states

is in fact a symmetrization of the SLC tensor indices Let us now study

what such a symmetrization implies

For every SLC tensor we have from the well known relation

1 B CD A

U U

AB

D C

B

Consider a link l and let U b e the paralell propagator of A along l

A

Consider the pro duct of two such propagators along the same link

BD B D

U U U

AC A C

This can b e written as the sum of its symmetrized and antisymmetrized

comp onents

(BD ) [BD ]

BD

U U U

AC

AC AC

However it is straightforward to show from the prop erties of two comp onent

spinors that

[BD ]

BD

U

AC

AC

so that we have the identity

(B D )

B D BD

U U U U

AC

A C

A C

If we write this in graphical tensor notation we have precisely equation

Following the same pro cedure it is easy to show that a pro duct of matrices

A A

A

1 2 n

U U U can b e decomp osed in a sum of terms each one formed

B B B

n

1 2

(A A )

A

1

k

2

by totally symmetrized terms U U times a pro duct of epsilon U

B B B

1 2

k

matrices

Of course what is going on here has a direct interpretation in terms

B

of SU representation theory Each matrix U lives in the spin

A

representation of SU the pro duct of n of these matrices lives in the nth

tensor p ower of the spin representation and this tensor pro duct can

b e decomp osed in the sum of irreducible representations The irreducible

representations are simply obtained by symmetrizing on the spin indices

The reason we have reconstructed the details of the decomp osition is that

this leads us to the precise relation b etween the tensorial expression of the

connection representation states and the lo op representation notation

In fact a fully antisymmetrized rop e of degree p is represented in matrix

notation by a fully symmetrized tensor pro duct of p parallel propagators

in the fundamental spin representation Therefore a rop e of degree p

corresp onds in the connection representation to a propagator in the spin

p representation The result that every lo op can b e uniquely expanded

in the spin network basis is equivalent to statements that the symmetrized

pro ducts of the fundamental representation of SL C gives all irreducible

representations

No des and 3j symb ols Explicit relations

The trivalent intersections b etween three rop es dene an SL C invariant

pro duct of three irreducible representations Clearly the fact that there is a

unique trivalent intersection in the lo op representation is the reection of the

fact that there is a unique way of combining three irreducible representations

to get the singlet representation or equivalently that there is a unique

decomp osition of the tensor pro duct of two irreducible representations In

this subsection we make the relation b etween the two formalisms explicit

for the sake of completeness

Consider a triple intersection with adjacent lines colored p q and r

These corresp ond to the representations with angular momenta l p

p

l q and l r The restriction on p q and r that p q r is even

q r

and none is larger than the sum of the other two corresp onds to the basic

tensor algebra relations of the algebra of the irreducible representations of

SL C namely the angular momentum addition rules In fact the two

conditions are equivalent to the following familiar condition on l once l

r p

and l are xed

q

l jl l j jl l j l l l l

r p q p q p q p q

Now let p q and r b e xed and let us study the corresp onding intersection

in the connection representation This is given by a summation over the

symmetrized indices of the three pro ducts of parallel propagators Let us

raise all the indices of the propagators adjacent to the no de Let us denote

A A

by U the spin propagator along the link colored p and by V and

B B

A

W the propagators along the links colored q and r where the propagators

B

are oriented towards he intersection so that the upstairs indices refer to the

end on the intersection Since the other index of each matrix is not going to

A A A

play any role we drop it and write simply U V and W We must have

at the intersection

C C B B A A

r q p

1 1 1

K W W V V U U

A A B B C C

p q r

1 1 1

where K is an invariant tensor symmetric in the rst p

A A B B C C

p q r

1 1 1

entries the middle q and the last r Since the only invariant tensor is

AB

K must b e a sum of pro ducts of s None of the s

A A B B C C AB AB

p q r

1 1 1

can have b oth indices among the rst p indices of K since

A A B B C C

p q r

1 1 1

is antisymmetric and the rst p indices are symmetrised Similarly for

AB

the middle q and the last r Thus we have s with an A index and a

AB

B index lets say we have a of them s with a B index and a C index

AB

b of them s with a C index and an A index c of them Clearly we

AB

must have a b p b c q and c a r Thus K may

A A B B C C

p q r

1 1 1

contain a term of the form

A B A B B C B C C A B A

p a q r c

c+1 1 a+1 1 1

b b+1

We have to symmetrize this in each of the three set of indices We obtain

pq r terms and it is not dicult to see that this sum is the only invariant

tensor with the required prop erties Thus

X

K

A A B B C C A B A B B A B C C A B C

p q r p a r c q

1 1 1 c+1 1 1 a+1 1

b b+1

where the sum is over all the symmetrizations of the A B and C indices

Now notice that if we read the graphical representation of the tensor as

representing the lo ops each of the terms in the sum corresp onds precisely

to the ro oting of a individual lo ops b etween the p and the q links and so

on Thus we obtain precisely the spin network vertex On other hand the

relation b etween the matrix K and the j symb ols is also

A A B B C C

p q r

1 1 1

clear For every representation with spin l let us intro duce the index m

p p

that takes the l values m l l And in the basis v in the

p p p p mp

representation space related to the fully symmetrized tensor pro duct of l

p

spinors we write

A A

p

1

A A

p

1

v

mp

(A A )

m

p

1

p

Then we can write the vertex in this basis as

C C B B A A

r q p

1 1 1

K K

A A B B C C m m m

m p q r m m p q r

1 1 1

r q p

By uniqueness this must b e prop ortional to the j symb ols of SU

l l l

p q r

K

m m m

p q r

m m m

p q r

Demonstration of the indep endence of the spin

network basis

We are nally in the p osition to prove the indep endence of the spin network

states jsi We will do this in the connection representation The indep en

dence of these states is a linear prop erty and it should therefore b e p ossible

to prove it using only the linear structure of the space However it is much

easier to construct a pro of using an arbitrary inner pro duct structure on

the state space Since linear indep endence is a linear prop erty once we have

proven indep endence using a sp ecic inner pro duct the result is indep endent

from the inner pro duct used

Ashtekar and Lewandowski have recently studied calculus on the

space of connections and have dened a measure d A on a suitable

AL

extension of the space of connections or equivalently a generalized mea

sure on the space of connections Lo op states and their pro ducts are all

measurable in this measure in fact the measure is dened using the tech

nology of cylindrical measures where the cylindrical functions are precisely

the lo op states Thus the measure denes a quadratic form or a scalar

pro duct on the linear span of the lo op states which is nite as long as

we consider only nite linear combination For our purp oses the measure

d A is convenient for several reasons First b ecause it is dieomor

AL

phism invariant Second b ecause it is under go o d control so calculations

with it are easy Here we will use the original Ashtekar Lewandowski mea

sure dened for SU connections The extension to SL C connections

is discussed in refs In the present context since lo op states as func

tionals on SL C connections are holomorphic functions of A and not A

they are determined by their restriction on the SU connections thus the

SU measure that we employ denes an Hilb ert space structure on these

functionals and this is all we need here In any case we refer the reader to

reference refs for a much more accurate treatement of this p oint

Let us thus consider functionals f A of the connection of the form

f A f U A U A

n

1

where f g g is a function on the nth p ower of SL C and thus

1 n

in particular on the nth p ower of SU An example is provided by the

lo op states A The AL measure can b e characterised as follows by

Z Z

d Af A f g g d g g

AL 1 n H 1 n

where d is the Haar measure on the nth p ower of SU The measure

H

denes an inner pro duct b etween lo op states via

Z

d A A A

AL

Now what we want to prove is that the spin network states are

s

linearly indep endent Supp ose that we can prove that they are all orthogonal

with resp ect to this scalar pro duct namely

s s

for every s and

0

for every s s

s

s

Then their linear indep endence follows b ecause if there were a linear com

bination of spin network states such that

X

c

m s s

m

m

we would have taking the scalar pro duct of the ab ove equation with

s

itself a vanishing right hand side and a nonvanishing left hand side Thus

to prove indep endence we have to prove and

Let us consider a given spin network s We have using denitions

Z

d A A A

s s AL s s

The spin network state is a sum over p ermutations m of pro ducts over

s

the single lo ops in the multiple lo op of traces of pro ducts over the

m

j

single links l covered by the lo op of holonomies of A Then

im a

im

Y Y X

n(m)+c(m)

U T r

j

s

l

im

m

j i

Therefore using the denition of the AshtekarLewandowski measure we

have

Z

X

n(m)+c(m)

0

dH g g L

s

s

m

Y Y X Y Y

0

0 0

j j

n(m )+c(m )

T r g l T r g l

0 0

im

i m

0 0 0

i j

m i j

where we have lab elled as L the links of the spin network Now we have

to use prop erties of the SU Haar measure The main prop erties we need

are

Z

d g

H

Z

B

d g U g

H

A

Z

BD D B

U g d g U g

AC H

C A

See reference for a detailed discussion

Let us analyze the eect of the integration graphically Pick a link l

R

and consider the corresp onding group integration d g l Let the colors

H

of this link b e p and p for s and s Then the integration over g l is the

integration over the pro duct of p U s The result is the pro duct of s

AB

describ ed ab ove Notice however that only the terms in which all the s

AB

have one index coming from one of the spin networks and one index from

the other can survive b ecause the others vanish due to the symmetry in the

spin network indices and the antisymmetry in the result of the integration

Therefore in each of the links involved in the integration there should b e

precisely the same numb er of segments in s and s This is sucient to see

that any two spin network states corresp onding to dierent spin networks

are orthogonal

Then integrating the link l gives a set of Let us now take s s

epsilons that connects the two copies of s one with the other We obtain

thus terms in which at every end of l the two other adjacent links can simply

b e retraced back plus terms which will vanish up on the next integration

Thus the two copies of s get completely retraced back leaving at the end

just pro ducts of integrations over the identity each giving Thus we have

shown that all spin network states are normalized This completes the pro of

for trivalent states The extension to higher valence intersections is simple

see also We may note that this result parallels the discussion of the

indep endence of the spin network basis for hamiltonian lattice gauge theory

given for example in Furmanski and Kowala Given that the Ashtekar

Lewandowski measure is built from the pro jective limit of inner pro ducts for

lattice gauge theories for all analytic emb eddings of lattices in it is not

surprising that the result extends from lattice gauge theory to this case

Relationship b etween the connection and the

lo op representation

In this section we review the relation b etween the lo op representation R

l

and the connection representation R which was intro duced in This

c

relation is simpler in the light of the spin network basis To see this we may

recall from that there exists a third relevant representation This is the

representation R dual to the connection representation The ket states in

c

R are by denition the brastates of the connection representation namely

c

they are linear functionals on the space of functionals A Equivalently

we may think of these states as measures on the space of the connections

R by jdi The op erators dening in R are We denote the states in

c c

R by their dual action immediately dened also in

c

In the absence of an inner pro duct there is no canonical map b etween the

state space of R and the state space of R On the other hand however the

c c

representation R is directly related to the lo op representation R In fact

c

l

by double duality any functional of the connection A denes a linear

map on the state space of R via

c

Z

h jdi hdj i dA A

In particular each lo op state j i which is dened in the connection rep

resentation R by hAj i T A determines a dual state h j in R

c c

via

Z

hdj i dA T A h jdi

By denition of the lo op representation these bra states h j in R are

c

to b e identied precisely with the lo op states hj

hj h j

In other words the lo op representation state is the state that is repre

R by the measure d where sented in the dual connection representation

c

Z

dAT A

This construction dep ends only on the linear structure of the quantum

theory namely it do es not dep end on a sp ecic scalar pro duct which may

or may not b e dened on the state space In the absence of a scalar pro duct

there is no canonical map b etween ketspace and braspace and therefore no

canonical asso ciation of a dual state a measure dA to a given connection

representation state A nor equivalently there is any canonical mapping

R b etween the connection representation R and lo op representation

c

l

If a scalar pro duct is given then we can map connection representation

states into lo op representation states b ecause given a connection represen

tation functional A there is a unique dual state d A such that for

every y

hd j i

And the lo op state corresp onding to the connection representation

state A is then given by

h jd i

In particular a scalar pro duct in the connection representation can b e as

signed by xing a measure dA on the space of connections so that

Z

dA A A

Then the induced map b etween the connection representation and the lo op

representation is the known expression for the lo op transform

Z

dA T A A

To implement this relation explicitely we must use a measure dA which

resp ects the invariances of the theory Nontrivial gaugeinvariant and dieo

morphisminvariant generalised measures on the space of connection have

recently b eing constructed and the AshtekarLewandowski measure we

used ab ove is the simplest of these The existence of this measure allows

us to establish a denite linear map b etween the connection and the lo op

representation Let us do so and study its consequences We will discuss

later the extent to which we can take the resulting scalar pro duct and thus

the resulting identication of the two representations as the physically

correct one

For every ket state A in the connection representation the Ashtekar

Lewandowski scalar pro duct asso ciates to it the bra state or measure

d A d A A Let us consider a spin network state A The

AL s

corresp onding bra state state is d A d A A By the prop erty

s AL s

of the spin network states under the AshtekarLewandowski integration we

have then the remarkable result

0 0 0

i hd j

s s

ss s s

The lo op representation state bra state hsj corresp onding to is dened

s

solely in terms of the linear prop erties of the representation The lo op

representation ket state jsi on the other hand is dened by

Z

d AT A

s s

and it follows immediately that it is the adjoint of hsj Thus the Ashtekar

Lewandowski inner pro duct b ecomes in the lo op representation

0

hsjs i

ss

a remarkable result indeed In other words the spin network basis is orthog

onal with resp ect to this inner pro duct It is imp ortant to notice that these

results do not hold for the lo op states themselves for which inner pro ducts

0

are inconsistent with the Mandelstam relations of the form hj i

Finally let us note that the condition on the theorys physical inner pro d

uct is that it makes the real observables hermitian While little is known

ab out the solution of this condition b ecause of the absence of explicit op era

tor expressions for physical observables we may say more at the kinematical

and spatially dieomorphism invariant level At the kinematical level we

may note that the volume and area op erators are enough to distinguish all

spin networks from each other This means that they must b e orthogonal

to each other in any kinematical inner pro duct If we imp ose the additional

conditions that the lo op op erators T b e hermitian which means that we

are describing Euclidean quantum gravityor a formulation such as that in

in which the connection is real then it is straightforward to show that

at this kinematical level the spin network states have unit norms An im

p ortant op en problem is whether there is a dierent choice of norms for the

spin network basis that realizes the Minkowskian reality conditions

At the dieomorphism invariant level we may note that we have at least

one explicit observable which is the volume of the spatial slice If this is

to b e hermitian then we know that the spin network states corresp onding

to dierent volumes must b e orthogonal to each other This is related to

the fact that b oth the area and volume are indeed symmetric with

resp ect to the AshtekarLewandowski inner pro duct We may note that

the dieomorphism invariant inner pro duct is physically meaningful in the

treatment of evolution describ ed in

Penrose diagram notation

Our nal task in this pap er is to exploit the results we have describ ed to in

tro duce a notation for the lo op states which simplies the graphical calculus

in the lo op representation We recall that we indicate as D the pictorial

representation of the lo op We are now going to dene a notation for the

lo op states which we denote as the Penrose notation or P notation This

was also called the binor formalism by Penrose This is done as follows In

P notation a lo op state ji is also represented by a certain pictorial represen

tation of the lo op itself which to distinguish it we will call P However

the representation takes into account the sign factor that we discussed in

previous sections Thus we cho ose the convention that the diagram of a

n()+1

lo op P in P notation is related to the lo op state ji by ji

where n we recall indicates the numb er of single lo ops or comp onents

of the multiple lo op In other words the P notation P of a lo op state

ji is dened by

n(a)+1

P D

The imp ortant asp ect of the P notation is that with these conventions the

the spinor identy is now lo cal In fact it now reads as

+ + = 0

Therefore equation holds rigorously whithin this notation Thus can

solve the spinor identity on trivalent states by restricting to the states in

which every rop e is in P notation fully antisymmetrized It is imp ortant

to note that the P notation is completely top ological in that a diagram

corresp onds to the same lo op state no matter how it is oriented ot drawn

This is a great advantage in calculations

For completeness we mention a variant of the P notation has b een used

in some published work The variant which we may denote Symmetric

Penrose notation corresp onds to what Penrose called the spinor calculus

as opp osed to binor calculus In this notation which we will refer to as SP

notation the diagram that corresp onds to a lo op will b e labled S It

is dened by

n()+c()+m()+1

S D

We recall that c and m are the numb er of crossings and the numb er

of minima in D Notice that the SP notation is not top ological in the

sense that the way the lo op a is drawn matters for the determination of the

sign adding a minumun and a maximum is equiavelent to change the sign

of the state Notice that the p ermutations of the lo ops along a rop e change

the numb er of crossings therefore the antisimmetrization in the P notation

corresp onds to a symmetrization in SP notation hence the name While

it is more cumb ersome for calculation the signicance of the SP notation is

that it has an immediate interpretation in terms of Penrose graphical tensor

calculus which we dened earlier in the connection representation Indeed

we have immediately that

SP G

Finally we mention the fact that the P notationcan b e obtained from

the graphical tensor notation by adding the immaginary unit to each and

adding a minus one for every crossing

Lo op op erators in Penrose notation

A most valuable asp ect of the Penrose diagram notation we have intro duced

is the simplication it allows in the calculus with the lo op op erators In

this section we describ e the action of the lo op op erators on the lo op states

expressed in Penrose notation

We recall that in terms of the standard lo op notation the action of

1 2

the lo op op erators T and T is given by

a a 1

h jT s sh j h j

X

r (i) ab a b

h j h jT s t s t

st i

i

a

The distributional factor s which do esnt play any role in the

diagramatics is dened in The geometrical part of the action of these

op erators can b e co ded in the grasping action shown in

2 a ()- l p

α β

However as is well known by anyb o dy who attempted to p erform complex

computations with these op erators the lo cal graphical action expressed in

eq do es not suce to compute the correct linear combination app earing

in the rhs of eq The diculty is given by the signs in front of the

various terms These signs are dictated by the global ro oting prop erties of

the lo op that are b eing grasp ed In particular the sign is determined in

by r i which is dened as the numb er of segments that have to

b e reversed in order to obtain a consistent orientation of the lo op after the

rero oting While complete this way of determining the sign is cumb ersome

and in computing the action of op erators as the area the Hamiltonian or the

volume the determination of the signs is the hardest part of the calculation

This diculty disapp ears using the Penrose diagram notation

1

Let us b egin by considering the action of T on a lo op state

a

T [ α ] ()s β

To compute this we express the rhs of in Penrose notation The result is

β β 2 a ( + ) l p

α α

Notice the plus sign due to the change of sign

β -1 = α ∗ β

α

a

This suggests that we indicate the op erator T s by

a a

T []β ()s β

To notate the action of the op erator we may then intro duce the following

fundamental grasping rule

2 a = l p

a

By using this rule we have immediately the correct action represented as

β β 2 e a = l p α

α

The go o d news are that this generalizes immediately to the higher order

2

lo op op erators For example let us represent the T lo op op erator as

e b e b α ( )

T [] s 1 s 2

Then we can use the fundamental grasping rule ab ove to compute the action

2

of T on a generic state We obtain

4 e b = l p

e b

By expanding the diagrams and taking into account the sign rules it is

straightforward to show that the rhs of this diagram represents the correct

linear combination of lo op states corresp onding to the rhs of equation

n

This result generalizes to higher order T op erators Thus by using

the Penrose notation the grasping rule of eq enco des automatically the

n

pattern of the signs in the action of the T op erators

n

These simplications extend to all the higher T op erators For example

in we showed how this this notation simplies the computation of the

3

action of the volume op erator which is dened in terms of a T op erator

This made much simpler the work of solving the corresp onding sp ectral

problem leading to the computation of the eigenstates and eigenvalues of

the op erator that corresp onds to the volume of an arbitrary region of space

Conclusion

We have dened a basis of indep endent states in the lo op and in the connec

tion representations of quantum gravity which solves the Mandelstam iden

tities This basis is lab elled by a generalization of Penrose spin networks It

is orthonormal in the scalar pro duct dened by the AsktekarLewandowski

measure and provides a simple relation b etween the connection and the

lo op representation We have intro duced a notation for the lo op states of

quantum gravity based on Penroses graphical tensor notation In this no

tation the action of the lo op op erators b ecomes lo cal and can b e expressed

in terms of the simple graphical rule given in equation

An intriguing suggestion on the p ossibility of mo difying the framework

we have presented in this pap er follows from the following observation Be

cause of the shortscale discretness of the geometry the only remain

ing divergences in nonp erturbative quantum gravity must b e infrared diver

gences analogous to the spikes or the uncontrolled proliferation of baby

universes seen in nonp erturbative numerical calculations employing dy

namical triangulations in b oth two and four dimensions In the present

context a source of such divergences may b e the sum over the colorings

of the spin networks which lab el the representations of SU This sug

gests that a natural invariant regularization of the theory could b e provided

by replacing SU with the quantum group SL Such a strategy has

q

b een successfully implemented in dimensions by Turaev and Viro

and there have b een attempts to extend it to four dimensional dieomor

phism invariant theories The use of qdeformed spin networks in the

lo op representation of quantum gravity is presently under investigation

Furthermore spin networks may make it p ossible to dene quantum gravity

on manifolds with a nite b oundary and to use the metho ds of top olog

ical eld theory to describ e the structure of the physical quantum gravity

state space in the presence of b oundaries In this context the level q of

SL turns out to b e related to the inverse of the cosmological constant

q

These investigations reinforce the conjecture that the q deformation

could play the role of infrared regulator The p ossible relevance of q defor

mations of the gauge group SU in quantum gravity is also suggested by

L

the imp ortant role quantum groups play in knot theory as well as by

the p ossibility of quantumgravity induced quantumgroup deformations of

the spacesymmetries

Finally we remark that the existence of a spin network basis for the

space of dieomorphism invariant states of the quantum gravitational eld

as well as the imp ortant role they seem to play in b oth practical calculations

and mathematical developments may b e seen as

vindicating the picture of a discrete combinatorial description of spacetime

geometry as well as the reasoning that led Roger Penrose to their original

construction

Acknowledgements

We thank Roger Penrose for intro ducing us to the details of the spin network

calculus and John Baez and Louis Kauman for explanations and various

suggestions ab out the use of spin networks We also thank Roumen Borissov

for many comments and suggestions Abhay Ashtekar Bernd Bruegmann

Louis Crane Junichi Iwasaki Renata Loll Seth Ma jor Jorge Pullin and

other memb ers of the Center for Gravitational Physics and Geometry for

discussions as well as for comments on an earlier draft of this pap er We

are grateful to Harlow Igims for help with the gures Finally ls would

like to thank Prof Chris Isham and the Newton institute for supp ort for

a visit during which this work was b egun and Prof Mauro Carfora and

SISSA for supp ort and hospitality where it was continued This work was

partially supp orted by the NSF under grants PHY PHY

PHY and INT and by research funds of Penn State Univer

sity

App endix

In this App endix we extend the denition of spin network states to inter

sections of any order The complication intro duced by higher order inter

sections is the fact that the order of the rop es entering the intersection do es

not determine the routing uniquely For instance in the simplest p ossi

ble fourth order intersection with all four rop es of order we have three

p ossible ro otings

out of which two are indep endent due to the spinor identity Neverthe

less with a small amount of additional technical machinery it is p ossible

to extend the spin network basis to include arbitrary intersections This

is b ecause given the order n of an intersection i and given the coloring

p p of the n rop es adjacent to i there is only a nite numb er of ways

1 n

of ro otings the lo ops through the intersection and therefore a smaller

nite numb er k p p of independent ro otings For completeness we put

1 n

k p p if a consistent ro oting through the intersection do es not

1 n

P

p exist for n rop es of orders p p this is for instance the case if

j 1 n

j

is o dd In the particular case of trivalent intersections n we have

k p p if the sum of three colors p is even and none of the three

1 3 j

is larger than the sum of the other two and k p p otherwise

1 3

In order to extend the denition of spin network states to nontrivalent

lo ops it is sucient to cho ose a unique way of lab eling the k p p

1 n

indep endent ro otings through an intersection i by means of an integer v

i

k p p Once this is done we dene a generalized spin network s

1 n

as an oriented imb edded graph with p ositive integers or colors p and

l

v assigned to each link l and to each of no de i satisfying the relations

i

v k p p p p b eing the colors of the links adjacent to the no de

i 1 n 1 n

i The construction of the corresp onding spin network quantum states hsj

is then as b efore

The task of lab eling indep endent spin networks can b e achieved as fol

(n)

lows For every n we cho ose a unique trivalent graph with n free ends

and no closed lo op for instance we may cho ose

Such a graph will have n links adjacent to the free ends and n internal

links which we denote as virtual links For every n and every set of colors

p p we consider the p ossible colorings q q of the virtual links

1 n 1

(n3)

(n)

of which are compatible with the colorings p p of its external links

1 n

under the spin networks vertex conditions We obtain in this way a family

(n) (n)

of colored trivalent spin networks with n external links

1

k (p p )

n

1

colored p p It is not dicult to see that these are linear combinations

1 n

of ro otings through the intersection which exhaust all p ossibilities up to the

spinor identities and which are indip endent from each other We lab el these

intersection with the integer v k p p If there is no way of

1 1 n

matching the coloring we put k p p

1 n

Let us work out an example of fourth order intersection

We arbitrarily pair the four rop es for instance let us pair the North and

West rop es and the South and East ones and expand the intersection by

intro ducing an additional virtual rop e b etween the joins of the paired

rop es We obtain

In this way the fourth order intersection is expanded into two trivalent

intersections Notice that in Figure the external rop es are symmetrized

while the internal one is not By using the spinor relations we can then

replace the diagram with a linear combination of diagrams in which the

internal rop e to o is symmetrized Thus we can represent the intersection as

p p

q i = c i q s i s

r r

where we have used the spin network notation The co ecients c dep end

i

on and can b e computed from the original ro otings in the fourth order

intersection The index i ranges from maxjp q j jr sj to minp q r s

Finally once the pairing is chosen it is clear that the decomp osition of

eq is always p ossible and unique and it reduces the spinor identities

completely Thus we have

k p q r s maxjp q j jr sj minp q r s

indep endent fourth order intersections b etween rop es of orders p q r s

and we have a simple way of ordering them in terms of the color of the

internal rop e

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