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20 Renate Loll

Represent ation Theory of Analytic Holonomy

C*-algebras

Abhay Ashtekar

Center for Gravitational Physics and Geometry,

Pennsylvania State University,

University Park, Pennsylvania 16802, USA

(email: ashtekar@phys.psu.edu)

Jerzy Lewandowski

Department of Physics, University of Florida,

Gainesville, Florida 32611, USA

(email: lewand@fuw.edu.pl)

Abstract

Integral calculus on the space A/G of gauge-equivalent connections

is developed. Loops, knots, links, and graphs feature prominently

in this description. The framework is well suited for quantization of

diﬀeomorphism-invariant theories of connections.

The general setting is provided by the Abelian C*-algebra of

functions on A/G generated by Wilson loops (i.e. by th e traces of

holonomies of connections around closed loops). The representation

theory of this algebra leads to an interesting and powerful ‘dual-

ity’ between gauge equivalence classes of connections and certain

equivalence classes of closed loops. In particular, regular measures

on (a suitable completion of) A/G are in 1–1 correspondence with

certain functions of loops and diﬀeomorphism invariant measures

correspond to (generalized) knot and link invariants. By carrying

out a non-linear extension of the theory of cylindrical measures on

topological vector spaces, a faithful, diﬀeomorphism-invariant mea-

sure is introduced. This measure can be used to deﬁne the H ilbert

space of quantum states in theories of connections. The Wilson loop

functionals then serve as the conﬁguration operators in the quantum

theory.

1 Introduction

The space A/G of connections modulo gauge transformations plays an im-

portant role in gauge theories, including certain topological theories and

general relativity [1]. In a typical set-up, A/G is the classical conﬁgu-

ration space. In the quantum theory, then, the Hilbert space of states

On leave from: Instytut Fizyki Teoretycznej, Warsaw University ul. Hoza 69, 00–689

Warszawa, Poland

22 Abhay Ashtekar and Jerzy Lewandowski

would consist of all square-integrable functions on A/G with respect to

some measure. Thus, the theory of integration over A/G would lie at

the heart of the quantization procedure. Unfortunately, since A/G is an

inﬁnite-dimensional non-linear space, the well-known techniques to deﬁne

integrals do not go through and, at the outset, the problem appears to be

rather diﬃcult. Even if this problem co uld be overcome, one would still

face the issue of constructing self-adjoint operators corresponding to phys-

ically interesting observables. In a Hamiltonian approach, the Wilson loop

functions provide a natural set of (manifestly gauge-invariant) conﬁgura-

tion observables. The problem of constr ucting the ana logous, manifestly

gauge-invariant ‘momentum observables’ is diﬃcult already in the classical

theory: these observables would cor respond to vector ﬁelds o n A/G and

diﬀerential ca lculus on this space is not well developed.

Recently, Ashtekar and Isham [2] (herea fter referred to as A–I) devel-

oped an algebraic appr oach to tackle these problems. The A–I approach

is in the setting of canonical quantization and is ba sed on the ideas in-

troduced by Gambini and Trias in the context of Yang–Mills theories [3 ]

and by Rovelli and Smolin in the co ntext of quantum general relativity

[4]. Fix an n-manifold Σ on which the connections are to be deﬁned and

a compact Lie group G which will be the gauge group of the theory un-

der consideration

. The ﬁrst step is the construction of a C*-algebra of

conﬁguration observables—a suﬃciently large se t of gaug e -invariant func-

tions of connections on Σ. A natural strategy is to use the Wilson loo p

functions—the tra c es of holo nomies of connections around closed loops on

Σ—to generate this C*-algebra. Since these are conﬁguration observables,

they commute even in the quantum theory. The C*-algebra is therefore

Abelian. The next step is to cons truct representations of this algebra.

Fo r this, the Gel’fand spec tral theory provides a natural setting since any

given Abelian C*-algebra with identity is naturally isomorphic with the

C*-algebra of continuous functions on a compact, Hausdor ﬀ space, the

Gel’fand spectrum sp(C

) of that algebra. (As the notation suggests,

sp(C

) can be constructed directly from the given C*-algebr a: its ele-

ments are homomorphisms from the given C*-algebra to the ?-algebra of

complex numbers.) Consequently, every (continuous) cyclic representation

of the A–I C*-algebra is of the following type: the carrier Hilbert space is

(sp(C

), dµ) for some regular measure dµ and the Wilson loop operators

act (on square-integrable functions on sp(C

)) simply by multiplication.

A–I pointed out that, since the elements of the C*-algebra are labelled by

loops, there is a 1–1 corres po ndence between regular measures on sp(C

)

In typical applications, Σ will be a Cauchy surface in an (n + 1)-dimensional space-

time in the Lorentzian regime or an n-dimensional spacetime in the Euclidean regime.

In the main body of this paper, n will be taken to be 3 and G will be taken to be

SU (2) both for concreteness and simplicity. These choices correspond to the simplest

non-trivial applications of the framework to physics.

Holonomy C*-algebras 23

and certain functions on the spac e of loops. Diﬀeomorphism-invaria nt mea-

sures correspond to knot and link invariants. Thus, there is a natural inter-

play between connections and loops, a point which will play an important

role in the present paper.

Note that, while the classical conﬁguration space is A/G, the domain

space of quantum states is sp(C

). A–I showed that A/G is naturally

embedded in sp(C

) and Rendall [5] showed that the embedding is in fact

dense. Elements of sp(C

) can be therefore thought of as generalized gauge-

equivalent connections. To emphasize this point and to simplify the nota-

tion, let us denote the spec trum of the A–I C* -algebra by

A/G. The fact

that the domain spa c e of quantum states is a completion of—and hence

larger than—the classical conﬁguration space may seem surprising at ﬁrst

since in ordinary (i.e. non-relativistic) quantum mechanics both spaces are

the same. The enlarg ement is, however, characteristic of quantum ﬁeld

theory, i.e. of systems with an inﬁnite number of degrees of freedom. A–I

further explored the structur e of A/G but did not arrive at a c omplete

characterization of its elements. They also indicated how one could make

certain heuristic considerations of Rovelli and Smolin precise a nd associate

‘momentum operators’ with (closed) strips (i.e. (n−1)-dimensional r ibbons )

on Σ. However, without further information on the measur e dµ on A/G,

one cannot decide if these operators can be made self-adjoint, or indeed

if they are even densely deﬁned. A–I did introduce certain measures with

which the strip operators ‘interact properly’. However, as they pointed out,

these measures have support o nly on ﬁnite-dimensional subspaces of A/G

and are therefore ‘too simple’ to be interesting in cases where the theory

has a n inﬁnite number of deg rees of freedom.

In broad terms, the purpose of this paper is to complete the A–I program

in several directions. More speciﬁcally, we will pres e nt the following res ults:

1. We will obtain a co mplete characterization of elements of the Gel’fand

sp e c trum. T his will provide a degree of control on the domain space of

quantum states which is highly desira ble: since A/G is a non-trivial,

inﬁnite-dimensional space which is not even locally compact, while

A/G is a compact Hausdorﬀ space, the completion procedure is quite

non-trivial. Indeed, it is the lack of c ontrol on the precise content

of the Gel’fand spectrum of physically interesting C*-algebras that

has prevented the Gel’fand theory from playing a prominent role in

quantum ﬁeld theory so fa r.

2. We will present a faithful, diﬀeomo rphism-invariant measure on A/G.

The theory under ly ing this construction sug gests how one might in-

troduce other diﬀeomorphism-invariant measur e s. Recently, Baez [6]

has exploited this general strategy to introduce new measure s. These

developments are interesting from a str ic tly mathematical standpoint

because the issue of existence of such measures was a subject of

24 Abhay Ashtekar and Jerzy Lewandowski

controversy until recently

. They are also of interest fro m a physical

viewp oint since one can, for example, use these measures to r e gulate

physically interesting operators (such as the constraints of general

relativity) and investigate their properties in a systematic fashion.

3. Together, these results enable us to deﬁne a Rovelli–Smolin ‘lo op

transform’ rigoro us ly. This is a map from the Hilbert space

(

A/G, dµ) to the space o f suitably regular functions of loops on

Σ. Thus, quantum states can be represented by suitable functions

of loops. This ‘loop representation’ is particularly well suited to

diﬀeomorphism-invariant theories of connections, including general

relativity.

We have also developed diﬀerential calculus on A/G and shown that the

strip (i.e. momentum) operators are in fact densely deﬁned, symmetric op-

erators on L

(A/G, dµ); i.e. that they interact with our measur e correctly.

However, since the treatment o f the strip o perators requires a number of

new ideas from physics as well as certain techniques fro m graph theo ry,

these results will be prese nted in a separate work.

The methods we use can be summarized as follows.

First, we make the (non-linear) duality between connections and loops

explicit. On the connectio n side, the appropriate space to consider is the

Gel’fand spectrum A/G of the A–I C*-algebra. On the loop side, the ap-

propriate object is the space of hoo ps—i.e. holonomica lly equivalent loops.

More precisely, let us consider based, piecewise analytic loops and regard

two as being equivalent if they give rise to the same holono my (evaluated

at the basepoint) for any (G-)connection. Call each ‘holonomic equivalence

class’ a (G-)hoop. It is straightforward to verify that the set of hoops ha s

the structure of a group. (It also carries a natural topo logy [7], which,

however, will not play a role in this paper.) We will denote it by HG and

call it the hoop group. It turns out that HG and the spectrum A/G can be

regarded as being ‘dual’ to each other in the following sense:

1. Every element of HG deﬁnes a continuous, complex-valued function

on A/G and these functions generate the algebra of all continuous

functions on A/G.

2. Every element of A/G deﬁnes a homomorphism from HG to the gauge

group G (which is unique mo dulo an automorphism of G) and e very

homomorphism deﬁnes an element of A/G.

In the case of topological vector spaces, the duality mapping respects

the topology and the linear structure. In the present case, however, HG has

the structure only of a gr oup and A/G of a topological space. The ‘duality’

mappings can therefore respect only these structures. Property 2 above

We should add, however, that in most of these debates, it was A/G—rather than i ts

completion

A/G—that was the center of attention.

Holonomy C*-algebras 25

provides us with a complete (alg e braic) characterizatio n of the elements

of Gel’fand spectrum

A/G while prope rty 1 speciﬁes the topology of the

sp e c trum.

The second set of techniques involves a family of projections from

the topological space A/G to certain ﬁnite-dimensional manifolds. For

each subgroup of the hoop group HG, genera ted by n independent hoops,

we deﬁne a projection from A/G to the quotient, G

/Ad, of the nth

power of the gauge group by the adjoint ac tion. (An element of G

/Ad

is thus an equivalence class of n- tuples, (g

, . . . , g

), with g

∈ G, where

, . . . , g

) ∼ (g

−1

, . . . , g

−1

) for any g

∈ G.) The lifts of func-

tions on G

/Ad are then the non-linear analogs of the cylindrical functions

on topological vector spaces. Finally, since each G

/Ad is a compact topo-

logical space, we c an equip it with measures to integrate functions in the

standard fashion. This in turn enables us to deﬁne cylinder measures on

A/G and integrate cylindrical functions ther e on. Using the Haar measure

on G, we then construct a natura l, diﬀeomorphism-invariant, faithful mea-

sure on A/G.

Finally, for completeness, we will summarize the strategy we have a dop-

ted to intro duce and analyze the ‘momentum operators’, although, as men-

tioned above, these results will be presented elsewhere. The ﬁrs t observa-

tion is that we can deﬁne ‘vector ﬁelds’ o n A/G as derivations on the ring of

cylinder functions. Then, to deﬁne the speciﬁc vector ﬁelds cor responding

to the momentum (or strip) operato rs we introduce certain notions from

graph theory. These speciﬁc vector ﬁelds are ‘divergence-free’ with respect

to our cylindrical measure on A/G, i.e. they leave the c ylindrical measure

invariant. Consequently, the momentum operators are densely deﬁned and

symmetric on the Hilbe rt space of square-integrable functions on A/G.

The paper is o rganized as follows. Section 2 is devoted to preliminaries.

In Section 3, we obtain the characterization of ele ments of A/G and also

present some examples of those elements which do not belong to A/G, i.e.

which represent genuine, gener alized (gauge equivalence classes of) connec-

tions. Using the machinery introduced in this characterization, in Section

4 we introduce the notion of cy lindrical measures on A/G and then deﬁne

a natural, faithful, diﬀeomorphism-invariant, cylindrical measure. Section

5 contains co ncluding remarks.

In all these considerations, the piecewise analyticity of loops on Σ plays

an important role. Tha t is, the speciﬁc constructions a nd proofs presented

here would not go through if the loops were allowed to be just piecewise

smooth. However, it is far from being obvious that the ﬁnal results would

not go through in the smooth category. To illustrate this point, in Appendix

A we restrict ourselves to U (1) connections and, by exploiting the Abelian

character of this group, show how one can obtain the main results of this

paper using piecewise C

loops. Whether simila r constructions are possible

in the non-Abelian case is, however, an open question. Finally, in Appendix

26 Abhay Ashtekar and Jerzy Lewandowski

B we consider another e xtension. In the main body of the paper, Σ is a

3-manifold and the gauge group G is taken to be SU(2). In Appendix B,

we indicate the modiﬁcatio ns required to incorporate SU (N) and U (N)

connections on an n-manifold (keeping, however, piecewise analytic loops).

2 Preliminaries

In this section, we will introduce the basic notions, ﬁx the notation, and

recall those results from the A–I framework which we will need in the

subsequent discussion.

Fix a 3-dimensional, real, a nalytic, connected, orientable manifold Σ.

Denote by P (Σ, SU(2)) a principal ﬁbre bundle P over the base space Σ

and the structure group SU(2). Since any SU(2) bundle over a 3-manifold

is trivial we take P to be the product bundle. Therefore, SU (2) connections

on P can be identiﬁed with su(2)-valued 1-form ﬁelds on Σ. Denote by A

the space of smooth (say C

) SU (2) co nnectio ns , equipped with one of the

standard (Sobolev) topologies (see, e.g., [8]). Every SU(2)-valued function

g on Σ deﬁnes a gauge transformation in A,

A · g := g

−1

Ag + g

−1

dg. (2.1)

Let us denote the quotient of A under the action of this group G (of

local gauge transformations) by A/G. Note that the aﬃne space structure

of A is lost in this projection; A/G is a genuinely non-linear space w ith a

rather complicated topological structure.

The next notion we need is that of closed loops in Σ. Let us begin by

considering continuous, piecewise analytic (C

) parametrized paths, i.e.

maps

p : [0, s

] ∪ . . . ∪ [s

n−1

, 1] → Σ (2.2)

which a re continuous on the whole domain and C

on the closed intervals

, s

k+1

]. Given two paths p

: [0, 1] → Σ and p

: [0, 1] → Σ such that

(1) = p

(0), we denote by p

◦ p

the natur al composition:

◦ p

(s) =



(2s), for s ∈ [0,

]

(2s − 1), for s ∈ [

, 1].

(2.3)

The ‘inverse’ of a path p: [0 , 1] → Σ is a path

−1

(s) := p(1 − s). (2.4)

A path which starts and ends at the same point is called a loop. We will

be interested in based loops. Let us therefore ﬁx, once and for all, a point

∈ Σ. Denote by L

the set o f piecewise C

loops which are based at

, i.e. which start and end at x

Given a connection A on P (Σ, SU(2)), a loop α ∈ L

, and a ﬁxe d point

ˆx

in the ﬁber over the point x

, we denote the corresponding element of the

Holonomy C*-algebras 27

holonomy group by H(α, A). (Since P is a product bundle, we will make

the obvious choice ˆx

= (x

, e), where e is the identity in SU(2).) Using

the space of connections A, we now introduce a key equivalence relation

on L

Two loops α, β ∈ L

will be said to be holonomically equivalent, α ∼ β, iﬀ

H(α, A) = H(β, A), for every SU(2) connection A in A.

Each holonomic equivalence class will be called a hoop. Thus, for example,

two loo ps (which have the same orientation and) which diﬀer only in their

parametrizatio n deﬁne the same hoop. Similarly, if two loops diﬀer just

by a retraced segment, they belong to the same hoop. (More precisely,

if paths p

and p

are such that p

(1) = p

(0) and p

(0) = p

(1) = x

so that p

◦ p

is a loop in L

, and if ρ is a path starting at the point

(1) ≡ p

(0), then p

◦p

and p

◦ρ ◦ρ

−1

◦p

deﬁne the same hoop.) Note

that in general the equivalence relation depends on the choice of the gauge

group, whence the hoop should in fact be called an SU(2 )-hoop. However,

since the group is ﬁxed to be SU (2) in the main body of this paper, we

drop this suﬃx. The hoop to which a loop α belo ngs will be denoted by ˜α.

The c ollection of all hoops will be denoted by HG; thus HG = L

/ ∼.

Note that, because of the prop e rties of the holonomy map, the space of

hoops, HG, ha s a natural group structure: ˜α ·

β =

α ◦ β. We will call it the

hoop group

With this machinery at hand, we can now introduce the A–I C*-algebra.

We begin by assigning to every ˜α ∈ HG a real-valued function T

˜α

on A/G,

bounded between − 1 and 1:

˜α

(

A) :=

Tr H(α, A), (2.5)

where

A ∈ A/G; A is any c onnection in the gauge equivalence cla ss

A; α

any loop in the hoop ˜α; and where the trace is taken in the fundamental

representation of SU (2). (T

˜α

is called the Wilson loop function in the

physics literature.) O wing to SU (2) trace identities, products o f these

functions can be expressed as sums:

˜α

˜α·

+ T

˜α·

−1

), (2.6)

where, on the right, we have used the composition of hoops in HG. There-

fore, the complex vector space spanned by the T

˜α

functions is closed under

the product; it has the structure of a ?-algebra. Denote it by HA and

call it the holonomy algebra. Since each T

˜α

is a bounded continuous func-

The hoop equivalence relation seems to have been introduced independently by a

number of authors in diﬀerent contexts (e.g. by Gambini and collaborators in their

investigation of Yang–Mills theory and by Ashtekar in the context of 2+1 gravity).

Recently, the group structure of HG has been systematically exploited and extended

by Di Bartolo, Gambini, and Griego [9] to develop a new and p otentially powerful

framework for gauge theories.

28 Abhay Ashtekar and Jerzy Lewandowski

tion on A/G, HA is a subalgebra of the C*-algebra of all complex-valued,

bounded, continuous functions thereon. The completion

HA of HA (under

the sup norm) has therefore the structure of a C*-alge bra. This is the A–I

C*-algebra.

As in [2], one can make the structure of HA explicit by constructing it,

step by step, from the space of loops L

. Denote by FL

the free vector

space over complexes g enerated by L

. Let K be the subspace of FL

deﬁned as follows:

∈ K iﬀ

˜α

(

A) = 0, ∀

A ∈ A/G. (2.7)

It is then eas y to verify that HA = FL

/K. (Note that the K-equivalence

relation subsumes the hoop equivalence.) Thus, elements of HA can be rep-

resented either as

˜α

or as

[α

]

. The ?-operation, the product,

and the norm on HA can be speciﬁed directly on FL

/K as follows:

[α

]

)

¯a

[α

]



[α

]





[β

]



([α

◦ β

]

+ [α

◦ β

−1

]

[α

]

k = sup

A∈A/G

˜α

(

A)|, (2.8)

where ¯a

is the complex conjugate of a

. The C*-algebra

HA is obtained

by co mpleting HA in the norm topology.

By construction, HA is an Abelian C*-algebra. Furthermore, it is

equipped with the identity element T

˜o

≡ [o]

, where o is the triv ial (i.e.

zero) loop. Therefore, we can directly apply the Gel’fand representation

theory. Let us summarize the main results that follow [2]. First, we know

that HA is naturally isomorphic to the C*-algebra of complex-valued, con-

tinuous functions on a c ompact Hausdorﬀ space, the spectrum sp(HA).

Further more, one can show [5] that the space A/G of connections mod-

ulo gaug e transfor mations is densely embe dded in sp(HA). Therefore, we

can denote the spectrum as A/G. Second, every continuous, cyclic rep-

resentation of HA by bounded operators on a Hilbert space has the fol-

lowing form: the repres e ntation space is the Hilbert spa c e L

(A/G, dµ) for

some regular measure dµ on A/G and elements of HA, regarded as func-

tions on A/G, act simply by (pointwise) multiplication. Finally, each o f

these representations is uniquely determined (via the Gel’fand–Naimark–

Segal construction) by a positive linear functional h·i on the C*-algebra

HA :

˜α

7→ h

˜α

i ≡

˜α

The functionals h·i naturally deﬁne functions Γ(α) on the space L

loops—which in turn serve as the ‘gener ating functionals’ for the r epresenta-