Baez J.C. (Ed.) Knots and Quantum Gravity (Oxford Lecture Series in Mathematics and Its Applications)

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The Loop Formu lation of Gauge Theory and Gravity 9

the variables T (γ) as a set of basic variables: on physical grounds, one

requires only physical observables O (i.e. in our case the Wilson loops, and

not the local ﬁelds A) to be represented by self-adjoint operators

O in the

quantum theory, with the classical Poisson relations {O, O

} going over to

the quantum commutators i¯h[

There is also a mathematical justiﬁcation for re-expressing the classical

theory in terms of loop variables. It is a well-known result that one can

reconstruct the gauge potential A up to gauge transformations from the

knowledge of all holonomies U

, for any structure group G ⊂ GL(N, C).

Therefore all that remains to be shown is that one can reconstruct the

holonomies {U

} fr om the set of Wilson loops {T (γ)}. It turns out that,

at least for compact G, there is indeed an equivalence between the two

quotient spaces [17]

{T (γ), γ ∈ ΩΣ/Diﬀ

)}

Mandelstam constraints

. (4.1)

By Ma ndels tam constr aints I mean a set of necessary and suﬃcient con-

ditions on generic complex functions f (γ) on ΩΣ/Diﬀ

) that ensure

they are the traces of holonomies of a group G in a given representation

R. They are typically algebra ic , non-linear constraints among the T (γ).

To obtain a complete set of such conditions (without making reference to

the connection space A) is a non-trivial problem that for general G to my

knowledge has no t b e en solved.

Fo r example, for G = SL(2, C) in the fundamental representation by

2 × 2-matrices, the Mandelstam constraints are

(a) T (point loop) = 1

(b) T (γ) = T (γ

−1

)

◦ γ

) = T (γ

◦ γ

) (4.2)

(d) T (γ) = T (γ

), γ

= γ ◦ µ ◦ µ

−1

(e) T (γ

)T (γ

) =



T (γ

◦ λ ◦ γ

◦ λ

−1

) + T (γ

◦ λ ◦ γ

−1

◦ λ

−1

)



The loop γ

in eqn (d) is the loop γ with an ‘appendix’ µ ◦ µ

−1

attached,

where µ is any path (not necessarily closed) starting at a point of γ (see

Fig. 1). In (e), λ is a path starting at a point of γ

and ending at γ

(see

Fig. 2). The circle ◦ denotes the usual path composition. It is believed

that the set (4.2) and all conditions that can be derived from (4.2) exhaust

the Mandelstam constraints for this particular gauge group and represen-

tation, although I am not aware of a formal proof. Still, owing to the

non-compactness of G, it can be shown that not every element of A/G can

be reconstructed from the Wilso n loops. This is the case for holonomies

that take their values in the so-called subgroup of null ro tations [18]. How-

ever, this leads only to an incompleteness ‘of measure zero’ of the Wilson

10 Renate Loll

Fig. 1. Adding an appendix µ ◦ µ

−1

to the loop γ does not change the

Wilson loop

Fig. 2. The loop conﬁgurations related by the Mandelstam constraint

T (γ

)T (γ

) =



T (γ

◦ λ ◦ γ

◦ λ

−1

) + T (γ

◦ λ ◦ γ

−1

◦ λ

−1

)



loops on A/G.

If one wants to restrict the group G to a subgroup of SL(2, C), there

are further conditions on the loop variables, which now take the form of

inequalities [23]. For SU (2) ⊂ SL(2, C), one ﬁnds the two conditions

−1 ≤ T (γ) ≤ 1,



T (γ

◦ γ

) − T (γ

◦ γ

−1

)



≤ 4(1 −T (γ

)

)(1 −T (γ

)

), (4.3)

whereas for SU(1, 1) ⊂ SL(2, C) one derives

T (γ

)

≤ 1 ∧ T (γ

)

≤ 1 ⇒



T (γ

◦ γ

) − T (γ

◦ γ

−1

)



≥ 4(1 − T (γ

)

)(1 − T (γ

)

), (4.4)

and no restrictions for other values of T (γ

) and T (γ

). Unfortunately,

The Loop Formu lation of Gauge Theory and Gravity 11

(4.3) and (4.4) do not exhaust all inequalities there are for SU(2) and

SU(1, 1) respectively. There are more inequalities corresponding to more

complicated conﬁgurations of more than two intersecting loops, which ar e

critically dependent on the geometry of the loop conﬁguration.

Let me ﬁnish this section with some further remarks on the Mandelstam

constraints:

• In cases where the Wilson loops form a complete set of variables,

they are actually overcomplete, due to the existence of the Mandel-

stam constraints (i.e. implicitly the non-trivial topological structure

of A/G). This is the overcompleteness I referred to in the previous

section.

• Note that one may interpret some of the constraints (4.2) as con-

straints on the underlying loop spa ce, i.e. the Wilson loops are eﬀec-

tively not functions on ΩΣ/Diﬀ

), but on some appropriate quo-

tient with respect to equivalence relations such as γ ∼ γ ◦µ◦µ

−1

. The

existence of such relations makes it diﬃcult to apply directly math-

ematical results on the loop space ΩΣ or the space ΩΣ/Diﬀ

)

of oriented, unparametrized loops (such as [12, 13, 14, 31]) to the

present case.

• One ha s to decide how the Mandelstam constraints are to be carried

over to the quantum theory. One may try to eliminate as many of

them as possible already in the cla ssical theory, but real progress

in this direc tion has only been made in the discretized lattice gauge

theory (see Section 5 below). Another possibility will be mentioned

in Section 6.

• Finally, let me emphas ize that the equivalence of the connection and

the loop description at the classical level does not necessarily mean

that quantization with the Wilson loops as a set of basic variables

will b e equivalent to a quantization in which the connections A ar e

turned into fundamental quantum op erators. One may indeed hope

that at some le vel, those two possibilities lead to diﬀerent results,

with the loop representation being better suited for the description

of the non-perturbative structure of the theory.

5 Some lattice results on loops

This section describes the idea of taking the Wilson loops as the set of basic

va riables, as applied to gauge theo ry, although some of the techniques a nd

conclusions may be relevant for gravitational theories too. The lattice

discretization is particularly suited to the loop formulation, because in it

the gauge ﬁelds are taken to be concentrated on the 1-dimensional links l

of the lattice. The Wilson loops are easily formed by multiplying together

the holonomies U

. . . U

of a set of contiguous, oriented links that form

12 Renate Loll

Fig. 3. γ is a loop on the hyperc ubic lattice N

, N = 6, d = 2.

a closed chain on the lattice (see Fig. 3),

T (γ) = Tr

. . . U

. (5.1)

We take the lattice to be the hypercubic lattice N

, with periodic

boundary conditions (N is the number o f links in each direction), and

the gauge group G to be SU (2), again in the fundamental re presentation.

Although the lattice is ﬁnite, there are an inﬁnite number of closed con-

tours one can form on it, and hence an inﬁnite number of Wilson loops. At

the same time, the number of Mandelstam constraints (4.2) is inﬁnite. The

only constraints that cause pro blems are tho se of type (e), because of their

non-linear and highly coupled character. Nevertheless, as I showed in [19],

it is pos sible to solve this coupled set of equations and obtain an explicit

local description of the ﬁnite-dimensional physical conﬁguration space

phy



lattice

{T (γ), γ a lattice loop}

Mandelstam constraints

(5.2)

in terms of an independent set of Wilson loops. The ﬁnal solution for

the indep e ndent degrees of freedom is surprisingly simple: it suﬃces to

use the Wilson lo ops of small lattice loops, consisting of no more than six

lattice links. At this point it may be noted that a similar problem arises

in 2 +1 gravity (in the connection formulation) on a manifold Σ

×R, with

a compact, 2-dimensional Riemann surface Σ

of genus g. O ne also has a

discrete set of loops, namely, the elements of the homotopy group of Σ

and can intro duce the corresponding Wilson loo ps. Again, one is interested

in ﬁnding a maximally reduced set of such Wilson loops, in terms of which

all the others can be expressed, using the Mandelstam constraints (see, for

example, [7, 27]).

The Loop Formu lation of Gauge Theory and Gravity 13

Of cours e it does not suﬃce to establish a set of independent loop

va riables on the lattice; one also has to show that the dynamics can be

expressed in a manageable form in terms of these variables. Some important

problems are the following.

1. Since the space C

phy

= A/G is topologically non-trivial, the set of

independent lattice Wilson loops cannot provide a good global chart

for C

phy

. It would therefore be desirable to ﬁnd a minimal embedding

into a spa c e of loop variables that is topologically triv ial (together

with a ﬁnite number of non-linear constraints that deﬁne the space

phy

2. What is the explicit form of the Jacobian of the transformation from

the holonomy link variables to the Wilson loops? From this one could

derive an eﬀective action/ Hamiltonian for gauge theory, and maybe

derive a prescription of how to translate local quantities into their

loop space analogues.

3. Is it possible to deﬁne a perturbation theory in loop space, once one

understands which Wilson loop conﬁgura tions contribute most in the

path integral, say? This would be an interesting alternative to the

usual perturbation theory in A or the

-expansion, and may lead

to new and more eﬃcient ways of performing calculations in lattice

gauge theor y.

4. Lastly, in ﬁnding a solution to the Mandelstam constraints it was

crucial to have the concept of a smallest loop size on the lattice (the

loops of link length 4 going around a single plaquette). What are the

consequences of this result for the continuum theory, where there is

no obstruction to shrinking loops down to p oints?

Some progr e ss on points 1 and 2 has been ma de for small lattices [22], and

it is clear that all of the above-posed problems are r ather non-trivial, even

in the discretized lattice theory where the number of degr e es of freedo m is

eﬀectively ﬁnite. One may be able to apply some of the techniques used

here to 3+1-dimensional gravity, where discrete structures appear too (the

generalized knot classes) after factoring out by the spa tial diﬀeomo rphisms.

6 Quantization in the loop approach

I will now return to the discussion of the continuum theory of bo th gravity

and gaug e theory, and review a few concepts that have bee n used in their

quantization in a loop for mulation. The contributions by Ashtekar and

Lewandowski [6] and Pullin [29] in this volume contain more details about

some o f the points raised here .

The main idea is to base the canonical quantization of a theory with

conﬁguration space A on the non-loc al, gauge-invaria nt phase space vari-

ables T (γ) (and not on some gauge-covariant local ﬁeld variables). This

14 Renate Loll

is a somewhat unusual procedure in the quantization of a ﬁeld theo ry, but

takes directly into account the no n-linearities of the theory.

You may have noted that so far we have only talked about conﬁgura-

tion space variables, the Wilson loops. For a cano nical quantization we

obviously need some momentum variables that depend on the genera lize d

electric ﬁeld

E. One choice of a generalized Wilson loop that also depends

E is

(A,

(γ, s) := Tr U

(s)

(γ(s)). (6.1)

This variable depends now on both a loop and a marked point, and it is still

gauge invariant under the transformations (2.7). Also it is strictly speaking

not a function on phase space , but a vector density. This latter fact may be

remedied by integrating T

over a 2-dimensional ‘ribbo n’ or ‘strip’, i.e. a

non-degenera te 1-parameter congruence of curves γ

(s) =: R(s, t), t ∈ [0 , 1]

[5]. In any case, the ‘momentum Wilson loop’ depends not just on a loop,

but needs an additional geometric ingredient. I will use here the unsmeared

version since it do es not aﬀect what I am going to say.

The crucial ingre dient in the canonical quantization is the fact that the

loop variables (T (γ), T

(γ, s)) form a closing algebra with respect to the

canonical symplectic structure on T

∗

A. This fact was ﬁrst established by

Gambini and Trias [16] for the gauge group SU(N ) and later rediscovered

by Rovelli and Smolin in the c ontext of gravity [30]. For the special case of

G = SL(2, C) (or any of its subgroups) in the fundamental re presentation,

the Poisson algebra of the loop variable s is

{T (γ

), T (γ

)} = 0,

(γ

, s), T (γ

)} = −∆

(γ

, γ

(s))



T (γ

◦

) − T (γ

◦

−1

)



(γ

, s), T

(γ

, t)} = (6.2)

−∆

(γ

, γ

(s))



(γ

◦

, u(t)) + T

(γ

◦

−1

, u(t))



+∆

(γ

, γ

(t))



(γ

◦

, v(s)) + T

(γ

◦

−1

, v(s))



In the derivation of this semidirect product algebra, the Mandelsta m con-

straints (4.2e) have been used to bring the right-hand sides into a for m

linear in T . The structure constants ∆ are distributiona l and depend just

on the geometry of loops and intersection po ints,

∆

(γ, x) =

dt δ

(γ(t), x) ˙γ

(t). (6.3)

The Poisson bracket of two loop variables is non-vanishing only when

an insertion of an electric ﬁeld

E in a T

-variable coincides in Σ with (the

holonomy of) another loop. For gauge group SU (N), the right-hand sides

of the Poisson brackets can no longer be written as linear expressions in T .

The Loop Formu lation of Gauge Theory and Gravity 15

By a ‘quantization of the theory in the loop representation’ we will mean

a representation of the loop algebra (6.2) as the commutator alge bra of a

set of self-adjoint operators (

T (γ),

(γ, s)) (or an appropriately smeared

version of the momentum ope rator

(γ, s)). Secondly, the Mandelstam

constraints must be incorporated in the quantum theory, for example by

demanding that relations like (4.2) hold also for the corresponding quan-

tum operators

T (γ). For the momentum Wilson loops there are analogo us

Mandelstam constraints.

The completion of this quantization pr ogram is diﬀerent for gauge the-

ory and gravity, in line with the remarks made in Section 2 above. Fo r

Yang–Mills theory, one has to ﬁnd a Hilbert space of wave functionals that

depend on either the Wilson loops T (γ) or directly on the elements γ of

some appropriate (quotient of a) loop space. The quantized Wilson loops

must be self-adjoint with respect to the inner product on this Hilbert space,

since they correspond to genuine observables in the classical theory. Then

the Ya ng–Mills Hamiltonian (2.8) has to be re-expres sed as a function of

Wilson lo ops, and ﬁnally, one has to solve the eigenvector equation

Y M

(T (γ), T

(γ))Ψ(α) = EΨ(α), (6.4)

to obtain the spectrum of the theory. Beca us e of the lack of an appropriate

scalar product in the continuum theory, the only progress that has been

made along these lines is in the lattice-regularized version of the gauge

theory (see, for example, [11, 15]).

Fo r the case of general relativity one does not require the loop vari-

ables (T (γ), T

(γ, s)) to be represented self-adjointly, because they do not

constitute observables on the classica l phase spac e. At least one s uch repre-

sentation (where the generalized Wilson loops are repres e nted as diﬀerential

operators on a space of loop functionals

Ψ(γ)) is known, and has be e n ex-

tensively used in quantum gravity. The completion of the loop quantization

program for gravity requires an appropriate inclusion of the 3-dimensional

diﬀeomorphism constraints C

and the Hamiltonian constraint C, cf. (2 .6),

which again means that they have to be re-expressed in terms of loop vari-

ables. Then, solutions to these quantum constraints have to be found, i.e.

loop functionals that lie in their kernel,

(T (γ), T

(γ))Ψ(α) = 0 ∧

C(T (γ), T

(γ))Ψ(α) = 0. (6.5)

Finally, the space of these solutions (or an appropriate subspace) has

to be given a Hilber t space structure to make a physical interpretation

possible. No suitable scalar product is known at the moment, but there

exist a large number of solutions to the quantum constraint equations (6.5).

These solutions and their relation to knot invariants are the subject of

Pullin’s chapter [29].

There is one more interesting mathematical structure in the loop ap-

16 Renate Loll

proach to which I would like to draw attention, and which tries to e stablish

a relation between the quantum connection and the quantum loop rep-

resentation(s). This is the so-called loop transform, ﬁrst introduced by

Rovelli and Smolin [30], which is supposed to intertwine the two types of

representations ac c ording to

Ψ(γ) =

A/G

[dA]

(γ) Ψ

phy

(A) (6.6)

and



T (α)



(γ) :=

A/G

[dA]

(γ)



T (α)Ψ

phy

(A)



, (6.7)

and similarly for the action of the

(γ). Indeed, it was by the heuristic

use of (6.6) and (6.7) that Rovelli and Smolin arrived at the quantum

representation of the loop algebra (6.2) mentioned above. The relation

(6.6) is to be thought of as a non-linear analogue of the Fourier transform

Ψ(p) =

√

2π

dx e

ixp

Ψ(x) (6.8)

on L

(R, dx), but does in general not possess an inverse. Again, for Yang–

Mills theory we do not know how to construct a concrete measure [dA]

that

would allow us to proceed with a loop quantization pro gram, although in

this case the loop transform can be given a rigor ous mathematical meaning.

Fo r the application to gravity, the domain space of the integration will

be a smaller space M, as explained in Section 2 above, but for this case we

know even less about the well-deﬁnedness of expressio ns like (6.6) and (6.7).

There is, however, a variety of simpler examples (such as 2+1-dimensional

gravity, e le c trodynamics, etc.) where the loop transform is a very useful

and mathematically well-deﬁned tool.

The following diagram (Fig. 4) summarizes the relations between con-

nection and loop dynamics, both classically and quantum-mechanically, as

outlined in Sections 4 and 6.

The Loop Formu lation of Gauge Theory and Gravity 17

Fig. 4.

18 Renate Loll

Acknowledgements

Warm thanks to John Baez for organizing an inspiring workshop.

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