is nothing else than the Haar measure of the gauge group. It re¬‚ects the presence

of variables (a3 ) which are built from elements of the Lie group and not of the

Lie algebra. Because of the topological equivalence of SU (2) and S 3 (cf. (52))

the Haar measure is the volume element of S 3

d„¦3 = cos2 θ1 cos θ2 dθ1 dθ2 d• ,

with the polar angles de¬ned in the interval [’π/2, π/2]. In the diagonalization

of the Polyakov loop (201) gauge equivalent ¬elds corresponding to di¬erent

values of θ2 and • for ¬xed θ1 are eliminated as in the example discussed above

(cf. (70)). The presence of the Haar measure has far reaching consequences.

Topological Concepts in Gauge Theories 79

Center Re¬‚ections. Center re¬‚ections Z have been formally de¬ned in (184).

They are residual gauge transformations which change the sign of the Polyakov

loop (185). These residual gauge transformations are loops in SU (2)/Z2 (cf.

(66)) and, in axial gauge, are given by

1

/2 iπ„ 3 x3 /L

Z = ieiπ„ e .

They transform the gauge ¬elds, and leave the action invariant

¦µ ’ ¦† ,

a3 ’ ’a3 , A3 ’ ’A3 , S[A⊥ a3 ] ’ S[A⊥ a3 ] . (205)

Z: µ µ µ

The o¬-diagonal gluon ¬elds have been represented in a spherical basis by the

antiperiodic ¬elds

1 3

¦µ (x) = √ [A1 (x) + iA2 (x)]e’iπx /L . (206)

µ µ

2

We emphasize that, according to the rules of ¬nite temperature ¬eld theory, the

bosonic gauge ¬elds Aa (x) are periodic in the compact variable x3 . For conve-

µ

nience, we have introduced in the de¬nition of ¦ an x3 -dependent phase factor

which makes these ¬eld antiperiodic. With this de¬nition, the action of center

re¬‚ections simplify, Z becomes a (abelian) charge conjugation with the charged

¬elds ¦µ (x) and the “photons” described by the neutral ¬elds A3 (x), a3 (x⊥ ).

µ

Under center re¬‚ections, the trace of the Polyakov loop changes sign,

1 1

tr P (x⊥ ) = ’ sin gLa3 (x⊥ ) . (207)

2 2

Explicit representations of center re¬‚ections are not known in other gauges.

9.2 Perturbation Theory in the Center-Symmetric Phase

The center symmetry protects the Z’odd states with their large excitation en-

ergies (199) from mixing with the Z’even ground or excited states. Any ap-

proximation compatible with con¬nement has therefore to respect the center

symmetry. I will describe some ¬rst attempts towards the development of a

perturbative but center-symmetry preserving scheme. In order to display the

peculiarities of the dynamics of the Polyakov-loop variables a3 (x⊥ ) we disregard

in a ¬rst step their couplings to the charged gluons ¦µ (206). The system of

decoupled Polyakov-loop variables is described by the Hamiltonian

δ2

1 L 2

’

2

h= d x⊥ + [∇a3 (x⊥ )] (208)

2

2L δa3 (x⊥ ) 2

and by the boundary conditions at a3 = ± gL for the “radial” wave function

π

ˆ

ψ[a3 ] boundary = 0 . (209)

80 F. Lenz

V[a3 ]

a3

Fig. 12. System of harmonically coupled Polyakov-loop variables (208) trapped by the

boundary condition (209) in in¬nite square wells

This system has a simple mechanical analogy. The Hamiltonian describes a 2 di-

mensional array of degrees of freedom interacting harmonically with their nearest

neighbors (magnetic ¬eld energy of the Polyakov-loop variables). If we disregard

for a moment the boundary condition, the elementary excitations are “sound

waves” which run through the lattice. This is actually the model we would

obtain in electrodynamics, with the sound waves representing the massless pho-

tons. Mechanically we can interpret the boundary condition as a result of an

in¬nite square well in which each mechanical degree of freedom is trapped, as

is illustrated in Fig. 12. This in¬nite potential is of the same origin as the one

introduced in (140) to suppress contributions of ¬elds beyond the Gribov hori-

zon. Considered classically, waves with su¬ciently small amplitude and thus

with su¬ciently small energy can propagate through the system without being

a¬ected by the presence of the walls of the potential. Quantum mechanically

this may not be the case. Already the zero point oscillations may be changed

substantially by the in¬nite square well. With discretized space (lattice spacing

) and rescaled dynamical variables

a3 (x⊥ ) = gLa3 (x⊥ )/2 ,

˜

it is seen that for L the electric ¬eld (kinetic) energy dominates. Dropping

the nearest neighbor interaction, the ground state wavefunctional is given by

1/2

2

ˆ˜

Ψ0 [ a3 ] = cos [˜3 (x⊥ )] .

a

π

x⊥

In the absence of the nearest neighbor interaction, the system does not support

waves and the excitations remain localized. The states of lowest excitation energy

˜

are obtained by exciting a single degree of freedom at one site x⊥ into its ¬rst

excited state

cos [˜3 (˜⊥ )] ’ sin [2˜3 (˜⊥ )]

ax ax

with excitation energy

3 g2 L

∆E = . (210)

82

Thus, this perturbative calculation is in agreement with our general considera-

tions and yields excitation energies rising with the extension L. From comparison

with (199), the string tension

3 g2

σ=

82

Topological Concepts in Gauge Theories 81

is obtained. This value coincides with the strong coupling limit of lattice gauge

theory. However, unlike lattice gauge theory in the strong coupling limit, here no

con¬nement-like behavior is obtained in QED. Only in QCD the Polyakov-loop

variables a3 are compact and thereby give rise to localized excitations rather

than waves. It is important to realize that in this description of the Polyakov

loops and their con¬nement-like properties we have left completely the familiar

framework of classical ¬elds with their well-understood topological properties.

Classically the ¬elds a3 = const. have zero energy. The quantum mechanical zero

point motion raises this energy insigni¬cantly in electrodynamics and dramati-

cally for chromodynamics. The con¬nement-like properties are purely quantum

mechanical in origin. Within quantum mechanics, they are derived from the

“geometry” (the Haar measure) of the kinetic energy of the momenta conjugate

to the Polyakov loop variables, the chromo-electric ¬‚uxes around the compact

direction.

Perturbative Coupling of Gluonic Variables. In the next step, one may

include coupling of the Polyakov-loop variables to each other via the nearest

neighbor interactions. As a result of this coupling, the spectrum contains bands

of excited states centered around the excited states in absence of the magnetic

coupling [88]. The width of these bands is suppressed by a factor 2 /L2 as com-

pared to the excitation energies (210) and can therefore be neglected in the

continuum limit. Signi¬cant changes occur by the coupling of the Polyakov-loop

variables to the charged gluons ¦µ . We continue to neglect the magnetic cou-

pling (‚µ a3 )2 . The Polyakov-loop variables a3 appearing at most quadratically

in the action can be integrated out in this limit and the following e¬ective action

is obtained

1

Se¬ [A⊥ ] = S [A⊥ ] + M 2 d4 xAa (x) Aa,µ (x) . (211)

µ

2 a=1,2

The antiperiodic boundary conditions of the charged gluons, which have arisen

in the change of ¬eld variables (206) re¬‚ect the mean value of A3 in the center-

symmetric phase

π

A3 = a3 + ,

gL

while the geometrical (g’independent) mass

π2 1

’2

M2 = (212)

L2

3

arises from their ¬‚uctuations. Antiperiodic boundary conditions describe the

appearance of Aharonov“Bohm ¬‚uxes in the elimination of the Polyakov-loop

variables. The original periodic charged gluon ¬elds may be continued to be used

if the partial derivative ‚3 is replaced by the covariant one

iπ 3

‚3 ’ ‚3 + [„ , · ] .

2L

82 F. Lenz

Such a change of boundary conditions is a phenomenon well known in quantum

mechanics. It occurs for a point particle moving on a circle (with circumference

L) in the presence of a magnetic ¬‚ux generated by a constant vector potential

along the compact direction. With the transformation of the wave function

ψ(x) ’ eieAx ψ(x),

the covariant derivative

d d

’ ieA)ψ(x) ’

( ψ(x)

dx dx

becomes an ordinary derivative at the expense of a change in boundary condi-

tions at x = L. Similarly, the charged massive gluons move in a constant color

π

neutral gauge ¬eld of strength gL pointing in the spatial 3 direction. With x3

compact, a color-magnetic ¬‚ux is associated with this gauge ¬eld,

π

¦mag = , (213)

g

corresponding to a magnetic ¬eld of strength

1

B= .

gL2

Also quark boundary conditions are changed under the in¬‚uence of the color-

magnetic ¬‚uxes

π

ψ (x) ’ exp ’ix3 „ 3 ψ (x) . (214)

2L

Depending on their color they acquire a phase of ±i when transported around the

compact direction. Within the e¬ective theory, the Polyakov-loop correlator can

be calculated perturbatively. As is indicated in the diagram of Fig. 13, Polyakov

loops propagate only through their coupling to the charged gluons. Con¬nement-

like properties are preserved when coupling to the Polyakov loops to the charged

gluons. The linear rise of the interaction energy of fundamental charges obtained

Fig. 13. One loop contribution from charged gluons to the propagator of Polyakov

loops (external lines)

Topological Concepts in Gauge Theories 83

in leading order persist. As a consequence of the coupling of the Polyakov loops

to the charged gluons, the value of the string constant is now determined by the

threshold for charged gluon pair production

4π 2

2

’2 ,

σpt = (215)

L2 3

i.e. the perturbative string tension vanishes in limit L ’ ∞. This de¬ciency

results from the perturbative treatment of the charged gluons. A realistic string

constant will arise only if the threshold of a Z’odd pair of charged gluons

increases linearly with the extension L (199).

Within this approximation, also the e¬ect of dynamical quarks on the Polya-

kov-loop variables can be calculated by including quark loops besides the charged

gluon loop in the calculation of the Polyakov-loop propagator (cf. Fig. 13). As

a result of this coupling, the interaction energy of static charges ceases to rise

linearly; it saturates for asymptotic distances at a value of

V (r) ≈ 2m .

Thus, string breaking by dynamical quarks is obtained. This is a remarkable and

rather unexpected result. Even though perturbation theory has been employed,

the asymptotic value of the interaction energy is independent of the coupling con-

stant g in contradistinction to the e4 dependence of the Uehling potential in QED

which accounts e.g. for the screening of the proton charge in the hydrogen atom

by vacuum polarization [89]. Furthermore, the quark loop contribution vanishes

if calculated with anti-periodic or periodic boundary conditions. A ¬nite result

only arises with the boundary conditions (214) modi¬ed by the Aharonov“Bohm

¬‚uxes. The 1/g dependence of the strength of these ¬‚uxes (213) is responsible

for the coupling constant independence of the asymptotic value of V (r).

9.3 Polyakov Loops in the Plasma Phase

If the center-symmetric phase would persist at high temperatures or small ex-

tensions, charged gluons with their increasing geometrical mass (212) and the