###### Abstract

We discuss the construction of the physical configuration space for Yang-Mills quantum mechanics and Yang-Mills theory on a cylinder. We explicitly eliminate the redundant degrees of freedom by either fixing a gauge or introducing gauge invariant variables. Both methods are shown to be equivalent if the Gribov problem is treated properly and the necessary boundary identifications on the Gribov horizon are performed. In addition, we analyze the significance of non-generic configurations and clarify the relation between the Gribov problem and coordinate singularities.

TPR-97-06

The Configuration Space of Low-dimensional Yang-Mills Theories

T. Pause^{a}^{a}ae-mail:
^{b}^{b}bSupported by the DFG-Graduiertenkolleg
Erlangen-Regensburg “Physics of Strong Interactions”
and T. Heinzl

*Universität Regensburg, Institut
für Theoretische Physik,*

*93040 Regensburg, Germany*

## 1 Introduction

Gauge field theories are at the heart of the standard model of the fundamental interactions. The weak coupling phase of the model is rather well understood in terms of standard perturbation theory. This is sufficient for the electro-weak theory where for the physically relevant scales weak and electromagnetic couplings are small. For the strong interactions, however, the situation is different. At small momentum transfer, or large distances, the associated gauge theory of color , quantum chromodynamics (QCD), is in the strong coupling phase and perturbation theory no longer works. One therefore has to develop nonperturbative techniques, the most elaborate one at the moment being lattice gauge theory [1, 2, 3].

An alternative approach, based on Bjorken’s idea of the femto-universe [4] has been initiated by Lüscher [5] and was later elaborated by van Baal and collaborators [6]. In this approach, one formulates QCD in a finite volume which in a first step is kept sufficiently small so that, due to asymptotic freedom, perturbation theory is still valid. Upon enlarging the volume, nonperturbative effects come into play, however, as is believed, in a controllable manner. Technically, one uses a Hamiltonian formulation of QCD or, neglecting quarks, pure Yang-Mills theory in the Coulomb gauge [7]. The way the nonperturbative effects show up is conceptually simple [8]. For small volumes, the wave functionals behave essentially as those in QED, i.e. they are concentrated around the classical vacuum. For larger volumes, the effective coupling increases, the wave functionals start to spread out in configuration space and become sensitive to its boundaries and nontrivial geometry [9]. It is therefore crucial for the understanding of these effects to learn as much as possible about the structure of the configuration space. Let us illuminate this reasoning with an example from quantum mechanics. For a particle in an infinitely deep square well of size there is a gap between the ground and first excited state of order . Obviously, the existence of the finite energy gap is directly related to the finite volume of the configuration space. Similar arguments have been given by Feynman to explain the origin of the mass gap for Yang-Mills theory in 2+1 dimensions [10] and are currently being re-investigated [11].

Let us discuss the case of non-Abelian gauge theories [12] in more detail. The configuration space of pure Yang-Mills theory is given in terms of the gauge fields (“configurations”) , which under the action of the gauge group transform as

(1) |

The set of all gauge equivalent points of a given
configuration constitutes the orbit of . Gauge invariance
requires physical quantities to take the same value for every
configuration on the orbit of . In this sense, the
description of gauge theories in terms of the potentials is
somewhat uneconomic as there is a huge redundancy associated with
these variables. One way to see this is the infinite volume factor
they contribute to the path integral measure. It is therefore
desirable to find the set of all gauge *inequivalent*
configurations, i.e. the space of gauge orbits

(2) |

which we will refer to as the *physical* configuration space
. The interesting question, of course, is, how to actually
find . A first hint can be obtained from (2),
which can naively be “solved” for yielding

(3) |

Though at this point it is unclear in which sense this identity really holds, it nevertheless suggests that the large configuration space of gauge potentials should be decomposed into gauge invariant quantities from and gauge variant ones parameterizing group elements . The decomposition indicated in (3) can explicitly be achieved using the transformation law (1): parameterize the group elements with an appropriate collection of angle variables such that

(4) |

Thus, any gauge potential carries an (implicit) label
which determines the position of on its orbit, in particular for . The identity (4) defines a
map which provides (at least locally)
the decomposition of an arbitrary configuration into a gauge
invariant representative and the gauge variant angles
. In general, this map will be a transformation from the
cartesian coordinates to *curvilinear* coordinates [7, 13].

Usually, the representative of the orbit is chosen via
*gauge fixing*, i.e. by defining functionals on such that

(5) |

This defines a hypersurface consisting
of all the representatives (or fields in the gauge ). There are two requirements that have to be met by an
admissible gauge fixing: *existence* and *uniqueness*.
Existence means that on *any* orbit there is a representative
satisfying the gauge condition. Thus, for any
there has to be a solution of the equation . The criterion of
uniqueness is satisfied if on each orbit there is only *one*
representative obeying the gauge condition. If, on the other hand,
there are (at least) two gauge equivalent fields, ,
, satisfying the gauge condition, the gauge is not
completely fixed. Instead, there is a residual gauge freedom given by
the gauge transformation connecting the copies, . In terms of the angles existence
and uniqueness mean that there is one and only one solution such that . As shown by Gribov
[14], for infinitesimal this amounts to the
condition that the Faddeev-Popov determinant, with

(6) |

should be non-vanishing. In this paper, we will concentrate on the transformation (4). Therefore, it is more natural to study the Jacobian of (4) instead of . The relation of both quantities is obtained via the chain rule,

(7) |

In what follows we will always work in a Hamiltonian formulation using the Weyl gauge, , which allows for a straightforward quantization [15]. The discussion above remains valid; one merely has to replace by its three-vector part .

For QED, the construction of the physical configuration space is rather straightforward, as gauge transformations are basically translations that preserve the cartesian nature of the coordinates. Explicitly, (4) becomes

(8) |

Thus, a natural representative is given by the transverse photon field (Coulomb gauge), and the angle is in one-to-one correspondence with the gauge variant longitudinal gauge field (for fields vanishing at spatial infinity). The physical configuration space consisting of transverse gauge potentials is Euclidean, i.e. flat and unbounded. This gives another explanation of why there is no mass gap for the photon so that it stays massless [10].

The situation becomes much more complicated for non-Abelian gauge theories. At variance with QED the decomposition (4) now involves curvilinear coordinates. It turns out that in this case (3) does not hold in a global sense as was first shown by Gribov [14] and Singer [16]. To be more specific consider the following example, which we will refer to as the Christ-Lee model [7, 13, 17, 18]. This model describes the motion of a particle in a plane with coordinates and which is the large configuration space, . Let the gauge transformations be the rotations around the origin. If we introduce polar coordinates, the radius and the angle , it is obvious that the radius is gauge invariant whereas , parameterizing the rotations, is gauge variant. The decomposition of (denoting ) is thus given by the transformation

(9) |

Accordingly, the physical configuration space is the non-negative real line

(10) |

Let us assume now that we are not as smart as to guess the gauge
invariant variable and proceed in a pedestrian’s manner via gauge
fixing. We gauge away , , and immediately
realize that this gauge selects *two* representatives on each
orbit at . There is a *discrete* residual gauge freedom
between the copies, , which constitutes the “Gribov
problem” for the example at hand. If we calculate the Faddeev-Popov
determinant,

(11) |

we find that it vanishes at , the “Gribov horizon”, which is just the point separating the two gauge equivalent regions and . Only if we fix the gauge completely by demanding that be non-negative we again have the non-negative real line as the physical configuration space and can identify with the radius . Denoting the representative satisfying as , we obtain the transformation analogous to (4),

(12) |

which is equivalent to the decomposition (9). In this simple case, the Jacobian of (12) is identical to the Faddeev-Popov determinant (11). We point out that has a boundary point, the origin, which is a fixed point under the action of the gauge group. In field theory such partially gauge invariant configurations [19] are called reducible [16]. For our simple models, however, we will use the term “non-generic” instead, to describe configurations that are invariant under subgroups of . Note that in the Christ-Lee model a single coordinate system suffices to parameterize the whole physical configuration space . This is not true in general as will be discussed in a moment.

The example above also raises another question. For gauge field theory several types of gauge invariant variables have been proposed [20, 21]). In the case of the Christ-Lee model we were able to “guess” a gauge invariant variable and after that found a gauge fixing and a representative corresponding to this particular choice of a gauge invariant variable. One might therefore ask whether it is generally true that to any construction of gauge invariant coordinates there corresponds a particular gauge fixing. We will address this question in the following sections.

It may also happen that the residual gauge freedom is continuous
instead of just discrete. In this case there are whole orbits
contained in the gauge fixing hypersurface, , which are
located at the Gribov horizon. A prominent example is provided by
axial-type gauges, , where the residual gauge freedom
consists of all gauge transformations independent of . To
proceed, one generally has to impose *additional* gauge
conditions to eliminate the continuum of Gribov copies. In this way
one identifies gauge equivalent points on the Gribov horizon. As the
latter seems to constitute (part of) the boundary of the physical
configuration space the described procedure is referred to as
“boundary identifications”
[22, 23, 24]. It is due to these
identifications that the nontrivial topology of the physical
configuration space comes into play, indicated by the fact that one
needs more than one coordinate system to cover . We will
discuss several examples where boundary identifications are necessary
and explicitely show how they are related to the topology of .

In general, we expect the features discussed above to also arise in Yang-Mills field theory. Of course there are additional complications due to the infinite number of degrees of freedom and the necessity of renormalization. Nevertheless, since Gribov’s original work there has been much progress in determining the physical configuration space, in particular by using the Coulomb gauge. In this particular case a certain distance functional turned out to be a very powerful tool to characterize [8, 23, 25, 26]. Due to the complicated nature of the functional, however, the set of gauge inequivalent configurations is only approximately known. A variant of the method also seems to work for the maximal abelian gauge [27] used to analyze the condensation of abelian monopoles and confinement due to a dual Meissner effect. Within lattice studies, in particular, the influence of Gribov copies on the dual superconductor scenario has been studied [28, 29].

At the moment, however, it is unclear how the same configuration space (which of course, by construction, has a gauge invariant meaning) can be obtained in different gauges. The method with the distance functional, for example, does not work in axial-type gauges. Furthermore, for the maximal abelian gauges, the physical configuration space has not been determined. We therefore consider it worthwhile to go back to quantum mechanics and a finite number of degrees of freedom. In the spirit of a recently presented soluble gauge model [30] we will address the question of finding the physical configuration space via (i) different types of gauge fixings, (ii) constructing gauge invariant variables without gauge fixing and (iii) relating these two methods.

The paper is organized as follows. In Section 2 we discuss a simple version of Yang-Mills quantum mechanics where the gauge group is reduced to . We will explicitly show the relation between the gauge fixing method and the method of gauge invariant variables. We will also perform the necessary boundary identifications and visualize the resulting physical configuration space by means of a suitable embedding into . Section 3 is mainly devoted to the study of non-generic configurations for the structure group . It will be shown, how these configurations give rise to a genuine boundary of the physical configuration space . As in Section 2 we will compare the spectra of the Hamilton operators defined on the gauge fixing surface and on , showing the equivalence of both. In Section 4 we will discuss Yang-Mills theory on a cylinder, which also reduces to a quantum mechanical model. We will apply the methods used in the preceding sections to construct the physical configuration space of this model and study the non-generic configurations.

## 2 Yang-Mills theory of constant fields

The first model we want to discuss is defined by the Lagrangian

(13) |

with the antisymmetric tensor . The special form of the kinetic term in (13) stems from the covariant time derivative in Yang-Mills theory for spatially constant fields. Since the lower indices and the upper indices of the basic variables only take the values and each, we will call our model the “-model”. For the time being we interpret as the Lagrangian describing the motion of two “particles” with position vectors and in a “color” plane with orthonormal basis vectors under the influence of the potential [17]. We choose the potential such, that it is invariant under

(14) |

where we parameterize the rotation matrix by a time-dependent angle

(15) |

For example, we may take a Yang-Mills type potential

(16) |

or the harmonic oscillator form

(17) |

Having chosen such a potential, we find, that is invariant under the combination of the transformations (14) and

(18) |

Hence in fact describes a gauge model with the abelian gauge group . Interpreting the transformations (14) as rotations of the coordinate system (), we realize that gauge invariance in our simple model means, that the physical motion of the two “particles” at positions , has to be independent of the (time-dependent) orientation of the coordinate axes. We will find that the correct implementation of this condition will eventually spoil our interpretation of and as the coordinates of independent particles.

As pointed out in the introduction, invariance under gauge transformations (14) and (18) implies, that the space of all configurations () contains redundant (unphysical) degrees of freedom. We will realize the reduction of to the physical configuration space using a Hamiltonian formalism. Denoting the momenta canonically conjugate to the coordinates by (the canonical momentum for vanishes) we get

(19) |

where we have put in the potential (16). The condition of gauge invariance is now expressed by the Gauß constraint equation , following from the Lagrangian equations of motion. In the particle picture we interpret as the total angular momentum, which has to vanish by gauge invariance. The variable , besides being the Lagrange multiplier of the constraint , may be interpreted as the angular velocity of a rotating coordinate system [31]. Because the physical quantities have to be independent of the rotation of the coordinate system, we are allowed to set (“body-fixed frame” [31]). This amounts to applying the gauge (fixing) transformations (14) and (18) with angle

(20) |

For Yang-Mills field theory this would correspond to the Weyl gauge , which does not fix gauge transformations constant in time. We will denote the group of time-independent gauge transformations as . In our model these residual transformations are parameterized by the undetermined time-independent integration constant in (20), which provides the orientation of the coordinate system at ,

(21) |

The “Weyl-gauge” Hamiltonian,

(22) |

only depends on the variables and . Assuming a Euclidean metric on , the form a pre-configuration space homeomorphic to with the Euclidean metric

(23) |

Therefore we can canonically quantize our model by replacing the Poisson brackets with quantum mechanical commutators,

(24) |

Accordingly, we promote functions on the phase space to operators acting on a Hilbert space, in particular . Within the Hamiltonian formalism, the Gauß constraint equation, , can only be realized weakly [32] on the Hilbert space of physical states ,

(25) |

In the Schrödinger representation the Hamilton operator acting on wave functions is given by the Laplacian on the Euclidean pre-configuration space and the Yang-Mills potential

(26) |

As discussed in the introduction there are several ways to eliminate the residual gauge symmetry and thus obtain the physical configuration space on which the physical wave functions are defined. To begin with, we will analyze a gauge condition which we will refer to as “axial gauge”.

### 2.1 Axial gauge

Since we have to eliminate one gauge degree of freedom, the most straightforward condition is to set one of the equal to zero. Thus, we demand the “axial gauge” condition

(27) |

which determines a three-dimensional gauge fixing surface . For any configuration there is a gauge (fixing) transformation which maps onto a point on . The inverse of this map is given explicitly by

(28) |

We interpret equation (28) as the transformation from the Euclidean coordinates to curvilinear coordinates . Multiplying equation (28) on both sides from the left with , we find that gets shifted to , whereas the variables remain unchanged. Therefore the transformation (28) explicitly realizes the separation of gauge variant from gauge invariant degrees of freedom. If this map was one-to-one, we would have found an homeomorphism , where we have identified the physical configuration space with the gauge fixing surface given in terms of the gauge invariant variables . However, it has been shown that in general it is impossible to write as a trivial fibre bundle [16]. Hence, let us study the map (28) in more detail by examining its Jacobian matrix , in particular the zeros of the Jacobian evaluated at ,

(29) |

We find that vanishes for indicating that the map (28) may not be one-to-one. In the following we will demonstrate how this is related to the existence of residual gauge copies. As in the case of the Christ-Lee model is equal to the Faddeev-Popov determinant modulo a possible sign change.

Intuitively the gauge condition (27) means that we rotate the coordinate system in color space such that the vector is collinear to the -axis. There are clearly two possibilities for this to happen: one where is parallel to and the other, where is anti-parallel. In terms of the transformation (28) we find that a given configuration may be represented by two sets of coordinates , since

(30) |

So if we discard the gauge variant variable , there are gauge equivalent configurations and related by a discrete residual gauge symmetry with the corresponding matrix . We may resolve this problem by restricting to positive values

(31) |

But what happens for , which implies, that there is no vector to rotate? The gauge condition (27) and define a hypersurface in , usually called the “Gribov horizon”. For the axial gauge, this is the plane . From the discussion above, we conclude, that the Gribov horizon separates regions on (“Gribov copies”), which are related by discrete residual gauge transformations. After the restriction to one Gribov copy (the “reduced gauge fixing surface”), demanding , the Gribov horizon seems to constitute a boundary of the configuration space. In order to see if this is in fact a boundary of the physical configuration space , let us have a closer look at configurations on the Gribov horizon. We find that every point on the gauge orbit of a horizon configuration does not only satisfy the gauge condition (27) but also :

(32) |

Hence, the Gribov horizon consists of complete gauge orbits. In the
generic case these orbits are non-degenerate and give rise to a
*continuous* residual gauge symmetry on the gauge fixing surface
. Therefore the complete reduction of the
configuration space requires an *additional* gauge condition for
the points on the Gribov horizon. We note that for horizon
configurations the “-model” is reduced to the “-model” of Christ and Lee [7] discussed
in the introduction. So by analogy to (12) we may proceed
with fixing the continuous residual gauge symmetry by imposing an
additional gauge condition on the configurations on the Gribov
horizon:

(33) |

As in the Christ-Lee model we take into account the residual discrete gauge symmetry, , by restricting the remaining degree of freedom to positive values,

(34) |

Note, that in the picture of independent particles, the problem arises because the total angular momentum is not well-defined, if one particle is at the origin. Nevertheless, it is possible to implement Gauß’ law by requiring the angular momentum of the other particle to vanish. This is exactly, what we have done in (33) and (34).

Now that we have eliminated all gauge symmetries, let us try to identify the physical configuration space . The axial gauge condition (27) reduces the pre-configuration space to a three dimensional gauge fixing surface parameterized by the coordinates . One might be tempted to regard this space as Euclidean. In order to check this, let us calculate the metric on the gauge fixing surface . Taking the Euclidean metric (23) on , we obtain by projecting tangent vectors in onto the horizontal subspace defined via Gauß’ law as shown by Babelon and Viallet [33, 34]. The projection onto the gauge fixing surface with coordinates finally yields

(35) |

with . The corresponding scalar curvature is given by , which is different from the zero curvature in the Euclidean case and even singular at the origin. We therefore have to conclude that the gauge fixing surface is not Euclidean.

To get some intuition for what happens let us embed parameterized by the coordinates into like depicted in Fig. 1. Note, however, that unlike for a Euclidean space the geodesics in this picture would no longer be straight lines, due to the nontrivial metric (35). We have also sketched the gauge orbits corresponding to the continuous residual gauge symmetry on the Gribov horizon at and the discrete residual gauge symmetry which interchanges with . Condition (31) eliminates the latter symmetry by restricting the gauge fixing surface to the upper half space, whereas the former symmetry reduces the Gribov horizon to the half line in accordance with the conditions (33) and (34). How can the upper half space and the half line be glued together to form the physical configuration space , which according to Singer [16] should be a smooth manifold, if non-generic configurations are discarded?

The answer is that we have to reconsider the additional gauge fixing on the Gribov horizon. Conditions (33) and (34) imply, that we have to identify every point on a residual gauge orbit with one point on the positive -axis. But we might as well identify such an orbit with the point () on the negative -axis in a continuous way. This identification is most easily performed by choosing spherical coordinates on and doubling the azimuthal angle. Just imagine the plane to be the surface of an opened umbrella. What we will do in the following is nothing but close the umbrella. We parameterize the gauge fixing surface with spherical coordinates , and ,

(36) |

Writing instead of guarantees that within the given range of we only parameterize the reduced configuration space defined by (31). With these new coordinates it is possible to define an embedding of the physical configuration space into , such that there are no residual gauge symmetries. Let , and denote cartesian coordinates in . Then we map any point in with coordinates () to via

(37) | |||||

where we have also specified the transformation in terms of the original variables (). Since any point with gets indeed mapped onto the negative -axis to the point (), we have accomplished the identifications on the Gribov horizon as required by the continuous residual gauge symmetry. In fact, these identifications are nothing but the “boundary identifications” discussed in the literature [23, 22], which are known to indicate a nontrivial topology of the physical configuration space . Since there are no residual gauge symmetries left, we can now identify the space obtained via (2.1) with the physical configuration space . Notice, that apart from a singular point at the origin, , the space is a smooth manifold (in particular without boundary).

To make this procedure more transparent we have represented it graphically in Fig. 2 focusing on a half plane in . To the left we have drawn the half plane () and () the half plane becomes the surface of a cone, which by a suitable embedding in may be represented as a plane. This “plane”, however, has non-vanishing curvature with a singularity at the origin corresponding to the tip of the cone. This is also true for the physical configuration space as a whole, as indicated by the scalar curvature which in spherical coordinates is given by . We note that the origin of the physical configuration space becomes a singular point analogous to the tip of a cone, because it is a fixed point under the operation of boundary identification. Therefore, has the structure of an orbifold [35]. However, the deeper reason for the origin to become a singular point of lies in the fact, that it is the only (non-generic) configuration with a nontrivial stability group: is invariant under the entire gauge group . ). After identification of the gauge equivalent configurations (

If we calculate the Jacobian for the transformation (28) in terms of the coordinates , using (2.1), we obtain

(38) |

with . Thus, in agreement with our previous considerations, there is only one zero of the Jacobian left, the one corresponding to the non-generic configuration, .

We will study possible physical consequences of the orbifold structure of in more detail at the end of this section. Before we can do so we have to determine the Hamiltonian on from (26) defined on the Euclidean pre-configuration space . This is most easily done in spherical coordinates combining the transformations (28) and (36). To find the Laplacian on in these coordinates we need the Jacobian matrix and its determinant

(39) |

Note the factors reflecting the non-generic singularity at the origin (38) and owing to the use of spherical coordinates. In particular we can now interpret the zero at as a pure coordinate singularity without any physical significance. Independent of the parameterization of the gauge fixing surface or the physical configuration space , the Gauß constraint is given by

(40) |

We solve (40) by requiring the wave functions not to depend on the gauge variant variable , so that we can discard all terms in containing derivatives with respect to . Thus we obtain the physical Hamiltonian in the axial gauge,

(41) |

This Hamiltonian only depends on the gauge invariant variables and acts on wave functions defined on the physical configuration space .

In the next subsection, we will compare the results obtained by choosing the gauge condition (27) with those, which we will get from a different procedure related to the method of gauge invariant variables.

### 2.2 Polar Representation

We notice that the Hamiltonian (22) has an additional symmetry generated by , which we write as

(42) |

with . Apart from being useful in the diagonalization of the Hamiltonian (as ), this symmetry can be further exploited to represent the matrix as

(43) |

The representation (43) is known as the polar decomposition of an arbitrary quadratic matrix into one diagonal and two orthogonal matrices [36]. This decomposition has been frequently applied to classical and quantum Yang-Mills mechanics [37, 38, 39, 40, 41]. Actually, upon inserting the transformation (43) into the classical Lagrangian (13), going to the Weyl gauge and setting , one would obtain the “xy-model”, a well-known playground for studying non-linear dynamics [42, 43]. For the case of field theory, Simonov proposed the closely related “polar representation” [20], whereas Goldstone and Jackiw applied the polar decomposition within the electric field representation [44].

By analogy with (28) we rewrite the representation (43) as

(44) |

and interpret (44) as the transformation to gauge invariant
variables , , and the gauge variant
coordinate . The crucial point of writing (43) in
this form is, that (44) can also be interpreted as a gauge
fixing transformation. The corresponding *non-linear* gauge
condition [45]

(45) |

can easily be read off from the first matrix on the right hand side of (44), where we denoted the corresponding matrix elements by as in (28). Hence the variables , and form a parameterization of the gauge fixing surface defined by . Expression (44) provides an explicit example for the equivalence between the method of gauge fixing and the use of gauge invariant variables from the outset.

From our experience with the axial gauge we anticipate the appearance of residual gauge symmetries. The calculation of the Jacobian for the transformation (44) yields

(46) |

which is zero for . Gauge equivalent configurations may be detected by investigating whether there are gauge copies of in the same gauge, i.e. ,

(47) |

We find that, apart from the zeros of the Jacobian (46), we have additional gauge copies related by , corresponding to discrete residual gauge symmetries. As in the case of the axial gauge we eliminate these discrete symmetries by restricting the values of the gauge invariant variables to and where we have to identify the points . The reduced gauge fixing surface can be embedded in as shown in Fig. 3, where the shaded region defined by is rotated around the -axis. The explicit embedding is given by , and with .

Let us turn to the configurations
corresponding to the zeros of the Jacobian (46). The set of
these configurations constitutes the Gribov horizon , which in the
embedding of Fig. 3 forms the surface of a double cone with
the -axis as its symmetry axis. Recalling the discussion of the
axial gauge we anticipate the existence of a continuous residual gauge
symmetry on this surface. And in fact we find, constructing a
relation similar to (32), that on the Gribov horizon
there are gauge orbits in the form of circles (cf. Fig.
3). This continuous residual gauge symmetry is directly
related to the fact, that the gauge invariant variable is not
well defined for . As in the case of the
axial gauge we have to fix this residual gauge symmetry by imposing an
*additional* gauge condition. Geometrically, we have to identify
all the points on a gauge orbit with one point. Choosing this point
to be on the -axis of , we may realize this
boundary identification by a similar “doubling of an azimuthal
angle” as in the case of the axial gauge. Define the angle
via and . Then map
every point of to via

(48) |

where are cartesian coordinates in . As there are no residual symmetries left, we have thus found an embedding of the physical configuration space into . In other words, (48) defines a coordinate system covering all of . Once again the physical configuration space has the structure of an orbifold with a singularity at the origin. The Jacobian of the resulting gauge transformation is proportional to . As in the case of the axial gauge we conclude that the only zero of the Jacobian which is not due to incomplete gauge fixing or coordinate singularities corresponds to the non-generic configuration . The significance of this configuration also follows from the scalar curvature , which we can calculate for the polar representation analogously to the axial gauge.

Expressed in terms of the spherical coordinates the Yang-Mills potential is given by

(49) |

Notice, that in polar representation does not depend on , due to the additional symmetry (42) for constant fields in the Weyl gauge. The corresponding generator is given in terms of spherical coordinates by

(50) |

which commutes with the Hamiltonian. We also note the difference between and the Gauß constraint . For the gauge symmetry generated by we require invariance of the wave function under the corresponding transformations of its arguments. This implies that wave functions are trivial representations of the gauge group. In the case of , however, the wave function may transform in an arbitrary representation of the corresponding symmetry group. Hence there is no restriction on the configuration space coming from the additional symmetry.

In order to establish the equivalence of the physical Hamiltonian obtained from the polar representation with the axial gauge Hamiltonian (41) we just need to redefine the angular variables and , which parameterize the physical configuration space by choosing another axis as the polar axis. It is a then a trivial exercise to show that one gets precisely the same results as in the axial gauge.

### 2.3 Natural coordinates

In the preceding subsection we have already demonstrated the equivalence of the choice of a gauge condition and the transformation to gauge invariant variables. Still, we want to consider yet another set of gauge invariant coordinates, which we will call “natural”, because they are the most obvious ones for our problem. Since the physical observables must not depend on the orientation of the coordinate axes with respect to the vectors and a natural choice of gauge invariant variables are the lengths of the two vectors and the angle between them. Once again we can write the transformation to this set of coordinates as a gauge transformation

(51) |

The Jacobian of the map is given by

(52) |

As in the familiar example of the transformation to polar coordinates in the plane, where the polar angle is not defined at the origin, the angle is not well defined when is zero for one . This is analogous to the case in the polar representation where the angle was not defined either. Likewise, we observe, that (51) can also be interpreted as the gauge transformation, relating an arbitrary configuration with , where the satisfy a certain gauge condition. In our case, the gauge condition can easily be read off from the first matrix on the right hand side of (51), and we get

(53) |

which is again non-linear. Let us for the moment assume, that the variables and just define a coordinate system on the gauge fixing surface corresponding to (53), ignoring e.g. the significance of the ’s as positive lengths. So let us take and . Then, using the language of gauge fixing, the zeros of the Jacobian (52) indicate the existence of Gribov horizons on the gauge fixing surface , separating different Gribov regions related by discrete residual gauge symmetries. How these residual symmetries can be found has been demonstrated before (e.g. in (47)). Thus, we only present the results of this analysis, which will be needed later on. The discrete symmetries relating different Gribov regions follow from

(54) |

whereas the continuous symmetries are (for arbitrary )