A Matrix Model for QCD: QCD Colour is Mixed
Abstract
We use general arguments to show that coloured QCD states when restricted to gauge invariant local observables are mixed. This result has important implications for confinement: a pure colourless state can never evolve into two coloured states by unitary evolution. Furthermore, the mean energy in such a mixed coloured state is infinite. Our arguments are confirmed in a matrix model for QCD that we have developed using the work of Narasimhan and Ramadas [3] and Singer [2]. This model, a dimensional quantum mechanical model for gluons free of divergences and capturing important topological aspects of QCD, is adapted to analytical and numerical work. It is also suitable to work on large QCD. As applications, we show that the gluon spectrum is gapped and also estimate some lowlying levels for and 3 (colors).
Incidentally the considerations here are generic and apply to any nonabelian gauge theory.
1 Introduction
The understanding of physical states in QCD is of fundamental importance. Conjectures regarding quark confinement and chiral symmetry breaking are based on speculations about their nature. It is also important for a nonperturbative formulation of QCD.
Gribov [1] showed many years ago that the Coulomb gauge in QCD does not fully fix the gauge and is inadequate for a nonperturbative formulation of QCD. Later, Singer [2] and Narasimhan and Ramadas [3] proved that the Gribov problem cannot be resolved by choosing another gauge condition since the gauge bundle on the QCD configuration space is twisted.
In this paper, we argue that as a consequence of the above twisted nature of the QCD bundle, coloured states restricted to the algebra of local observables are necessarily mixed: they carry entropy. This argument is confirmed in a matrix model for gluons we also propose here. This model is dimensional and free of the technical problems of quantum field theory.
The matrix model, being a quantum mechanical model of real matrices, for colours, and capturing certain essential topological aspects of QCD offers a new approach to QCD calculations. It is also suitable for the study of ’t Hooft’s large limit. As an explicit illustration of the power of our approach, we show that the gluon spectrum has a gap in our model. In lattice calculations this is taken as a signal for confinement.
For and 3, we also use simple variational calculations to estimate lowlying glueball masses. Detailed numerical work is on progress.
Just as in a soliton model, it is necessary to quantise the excitations around our matrix model solutions in a full quantum field theory. In this connection, we note that the matrix model contains the vacuum sector where the gluon potential is gauge equivalent to the zero field. We also indicate how to construct multiparticle levels for our gluon levels adapting standard techniques in soliton physics [4].
In a paper under preparation, we will argue that QCD has different phases, and also calculate the glueball spectrum in these phases. The Dirac operator in the matrix model approach will also be discussed.
2 The Gauge Bundle in QCD
Let , with and being GellMann matrices, denote the QCD vector potentials (in our convention, , with ) in the temporal gauge. Its gluon configuration space is based on the space of their connections. The QCD gauge group is the group of maps from to with the asymptotic condition (time argument is suppressed)
(2.1) 
(See also the Sky group in this respect [5].) The group acts on according to
(2.2) 
There are two normal subgroups of of importance here,
(2.3)  
(2.4) 
As discussed elsewhere [5, 4], the Gauss law generates which therefore acts trivially on the physical quantum states.
The group is normal in and
(2.5) 
Its representations characterise the states of .
The colour group is
(2.6) 
All observables commute with the Gauss law, that is, . In quantum physics, observables are also local [6], that is, they are obtained by smearing standard quantum fields with test functions with supports in compact spacetime regions. In the canonical formalism, that means that local observables are obtained from smeared quantum fields over a compact^{4}^{4}4Instead of a fixed time slice, if one considers a time average of a field over given arbitrarily small, but finite time slices, matrix elements of fields become smooth functions on [7]. spatial region . Call such a field . The action of on depends only on the restriction of to . But can be smoothly extended beyond to a gauge transformation . There are many ways of doing so and for each , by Gauss law, if is an observable, then
(2.7) 
Hence,
(2.8) 
so that all local observables commute with elements of .
The configuration space for local observables is thus associated with and not as naive considerations using the Gauss law would suggest.
Quantum vector states instead can be built from maps from to which are annihilated by the Gauss law:
(2.9) 
Hence wave functions are sections of vector bundles built on . It follows that we have the fibre bundle structure
(2.10) 
for the group
(2.11) 
Any function on is invariant under gauge transformations, and is hence a colour singlet.
The bundle (2.10) is twisted. Otherwise we would conclude that , which is false since is connected. This last statement follows from the fact that itself is connected.
This argument however must be sharpened since does not act freely on . Indeed, an element of is
(2.12) 
The action of on this element is
(2.13) 
since is normal in . To see explicitly that the action is not free, choose and with Lie algebra basis to find that leaves invariant. Hence does not act freely on .
The centre of leaves all vector potentials invariant, so we can change to , and correspondingly define and .
We next consider generic connections with holonomy at any point being . Then the above groups act freely on [2, 3], so that we obtain the principal fibre bundle
(2.14) 
Previous authors [2, 3] had shown that this bundle is twisted, that is, nontrivial,
(2.15) 
A quick proof is due to Singer, see his Theorem 2 in [2]. He starts with the fact that , for any where is the th homotopy group of . In particular, since , then . But on the RHS of (2.15), we have that . Also, , since , while on the RHS we have . Thus, since the LHS and RHS of (2.15) have different homotopy groups, we conclude that they cannot be equal. For a related discussion of the relevant cohomologies, see [10].
The nongeneric connections lead to some sort of boundary points. More precisely, these “boundary points” give a “stratified” manifold [11].
A similar situation is already known to happen in a different context. Recall the treatment of identical particles on [4]. In this case, the bundle space is
(2.16) 
whereas the configuration space is
(2.17) 
where acts by permutations of ’s and is an unordered set
(2.18) 
But if , for some , then is invariant under the transformation , so that the action of on is not free. Hence to get a genuine fibre bundle, we exclude coincidence of any two points and work with
(2.19) 
Then
(2.20) 
gives a principle fibre bundle. This bundle is also twisted.
Given an operator like the Laplacian on , the points of with turn up as “boundary points” where suitable boundary conditions have to be imposed.
Likewise, the nongeneric connections may have to be treated by suitable conditions in an appropriate setting. They are conjectured to lead to different phases of QCD. We will take up these issues in another paper. But we will not encounter the need for such conditions in the approach taken here.
Since the bundle (2.10) is twisted, previous works [2, 3] infer that (or ) gauge theories do not admit global gauge conditions.
In conclusion, we have the twisted bundle (2.10) in QCD. Wave functions are functions on which under transform by one of its unitary irreducible representations (UIR’s). Local observables instead are colour singlets.
3 How Mixed States Arise
The UIR’s of lead to states. We will remark on them in Section 8.
For now, we focus on . Hence consider the wave functions
(3.1)  
transforming as the component of the UIR of
(3.2) 
The corresponding density matrix, from which the state on the space of observables is defined, is
(3.3) 
where we assume for simplicity that the kets are normalised to 1 in a suitable scalar product. (Actually, we must really consider wave packets in ).
The observable algebra we work with is the algebra of colour singlet operators. They are associated with . contains . We assume that it is a algebra, though this point does not enter the formal considerations here. The algebra , the algebra of local observables, is a subalgebra of , so that a mixed state on remains mixed when restricted to . In what follows, we work with itself.
If , then its mean value in the state (3.3) is
(3.4) 
If is the colour singlet representation, the state (3.3) restricted to is pure. But that is not the case if is a nontrivial UIR. We now show this result using the GNS construction. The argument is modelled on our previous work on ethylene [8].
Suppose now that is a nontrivial UIR. We introduce the vector states
(3.5) 
and the inner product
(3.6) 
We emphasize that the GNS inner product is different from .
Consider the projector
(3.7) 
which is a colour singlet and hence is an element of . Further if
(3.8) 
then is a null vector, that is,
(3.9) 
Thus we introduce the equivalence classes
(3.10) 
and the vector , so that
(3.11) 
There are no nonzero null vectors among . The completion of in the scalar product (3.6) gives the Hilbert space .
The representation of on is
(3.12) 
The vector is cyclic in , so that all of can be obtained from the action of the elements of (and its completion in the norm), and
(3.13) 
Now, the representation (3.12) is reducible showing that is not pure. We can see this as follows. Since ,
(3.14) 
Since is an singlet, its action does not affect . Hence as a state,
(3.15)  
(3.16) 
On each , regarded as a cyclic vector, we can build a representation of :
(3.17) 
Thus restricted to is a mixture of pure states ( being the dimension of ) and is mixed for .
As discussed elsewhere [9, 8], the decomposition (3.15) is not unique. If is the rank of , and
(3.18) 
then
(3.19) 
This ambiguity introduces ambiguities in entropy.
The group algebra restricted to the representation and coincide. Thus the entropy ambiguities emerge from unobserved colour. If colour were part of , the state (3.3) would remain pure.
The following point is important. Since observables are colour singlets, we can observe only and not or . Hence while we can prepare the vector by observing , we cannot prepare or . This with (3.15) shows another way to understand how mixed states arise in QCD.
4 The Matrix model
4.1 The Case of Two Colors: A Review
The basic work leading to this model is that of Narasimhan and Ramadas [3]. They consider the colour group and the spatial slice . We remark that as for fuzzy spheres, we can recover from by suitable limits.
Narasimhan and Ramadas rigorously prove that for , the gauge bundle
(4.1) 
is twisted and does not admit a global section (that is, a gauge fixing). For proving this result, they reduce the problem to one of studying the special leftinvariant connections
(4.2) 
where are the Pauli matrices, and is a real matrix. The connection on spatial is obtained by diffeomorphically mapping onto and pulling back . The submanifold of such is preserved only by the global adjoint action
(4.3) 
or
(4.4) 
where is the image of under the homomorphism . The action of on the space of real matrices of rank is free and leads to an fibration
(4.5) 
From this result, they deduce that the gauge bundle is also twisted.
4.2 The Case of Three Colors
We now adapt the preceding discussion to .
We start with the leftinvariant oneform on ,
(4.6) 
where is a real matrix and is in the fundamental representation of . These ’s parametrize a submanifold of connections which captures the essential topology of current interest.
In , , , generate an subgroup. We map spatial diffeomorphically to ,
(4.7) 
with a distinguished point having the image . A convenient choice is the Skyrme ansatz [4]
(4.8) 
Although , , so that gives a mapping from to .
Now, if are vector fields on representing for the right action , then , and
(4.9) 
Thus on identifying spatial vector fields with , , one has for the vector potentials on the spatial slice,
(4.10) 
Here has no spatial dependence whereas acting on will introduce such dependence, except at identity (since ), and will not preserve the form of . This submanifold is thus gauge fixed with respect to (Such gauge fixation is not possible for the space of all since ).
But of colour acts on . If ,
(4.11) 
Remark: For later use, we now show that the action (4.11) is not necessarily free. This result will not be of importance in this paper.
There are four linearly independent vectors in the octet representation of which are singlets under hypercharge , since
(4.12) 
These correspond to the pions and the eta meson . Hence if the columns of are spanned by and , then
(4.13) 
It follows that the action on is not free if its rank is .
But does act freely on , the space of matrices of rank . We can see this as follows. Let and map the columns of to the Lie algebra according to
(4.14) 
The action on is equivalent to its adjoint action on . So we focus on the vector space spanned by on which acts by conjugation. of
Now if an element leaves and invariant under conjugation, it also leaves their product invariant. So the set of such vectors left invariant under conjugation forms an algebra. So does their complex linear span. Let denotes this complex algebra. This algebra is a algebra with the defined by hermitian conjugation being unitary. It is then a standard result that is the direct sum of full matrix algebras. As acts on , we can conclude that
(4.15) 
We already found an algebra fixed by hypercharge, namely
(4.16) 
the being generated by while can be obtained from and .
This is maximal if its stabiliser is not a multiple of . For the only bigger is , and if commutes with all of , then lies in the centre of . Then is identity.
We have thus proved that acts freely on .
4.3 The Matrix Model Bundle is Twisted
Now, the dimension of is . The dimension of matrices of rank is . Hence their codimension is also . Furthermore, since is contractible, for all . Hence by Remark 3 to Theorem 6.2 in Narasimhan and Ramadas [3], .That is enough to show that
(4.18) 
since .
We thus conclude that the bundle
(4.19) 
captures the twist of the exact theory.
Narasimhan and Ramadas in their proof of Theorem 6.2 also show that for , the bundle
(4.20) 
is twisted. This result is important for us as we also consider explicitly in Sections 5.1 and 6.
4.4 The Hamiltonian for
Recall that the YangMills action is
(4.21) 
Upon rescaling , we recover the form used in perturbative QCD.
From the Hamiltonian
(4.22) 
of (4.21), we can easily write down the Hamiltonian for the reduced matrix model, which we will do in the next section.
As the configuration space variables for the matrix model are , it is natural to take the after Legendre transformation as the conjugate of . In QCD, the conjugate to the connection is the chromoelectric field. So we identify this conjugate operator with the matrix model chromoelectric field . On quantising the reduced model, these satisfy
(4.23) 
5 Matrix Model for gauge theory
In the matrix model, plays the role of the vector potential. From its curvature , we get
(5.1) 
where are structure constants.
In the reduced matrix model, the term plays the role of the potential :
(5.2)  
(5.3) 
The reduced matrix model Hamiltonian is thus
(5.4) 
We have introduced an overall factor of for dimensional reasons, having the dimension of length.
Notice that in the limit , the potential term dominates, while the kinetic term dominates in the limit .
As a quantum operator, is thus given by
(5.5) 
It acts on the Hilbert space of functions of with scalar product
(5.6) 
Previous work on Related Models:
Savvidy has suggested a matrix model for YangMills quantum mechanics [12], which has been explored by many researchers. However, their arguments for arriving at the matrix model differ from ours, as does their potential.
Other investigations of YangMills quantum mechanics involve approximating the gauge field by several (unitary or hermitian) matrices. The potential has interesting properties in the large limit, and several investigations have been carried out by [13, 14, 15, 16]. Again, these models differ from our model, in that our model (5.5) is based on a single real matrix with a kinetic energy term.
5.1 Simplification of Potential and its Extrema: Case
Let us specialise to the case of gauge theory. Then . Hence
(5.7) 
Let us do the singular value decomposition (SVD) of : , where is a diagonal matrix with nonnegative entries , and and are real orthogonal matrices. By applying extra rotations to the right of or , we can assume that . With this decomposition,
(5.8) 
Note that under gauge transformations, (with ), so is invariant under gauge transformations.
The potential is zero for , and . These two are gaugerelated by a large gauge transformation, because is a winding number 1 transformation and for , is the gauge transform of the zero connection by a winding number 1 transformation.
The minima of are given by
(5.9) 
and similar equations from . Symmetry of the equations under suggests that all are equal at the extremum. Putting immediately gives as the extrema.
We can look at the Hessian matrix :
(5.10) 
This is positive definite at with eigenvalues . It is also positive definite at with eigenvalues . Even though and are related by a (large) gauge transformation, the Hessian has a very different spectrum. The physical consequences of this is unclear to us.
The Hessian at has eigenvalues . So this extremum is a saddle point. Again, we need to understand the physical interpretation of this saddle point.
Separation of Variables in :
6 Spectrum of the Hamiltonian
We will work with the Hamiltonian (5.5) and limit ourselves here to qualitative remarks and estimates about its spectrum for and . Detailed work is in progress with S. Digal.
The potential grows quadratically in as , while it is smooth elsewhere. It follows immediately that the spectrum is gapped as required by colour confinement, and is discrete as well.
The potential resembles that of the anharmonic quartic oscillator. In the latter case, the anharmonic term is known to be a singular perturbation which cannot be treated using perturbation theory [17, 18, 19].
We will use variational methods to estimate energy levels. We will be guided by
(6.1) 
in our choice of the variational ansatz.
The eigenfunctions of are of the form , where are products of Hermite polynomials in variables .
For the variational ansatz for the ground state, we take
(6.2) 
and minimise with respect to the parameter .
We find
(6.3) 
Minimizing with respect to gives the variational ground state energy . It is plotted in Figure 1 as a function of t’Hooft coupling .
Similarly, we can take the ansatz
(6.4) 
for the first excited state. This is an impure state because the colour index is not ”soaked up”. We then calculate
(6.5) 
to find
(6.6) 
Its minimum is plotted against in Figure 2.
Notice that both these trial wave functions are insensitive to the term in the Hamiltonian. The simplest ansatz that is sensitive to this term is
(6.7) 
This has three variational parameters: and . The variational energy for this ansatz is shown in Figure 3.
Our variational energy estimate is rather crude, and is presented here for representational purposes only. We expect that the variational estimate differs significantly from the true energy for large values of t’Hooft coupling . Much better numerical estimates may be obtained by taking more sophisticated (or complicated!) variational ansatz for the wavefunctions. We will not do it here.
7 On Mixed States in the Matrix Model
Considerations using in sections 2 and 3 were formal, whereas the matrix model for is that of a particle with degrees of freedom. It is a welldefined quantum mechanical model, which captures the colour twist topology of .
The algebra of the observables are made up of colour singlets. It contains colour singlet functions of . (More precisely, we consider only bounded operators of this sort). The full is generated by such operators.
We can now adapt section 3 to show that coloured states restricted to are not pure.
8 Final Remarks
The one definite result we have in the work is the conclusion that coloured states in QCD are mixed. That will affect correlators and partition functions and hence physical predictions. Calculations in this directions have not been done.
In addition, we have developed a matrix model for pure QCD which gives a gapped spectrum and discrete levels for glueballs.
Our present work can be generalised to other gauge groups.
We conclude with a few further remarks on the matrix model.

We can couple quarks to by using covariant derivative in the Dirac operators, this being its only modification in the gauge.

We can construct QCD states as follows. The ChernSimons form gives the field theory action
(8.1) (8.2) which in the matrix model becomes, on using (4.10) and (5.1),
(8.3) The overall is fixed by requiring that for a pure gauge, where , where is in the subspace, the RHS becomes the winding number . Then under a gauge transformation