# Confinement and Deconfinement
in
Gauge Theories: A Quantum Field Theory

A. P. Balachandran  Physics Department, Syracuse University, NY 13244-1130,
USA

###### Abstract

After a brief discussion of small and large gauge transformations and the
nature of observables, we discuss superselection sectors in gauge theories.
There are an infinity of them, classified by large gauge transformations.
Gauge theory sectors are labelled by the eigenvalues of a complete commuting
set (CCS) of these transformations.

In QED, the standard chemical potential is one such operator generating global
U(1). There are many more given by the moments of the electric field on the
sphere at infinity. In QCD, the CCS are constructed from the two commuting
generators spanning a Cartan subalgebra.

We show that any element of a large gauge transformation can be added to the
standard Hamiltonian as a chemical potential without changing field equations
and that in QCD, they lead to confined and deconfined phases . A speculation
about the physical meaning of these chemical potentials is also made.

Comment: This note is based on seminars by the author. So only a limited
number odreferences are given, from which further literature can be traced.

A paper is under preparation.

## Introduction

We propose a new mechanism for confinement-deconfinement transitions in gauge
theories. It is based on a chemical potential in the Hamiltonian which was
discussed briefly by Balachandran et al.[1] and which is associated with
‘large gauge transformations’. It defines both superselection sectors, and in
non-abelian gauge theories, their evolution as well and suggests several new
phenomena.

Let us first review the basics of gauge transformations as formulated by our
group and reviewed in [2]. In later sections, we will develop the announced
results.

The spacetime unless otherwise stated is 3+1 dimensional Minkowski $M^{3,1}$
and the metric will be $(1,-1,-1,-1)$ diag. The gauge group $\mathcal{G}$ is
the group of maps from spacetime to a compact Lie group $G$. On a spatial
slice, if $g$ belongs to $\mathcal{G}$, then $g(\vec{x})$ approaches an
element $h(\hat{x})$ of $G$ as $|\vec{x}|$ goes to infinity.

The case where $|\vec{x}|$ going to infinity of $g(\vec{x})$ does not exist is
not considered here.

%beginframe

The Lie algebra-valued connection will be denoted by $A_{\mu}$ and its
conjugate electric field will be $E^{\mu}$. If $\lambda_{\alpha}$ form a (
hermitean ) basis for the Lie algebra of $G$ with the normalisation
$\mathrm{tr}\lambda_{\alpha}\lambda_{\beta}=2\delta_{\alpha,\beta}$ in a
defining representation, an $N\times N$ one for $SU(N)$, then we can write
$A_{\mu}=A_{\mu}^{\alpha}\lambda_{\alpha}$ and
$E_{\mu}=E_{\mu}^{\alpha}\lambda_{\alpha}$. For $U(1)$, we can put electric
charge for $\lambda_{\alpha}$.

In a gauge theory, an infinitesimal gauge transformation can be written on a
spatial slice as

$Q(\Xi)=\int_{0}^{\infty}d^{3}x\langle\left(D_{i}E_{i}+J_{0}\right)\Xi\rangle$

where

a) $\langle\cdots\rangle$ indicates trace over Lie algebra indices and $D_{i}$
is covariant derivative. In the $U(1)$ case, there is no need for trace and
$D_{i}$ becomes ordinary derivative $\partial_{i}$.

b) $\Xi$ is a Lie algebra-valued test function. In the non-abelian case, we
can write it at a point $\vec{x}$ as $\Xi^{\alpha}(\vec{x})\lambda_{\alpha}$
where the functions $\Xi^{\alpha}$ approach functions of
$\hat{\vec{x}}=\vec{x}/|\vec{x}|$ as $|\vec{x}|$ goes to infinity. The test
function can be any smooth function so that all local gauge transformations
can be generated.

If all $\Xi^{\alpha}$ go to zero at infinity adequately fast , we can
partially integrate without surface terms and write

$Q(\Xi)=\int_{-\infty}^{+\infty}d^{3}x\langle(-E_{i}D_{i}\Xi+J_{0}\Xi)\rangle$

and that is the standard Gauss law. So it vanishes on the quantum states and
maps to zero in the GNS construction.

But if $\Xi$ do not become zero functions at infinity, $Q(\Xi)$ need not
vanish on quantum states. They generate the Sky group of Balachandran and
Vaidya. In either case, regardless of the asymptotic behaviour of $\Xi$, they
commute with the algebra of all observables $\mathcal{A}$ due to locality.

We have argued for this fact many times elsewhere. (Chern-Simons papers, [2]
). This result has many consequences as we outline below.

## Novel Chemical Potentials

Let $H_{0}$ be the standard full gauge theory Hamiltonian without any sort of
chemical potential which generates equations of motion. Pick a $\Xi$ and a
constant $\mu$ with energy dimension and consider the Hamiltonian

$H=H_{0}+\mu Q(\Xi):=H_{0}+H_{1}.$

The new term will not affect equations of motion as it commutes with all local
observables. It will change the fields in equations of motion only by local
gauge transformations which will not affect equations of motion. It is the
non-abelian chemical potential.

Non-abelian chemical potentials have been discussed in previous work on finite
temperature field theories [3].

Later we will interpret it as the coupling of the experimental setup with the
system and suggest a scheme to resolve the observation problem in quantum
physics. For $U(1)$ and constant $\Xi$, it becomes the standard chemical
potential. The constant $\mu$ sets the scale at which $Q(\Xi)$ becomes
significant.

There is no point in making $\mu$ functions of $\vec{x}$ as $\Xi$ are already
spatial functions.

The exponentials of $Q(\Xi)$ for different test functions generate the Sky
group $\mathcal{G}$ and its group algebra $\hat{\mathcal{G}}$.

For non-abelian target group $G$ such as $SU(3)$, this algebra is non-
commutative since

$[Q(\Xi_{1}),Q(\Xi_{2})]=Q([\Xi_{1},\Xi_{2}]).$

As Dirac taught us, an irreducible representation of $\mathcal{A}$ is
characterised by a vector state diagonal in a complete commuting set. This
vector state may or may not be preserved under the time evolution induced by
$\mu Q(\Xi)\in H$ ($H_{0}$ will not affect it). We will interpret the former
as a colour deconfining state and the latter as colour-confining state for
reasons given below. In the latter case, for generic situations, the orbit of
the vector state is ergodic as we shall also see below.

The term $Q(\Xi)$ is already present in the Hamiltonian when it is derived
from the gauge theory Lagrangian by Legendre transformation, as was observed
in an earlier paper ( Balachandran, Nair, Pinzul, Reyes-Lega, Vaidya). It is
the term

$\int_{-\infty}^{+\infty}d^{3}x<(-D_{i}A_{0}E_{i}+A_{0}J_{0})>$

where $\mathrm{tr}~{}\langle\cdots\rangle$ is over the Lie algebra indices as
usual and $J_{0}$ is the Lie algebra valued charge density.

It is usually discarded by treating $A_{0}$ as a classical field vanishing
fast towards infinity and doing a partial integration. Then it becomes Gauss’s
law and hence zero as an operator.

But $A_{0}$ need not vanish at infinity. It has become $\Xi$ in the current
notation.

The extra term in $H$ will not spoil the commutativity of spatial translation
operators with $H$. But with $H$ as $P_{0}$ and with the usual unaltered
spatial translation generators, $P_{\mu}+\mu Q(\Xi)\delta_{\mu,0}$ will not
transform as a four vector under Lorentz transformations : Lorentz symmetry is
broken at the operator level But as the extra term does not affect equations
of motion, the latter will maintain their covariance.

However, it has long been known that Lorentz invariance is broken by infrared
effects and gauge transformations [ Fröhlich,Morchio and Strocchi; Mund,
Rehren and Schroer and references therein], so that there is no need for
concern about Lorentz breaking from this chemical potential.

## Dynamics from $H$

This is determined by the expectation values in the vector states of the
observables. The latter in turn are classified by the eigenvalues of the
superselection operators on the vector states. In a particular superselection
sector, a complete set of commuting superselection operators( complete
commuting set, CCS) are diagonal.

The CCS is an abelian subalgebra, preferably a maximal abelian subalgebra, of
the Sky algebra with generators ${Q(\Xi)}$. For example, in QED, the
conventional choice for $Q(\Xi)$ is the multiple of the electric charge
operator $Q$. The test function $\Xi$ in this case goes to a constant on the
celestial sphere $S^{2}$ ( with coordinates $\hat{x}$).

But it can also go to any smooth function of $\hat{x}$. Then the superselected
operators become infinite dimensional.

In QCD and in non-abelian gauge theories, CCS is much richer. First we choose
a CCS from the enveloping algebra of $SU(3)$ Lie algebra. That is spanned by
its quadratic and cubic Casimir operators , a Casimir of an ‘isospin’
$SU(2)\in SU(3)$ , a ‘third component ‘ $I_{3}$ of its isospin and the
hypercharge Y commuting with $I_{3}$. Then in addition to the above Casimirs,
we can diagonalise the commuting Sky operators $Q(\tilde{\Xi})$ where
$\tilde{\Xi}$ is a linear combination $I_{3}$ and $Y$ with coefficients
becoming functions of $\hat{x}$ at infinity.

For now, let us consider the case where the coefficient functions are
constants at infinity. Then the superselection sector is labelled by $I_{3}$
and $Y$ in an irreducible representation ( IRR) of $SU(3)$ such as a triplet
or octet. But a generic $\mu Q(\Xi)=H_{1}$ will not commute with
$Q(\tilde{\Xi})$ and will change the superselection sector. Local observables
and evolution by the standard $H_{0}$ will preserve it, but not so the
chemical potential.

This is the new feature in QCD : a generic non-abelian chemical potential will
not preserve the superselection sector. We will discuss the orbit of the
superselection sector below, indicate its ergodic features and argue that the
mean values of coloured observables in this situation are all $0$. This result
substitutes for colour confinement. But the expectation values of Casimir
operators of $G$ are constants of motion for $H$ and can certainly be
measured. Hence the representation of the coloured sector, if it is
$\mathbf{3},\mathbf{3*},\mathbf{8},\cdots$ can be determined.

But we can also choose a $\Xi$ commuting with $\tilde{\Xi}$, the diagonalised
CCS. It will affect evolution by an $\hat{x}$-dependent phase, but will not
change the superselection sector at all. We can call such states as
characterising colour deconfined phases.

We need explicit examples to which we now turn.

##  The Colour-Confined and Deconfined Phases

The title is misleading. Presumably in the popular literature, confinement is
supposed to mean that coloured states are not in the domain of the
Hamiltonian. It is not clear if numerical work based on Wilson lines
accomplish such a result.

What emerges from our analysis leads to a related result : the mean values of
coloured observables averaged over the interaction time are 0, in this phase.
All the same, as Casimirs are colour singlets, they can be determined as
previously remarked.

Here is a worked ergodic example for $SU(3)$. The superselection sector is one
with $\lambda_{3,8}$ ( and Casimirs) diagonal and in the triplet
representation. Their joint eigenvalues are (1,1),(-1,1) and (0,-2). Any one
of them gives a density matrix on the coloured observables. Their time
evolution is given by the chemical potential, assumed time independent.

If the picked vector state is an eigenstate of the chemical potential, it will
be preserved in time, and coloured expectation values too will be preserved in
time : these are the sectors with colour deconfinement.

So let us pick a chemical potential which does not preserve them, say

$H_{1}=Q(\Xi),\Xi=\mu_{1}\lambda_{1}+\mu_{8}\sqrt{3}\lambda_{8},\qquad\mu_{1}/\mu_{8}\mathrm{~{}~{}~{}irrational}.$

The $\mu$’s have energy dimension which sets the scale of this term. What sets
this scale ? It seems to be the same effect which sets the scale of the
standard chemical potential in finite temperature field theories.

This operator evolves colour (but not local observables) and is our $H_{1}$.

In this example , the two terms in $H$ commute , making computations easy. But
$|(0,2)\rangle$ is its eigenstate, so ergodicity can show up only in the
remaining states. But that is fine to illustrate the phenomenon.

A calculation shows that

$e^{it\Xi}=e^{it\mu_{1}\lambda_{1}}e^{it\mu_{8}\sqrt{3}\lambda_{8}},$

which gives the action of $e^{itH_{1}}$ on a chosen vector state.

The vector ( 0,0,1) in the basis indicated above is an eigenstate of $H_{I}$
and defines a deconfined phase. $H_{1}$ has eigenvalue $-2\mu_{8}$ and
$e^{itH_{1}}$ has periodicity $\pi/\mu_{8}$. But under time evolution, the
other two vectors never come back to the starting value as claimed.

That is because $\frac{\mu_{1}}{\mu_{8}}$ is irrational by choice. So too are
the ratios of periods.

If the ratios are rational, the orbit of vector states are periodic.

The strong interaction time is about $10^{-23}$ secs. No experiment can probe
the evolution of the system during this time. What is observed is the average
of an operator during this time. But such an average is expected to be zero,
especially if the scales $\mu_{i}$ are large compared to QCD scales.

For the vector state, we can also choose an eigenstate of $H_{1}$ such as the
one given by the choice $(0,0,1)$. It changes just by a phase under time
evolution and hence the corresponding density matrix does not change at all.
Expectation values of colured observables are then time-independent and there
is no theoretical issue in observing them. So we call them colour-deconfined
phases.

## Open Question

It is reasonable to ask if the colour-deconfined phases are attractors for the
confined ones. That will involve perturbing them and observing if they will
relax back to the deconfined phase.

But in quantum theory, a unitary perturbation of short duration seems
inadequate for this task. After it is switched off, the system will keep
evolving unitarily after the switch-off time from wherever it finds itself. A
perturbation with a POVM also seems not to help. A better formulation of the
question seems needed. Maybe one should switch on the perturbation
adiabatically and switch it off in the same way.

## An Interpretation of the Chemical Potential

One supposes that it has an interpretation similar to the standard one:
$\lambda XN$,$N$ = number operator. In QED, it is the generator of $U(1)$
gauge transformations so that it commutes with all observables, adding it to
the QED Hamiltonian $H_{0}$ as the $H_{1}$ above will not affect equations of
motion. It is like the abelian $Q(\Xi)$ of QED discussed above. It is in fact
what one gets for the choice $\Xi(x)=\Lambda$ for all $x$.

When $\Xi$ has an $\hat{x}$ dependent limit for $|\vec{x}|$ going to $\infty$,
$Q(\Xi)$ generates the Sky group yielding more general chemical potentials.
One can consider $\Xi$ approaching definite combinations of spherical
harmonics at infinity getting any number of novel chemical potentials. Just
like the phase diagram of temperature $T$ versus $\lambda$, one can also
consider multidimensional phase plots.

In the non-abelian case, the nature of the chemical potential changes : the
non-abelian coloured sector is labelled by the eigenvalues of a basis of a
Cartan subalgebra of the Sky group.

But $H_{1}$ can be any element of the Sky group and need not commute with the
diagonalised element. So in general it will generate an ergodic orbit in the
space of states as discussed above.

Note that the evolution is via superselection sectors. It is like the problem
studied by [4]. and the more recent one [5].

Elsewhere we have argued that quantum observations are done by coupling
experimental operators which form an abelian algebra. We next show that the
superselection sectors naturally generate an abelian algebra and that our
states from CCS are on this abelian algebra. It is this abelian algebra that
is observed.

When $H_{1}$ is switched off, the superselection sector no longer evolves and
the expectation values of the abelian algebra must be giving information about
the system. The suggestion here is similar to that of Fiorini and Immirzi,
Wightman and probably others like Bohr and Landsman.

Therefore we can suggest that $H_{1}$ is the evolution Hamiltonian for
superselection sectors and reflects the coupling of the experimental apparatus
to the system.

Thus let us suppose that the experiment is initiated at time $t=0$. At this
time, when the state is mixed in colour with zero mean value for coloured
observables, which the experiment knows, $H_{1}$ makes its appearance in $H$.
After a time $T$, when the experiment ceases, and $H_{1}$ becomes $0$, the
evolved mixed state no longer further evolves. Colour singlets can then be
measured and the evolution of their expectation values from $0$ to $T$ should
reveal information about the operator $H_{0}$ and what has happened to the
original state.

It has been assumed the original state is mixed in colour. If it is pure for
colour and evolves at time T to a pure state, that would be appropriate for
colour-deconfined state.

We have to show the emergence of a commutative algebra from superselection
operators. The conjectures regarding them now follow.

## The Commutative Algebra at Infinity in QED

Let us first recall how to create a charge $q$ state localised at $e_{\infty}$
in QED from the vacuum vector $|\Omega\rangle$, assumed to be invariant under
all gauge transformations.

Let $W(x,e)$ be the Wilson line from $\vec{x}$ to infinity along $e$ , it is
$exp(iq\phi(x,e))$ in terms of the escort field $\phi$.

Let $\psi(x)$ be any charge $q$ field. Then

$exp(iq\phi(x,e))\psi(x)|\Omega\rangle$

has no local charges and a charge $q$ blip at $e_{\infty}$. That is the case
for any $\vec{x}$.

If $\pi_{q,e}(\mathcal{A})$ is the charge $q$ representation of the algebra
$\mathcal{A}$ of local observables realised on (1), it acts on (1) by left
multiplication.

The vector states $H_{q,e_{\infty}}$ are
$\pi_{q,e}(A)exp(iq\phi(x,e))\psi(x)|\Omega\rangle$.

Under a large gauge transformation $exp(iq\Lambda)$, every vector in this
sector transforms with the fixed phase $exp(iq\Lambda(e_{\infty}))$ where this
has been defined as

$exp(iq\lim_{\tau\rightarrow\infty}\Lambda(x+\tau e)).$

So, this charge $q$ superselection sector is labelled by the representation
$Q(\Xi\rightarrow\Xi(e_{\infty}))$ where the latter is defined as in above for
$\Lambda$.

For

$exp(iq\phi(x,e^{\prime}))\psi(x)|\Omega\rangle,\qquad e^{\prime}\neq
e,\qquad\qquad(2)$

since there are certainly $\Xi$ with different values at $e$ and $e^{\prime}$,
no small gauge transformation or local observable can map the superselection
sector defined by (1) to that defined by (2).

Hence

$\pi_{q,e}((a)exp(iq\phi(x,e))\psi(x)|\Omega\rangle\qquad(3)$

$\pi_{q,e^{\prime}}((b)exp(iq\phi(x,e^{\prime})\psi(x)|\Omega\rangle\qquad(4)$

for any $a,b$ in $A$ are eigenvectors with different eigenvalues for such
$Q(\Xi)$ and hence orthogonal. The Hilbert spaces $H_{q,e,e^{\prime}}$ built
on (3) and (4 ) are orthogonal.

But $e$’s can be any points on a de Sitter space and take continuous values.

So we conclude that the direct integral over $e$ of these Hilbert spaces is
non-separable. ($https://en.wikipedia.org/wiki/Direct_{i}ntegral.$
$https://math.stackexchange.com/questions/2163261/non-seperable-hilbert-
spaces$ )

Let $P_{q,e}$ be the ‘projection’ operator for $H_{q,e}$ , like
$|x\rangle\langle x|$ for position in quantum mechanics. Then

$P_{q,e}P_{q,e^{\prime}}=0\qquad\mathrm{if}~{}~{}e\neq e^{\prime},\qquad(5)$

The set of $e$’s form a de Sitter space.

Hence

$RHS\mathrm{~{}of~{}(5)}=\delta_{e^{\prime},e}P_{q,e}(6)$

where $\delta_{e,e^{\prime}}$ is the Dirac delta functtion on de Sitter space
defined by

$\int de~{}f(e)\delta(e^{\prime},e)=f(e^{\prime})\qquad(7)$

where $de$ is the Lorentz invariant volume on the de Sitter space and $f$ is
any smooth function thereon.

It follows that

$I(f)=\int de~{}f(e)P_{q,e}\qquad(8)\ $

obeys the abelian algebra

$I(f)I(g)=I(f\,g)\qquad(9)$

where $f\,g$ is defined by the pointwise multiplication of $f$ and $g$.

With $*$ as complex conjugation and with sup norm, we get a commutative
C*-algebra $\mathcal{C}$ with spectrum as the de Sitter space. Elements of
this algebra label the superselection sectors associated in a charge $q$
sector.

## The Commutative Algebra at Infinity in QCD

The remarks above can be adapted to any non-abelian gauge theory.

## Epilogue

There are many issues that remain open and to be addressed.

## Acknowledgement

I have beniefited by discussions with Manolo Asorey,Bruno Carneiro da Cunha,
Arshad Momen, Paraneswaran Nair, Sasha Pinzul , Amilcar Queiroz, Babar Qureshi
and Sachin Vaidya.Much of this work was done at the Institute of Mathematical
Sciences. I thank my hosts Ravidran,Sanatan Digal and other colleagues there
for their wonderful hospitality.

## References

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