# Wilson loops in the Hamiltonian formalism

Robert D. Pisarski  Department of Physics, Brookhaven National Laboratory,
Upton, NY 11973

###### Abstract

In a gauge theory, the gauge invariant Hilbert space is unchanged by the
coupling to arbitrary local operators. In the presence of Wilson loops,
though, the physical Hilbert space must be enlarged by adding test electric
charges along the loop. I discuss how at nonzero temperature Polyakov loops
are naturally related to the propagator of a test charge. ’t Hooft loops
represent the propagation of a test magnetic charge, and so do not alter the
physical Hilbert space.

Since the seminal work of Wegner, Wilson, and Creutz, simulations of lattice
gauge theories using Monte Carlo methods on classical computers has given us
invaluable information about Quantum ChromoDynamics (QCD). For the phase
diagram at a nonzero temperature $T$ and quark chemical potential $\mu$, at
$\mu=0$ the order parameter for chiral symmetry exhibits crossover behavior at
a temperature of $T_{\chi}\approx 156\pm 1.5$ MeV Bazavov  _et al._ (2019);
Borsanyi  _et al._ (2020); Guenther (2022) 111This accuracy should not obscure
the fact that the crossover in QCD is rather broad, over tens of MeV; the
lattice precisely measures the maximum of a slowly varying function Guenther
(2022).. These methods have been extended to quark chemical potentials less
than the temperature, $\mu\leq T_{\chi}$ Borsanyi  _et al._ (2020); Guenther
(2022); Bazavov  _et al._ (2020); Bollweg  _et al._ (2021); Borsányi  _et al._
(2021); Ratti and Bellwied (2021).

This leaves many quantities of direct experimental significance which have not
yet been computed. For example, the diffusion coefficient for a heavy quark
has been computed in the $SU(3)/Z(3)$ gauge theory, without dynamical quarks
Brambilla  _et al._ (2020); Altenkort  _et al._ (2021). The computation of
other transport coefficients, notably the shear and bulk viscosities with
dynamical quarks in QCD, is conceivable with much larger classical computers
222S. Mukherjee, private comment..

Many other quantities, such as correlation functions in real time, and the
properties of cold, dense QCD, are only possible with quantum computers. While
large scale quantum computers with logical qubits lie well in the future, it
is useful to consider the questions of principle which are unique to a gauge
theory.

A classical computer deals with the Lagrangian. If the chemical potential
vanishes, at any temperature the action is real, and sophisticated techniques,
including those for nearly massless quarks, have been developed. While in
principle many states contribute to the partition function, the Metropolis
algorithm automatically selects the most important. The difficulty is the sign
problem: at nonzero chemical potential the action is no longer real, and
standard techniques fail.

In contrast, for a quantum computer it is best to deal with the Hamiltonian.
The partition function is

${\mathcal{Z}}=\sum{\rm e}^{-{\mathcal{H}}/T-\mu{\mathcal{N}}}\;,$ (1)

where ${\mathcal{H}}$ is the Hamiltonian, ${\mathcal{N}}$ the quark number
density, and $\sum$ is the sum over all states. Because everything is real,
there is no sign problem when $\mu\neq 0$. The difficulty is that
exponentially many states contribute, and even for states near the ground
state, it is not obvious how to choose the most important. Strategies to solve
this have been developed in condensed matter systems, and include the density
matrix renormalization group, matrix product states, and projected entangled
pair states Cirac  _et al._ (2021); Shachar and Zohar (2021).

While some generalization of these methods will be essential in QCD, the
purpose of this paper is to make an elementary point about how the Hamiltonian
form of a gauge theory changes in the presence of Wilson loops.

For a theory without gauge fields, the most general correlation functions are
given by adding sources for arbitrary local operators to the Lagrangian.
Multiple insertions of composite operators induce new counterterms to the
theory, but this is standard Zinn-Justin (2002); Skokov  _et al._ (2010);
Pisarski and Skokov (2016a), and going from the Lagrangian to the Hamiltonian
formalism is direct.

With gauge fields, however, there are gauge invariant non-local operators,
such as the Wilson loop,

${\mathcal{W}}_{\mathcal{C}}={\rm
tr}\,{\mathcal{P}}\exp\left(ig\int_{\mathcal{C}}A_{\mu}\,dx^{\mu}\right)\;.$
(2)

Here $g$ is the gauge coupling, $A_{\mu}$ is the vector potential for the
gauge field, ${\mathcal{P}}$ denotes path ordering along a closed curve
${\mathcal{C}}$, and the trace is over color.

My basic point is simple. Dynamical quarks contribute to Gauss’s law at each
point in space. If the quark mass is sent to infinity, so all that remains is
one test quark propagating along ${\mathcal{C}}$, then Gauss’s law must
include the contribution of that test quark Gervais and Sakita (1978);
Echevarria  _et al._ (2021). I show in this paper how the sum over states of
the test quark generates the Wilson loop, ${\mathcal{W}}_{\mathcal{C}}$.

I begin with the Lagrangian formalism, where the analysis is transparent, and
use that to proceed to the Hamiltonian form. While I consider systems which
are independent of time, with the Hamiltonian formalism it is possible to
perturb a gauge theory with a gauge-invariant, time dependent source, and then
measure the evolution of Wilson loops in time. This validates computing the
“holonomous” potential for the eigenvalues of the thermal Wilson line Gross
_et al._ (1981); Weiss (1981); Bhattacharya  _et al._ (1991); Belyaev (1991);
Bhattacharya  _et al._ (1992); Gocksch and Pisarski (1993); Korthals Altes
(1994); Korthals Altes  _et al._ (1994, 1997, 2000); Giovannangeli and
Korthals Altes (2001, 2005a, 2005b); Dumitru  _et al._ (2014); Guo (2014);
Nishimura  _et al._ (2018); Guo and Du (2019); Korthals Altes  _et al._
(2020a, b); Hidaka and Pisarski (2021); Guo and Kuang (2021), and using it to
construct effective theories for the deconfining Polyakov (1978); Susskind
(1979) and chiral Pisarski and Wilczek (1984) phase transitions Pisarski
(2000); Dumitru and Pisarski (2001, 2002); Dumitru  _et al._ (2004, 2005a,
2005b); Oswald and Pisarski (2006); Pisarski (2006); Hidaka and Pisarski
(2008, 2009a, 2009b, 2010); Dumitru  _et al._ (2011, 2012); Kashiwa  _et al._
(2012); Pisarski and Skokov (2012); Kashiwa and Pisarski (2013); Lin  _et al._
(2013); Bicudo  _et al._ (2013); Smith  _et al._ (2013); Lin  _et al._ (2014);
Bicudo  _et al._ (2014); Gale  _et al._ (2015); Hidaka  _et al._ (2015);
Pisarski and Skokov (2016b, a). Incidentally, it alleviates some concerns
Belyaev  _et al._ (1992); Chen  _et al._ (1992); Kogan (1994); Smilga (1994);
Hansson  _et al._ (1995); Kiskis (1995); Smilga (1997); Korthals-Altes  _et
al._ (1999); Korthals-Altes and Kovner (2000); de Forcrand  _et al._ (2001);
Cohen (2022) about holonomous potentials

A Wilson loop alters the Hilbert space because it represents the propagation
of a test electric charge. In contrast, the ’t Hooft loop ’t Hooft (1979,
1980); Korthals-Altes  _et al._ (1999); Korthals-Altes and Kovner (2000); de
Forcrand  _et al._ (2001); de Forcrand and von Smekal (2002); de Forcrand and
Noth (2005); Reinhardt (2003); Reinhardt and Epple (2007) represents the
propagation of a test magnetic charge, and so doesn’t modify the Hilbert
space. I also show how to compute ’t Hooft loops in the Hamiltonian formalism.

Lagrangian formalism: Consider a quark of mass $M$ as $M\rightarrow\infty$.
Then we can neglect the spin of the quark, as spin-dependent effects are
uniformly suppressed as $\sim 1/M$. Similarly, we can consider either a quark,
propagating forward in time, or an anti-quark, propagating backwards in time.
In either case, since it is too heavy to move, the quark (or anti-quark), just
sits at some point in space 333Boosting to a moving frame gives a test quark
moving at constant velocity.. The gauge invariant effective Lagrangian is then

${\mathcal{L}}=\psi^{\dagger}\,D_{0}\,\psi\;.$ (3)

I assume that $\psi$ lies in the fundamental representation, where the
covariant derivative $D_{0}=\partial_{0}-igA_{0}$. Later I generalize to
arbitrary representations.

I introduce the Wilson line, running from a point $x$ in space-time to $y$:

${\bf
L}(y;x)={\mathcal{P}}\exp\left(ig\int^{y}_{x}A_{\mu}(z)\,dz^{\mu}\right)\;,$
(4)

uniformly taking a straight line path between the two. Regardless of the path,
the Wilson line transforms homogeneously under a gauge transformation
$\Omega(x)$,

${\bf L}(y;x)\rightarrow\Omega^{\dagger}(y)\,{\bf L}(y;x)\,\Omega(x)\;.$ (5)

Since $D_{0}\,{\bf L}(\vec{x};t^{\prime},t)=0$, the propagator for a test
quark sitting at a point $\vec{x}$ is proportional to ${\bf
L}(\vec{x};t^{\prime},t)={\bf L}(\vec{x},t^{\prime};\vec{x},t)$, using an
obvious abbreviated notation.

A test meson is constructed by putting a test quark at one point,
$(\vec{x},0)$, and tying it with a spatial Wilson line to a test anti-quark at
$(\vec{0},0)$. A rectangular Wilson loop represents the propagation of this
test meson up in time, until it is annhilated by a test anti-meson.

At nonzero temperature it is possible to construct a gauge invariant operator
for a single test quark. In most gauges the gauge fields can be taken to
periodic in imaginary time, $A_{\mu}(\vec{x},1/T)=+A_{\mu}(\vec{x},0)$. The
Polyakov loop is the trace of the full thermal Wilson line, which runs from
$\tau:0\rightarrow 1/T$,

${\mathbb{P}}(\vec{x})={\rm tr}\;{\bf L}(\vec{x};1/T,0)\;.$ (6)

This is invariant under strictly periodic gauge transformations. In a gauge
theory without dynamical quarks, though, the gauge symmetry is $SU(N)/Z(N)$,
and it is also necessary to consider global $Z(N)$ transformations. These form
the center of the gauge group, $\omega_{j}={\rm e}^{2\pi ij/N}\,{\bf 1}$,
$j=1\ldots N$, where the $\omega_{j}$ commute with all elements of the group.
Then more general gauge transformations are possible, which are periodic only
up to a factor of $\omega_{j}$,

$\Omega(\vec{x},1/T)=\omega_{j}\,\Omega(\vec{x},0)\;,$ (7)

Since the $\omega_{j}$ commute with all group elements, the gauge fields
remain strictly periodic. The Polyakov loop, though, transforms linearly,

${\mathbb{P}}(\vec{x})\rightarrow\omega_{j}\,{\mathbb{P}}(\vec{x})\;.$ (8)

The spatial average of the vacuum expectation value, $\int
d^{3}x\,\langle{\mathbb{P}}(\vec{x})\rangle$, is an order parameter for the
spontaneous breaking of a global, one-form Gaiotto  _et al._ (2015) $Z(N)$
symmetry Polyakov (1978); Susskind (1979). Without dynamical quarks, this is
an exact symmetry in the confined phase, which is spontaneously broken in the
deconfining phase. With dynamical quarks the Polaykov loop is still a gauge
invariant operator, but the global $Z(N)$ symmetry is lost.

Averaging over space, the expectation value of the Polyakov loop is

$\langle{\mathbb{P}}\rangle=\int\frac{d^{3}x}{V}\;\int{\cal D}A_{\mu}\;{\rm
e}^{-\,{\cal S}(A_{\mu})}\;\langle{\mathbb{P}}(\vec{x})\rangle\;.$ (9)

This is the path integral over the gauge field, with action ${\cal
S}(A_{\mu})$, averaged over the spatial volume $V$. This average is not a free
energy, as sometimes stated Belyaev  _et al._ (1992); Chen  _et al._ (1992);
Kogan (1994); Smilga (1994); Hansson  _et al._ (1995); Kiskis (1995); Smilga
(1997); Korthals-Altes  _et al._ (1999); Korthals-Altes and Kovner (2000); de
Forcrand  _et al._ (2001). Assume that $\langle{\mathbb{P}}\rangle={\rm
e}^{-{\mathcal{F}}_{\infty}/T}$. If ${\cal F}_{\infty}$ were a free energy,
then it must be real, with an exponential which is real and positive. For two
colors, however, ${\mathbb{P}}(\vec{x})$ can be negative, while for three or
more colors, it is complex.

As the color trace of a propagator, though, there is nothing amiss if the
Polyakov loop is either negative or complex. Consequently, the Polyakov loop,
or more properly the eigenvalues of the thermal Wilson line Gross  _et al._
(1981); Weiss (1981); Bhattacharya  _et al._ (1991); Belyaev (1991);
Bhattacharya  _et al._ (1992); Gocksch and Pisarski (1993); Korthals Altes
(1994); Korthals Altes  _et al._ (1994, 1997, 2000); Giovannangeli and
Korthals Altes (2001, 2005a, 2005b); Dumitru  _et al._ (2014); Guo (2014);
Nishimura  _et al._ (2018); Guo and Du (2019); Korthals Altes  _et al._
(2020a, b); Hidaka and Pisarski (2021); Guo and Kuang (2021), are not an
artifact of Euclidean spacetime Belyaev  _et al._ (1992); Chen  _et al._
(1992); Kogan (1994); Smilga (1994); Hansson  _et al._ (1995); Kiskis (1995);
Smilga (1997); Korthals-Altes  _et al._ (1999); Korthals-Altes and Kovner
(2000); de Forcrand  _et al._ (2001); Cohen (2022), but perfectly sensible
physical quantities.

At nonzero temperature, the potential between a test quark and anti-quark is
given by the two point function of Polyakov loops,

${\rm
e}^{-{\mathcal{V}}_{\infty}(\vec{x})/T}=\langle{\mathbb{P}}^{\dagger}(\vec{x}){\mathbb{P}}(\vec{0})\rangle-|\langle{\mathbb{P}}\rangle|^{2}\;.$
(10)

Another gauge invariant quantity is the thermal Wilson loop,

${\mathcal{W}}_{T}={\rm tr}\,{\bf L}(\vec{0};1/T,0){\bf
L}(\vec{x},\vec{0};1/T){\bf L}(\vec{x};0,1/T){\bf L}(\vec{0},\vec{x};0)\;.$
(11)

Because of the spatial Wilson lines from $\vec{0}$ to $\vec{x}$ at $\tau=0$
and $1/T$, contributions which don’t appear in
${\mathcal{V}}_{\infty}(\vec{x})$ enter Laine  _et al._ (2007).

Hamiltonian formalism: It is necessary to transform to $A_{0}=0$ gauge. I
ignore technicalities, such as fixing the residual degrees of freedom for the
$A_{i}$ fields Gervais and Sakita (1978); Goldstone and Jackiw (1978); Jackiw
(1980); Christ and Lee (1980), as these do not affect my analysis. Under a
gauge transformation,

$A_{\mu}(\vec{x},\tau)\rightarrow\frac{1}{-ig}\Omega^{\dagger}(\vec{x},\tau)D_{\mu}\Omega(\vec{x},\tau)\;.$
(12)

The gauge transformation which implements $A_{0}=0$ gauge is just
$\Omega(\vec{x},\tau)={\bf L}(\vec{x};\tau,0)$. Since in general
$\Omega(\vec{x},1/T)\neq\Omega(\vec{x},0)$, when $A_{0}(\vec{x},\tau)=0$ the
$A_{i}(\vec{x},\tau)$ are no longer periodic in $\tau$.

In the Hamiltonian formalism, the basic variables are the spatial gauge
fields, $\vec{A}$, whose conjugate momenta are the color electric fields,
$\vec{E}=\partial_{0}\vec{A}$. For a quark field $\psi$ the conjugate momentum
is $\overline{\psi}$ Gervais and Sakita (1978); Goldstone and Jackiw (1978);
Jackiw (1980); Christ and Lee (1980). The Hamiltonian density is

${\mathcal{H}}(\vec{x})={\rm
tr}\left(\vec{E}^{2}(\vec{x})+\vec{B}^{2}(\vec{x})\right)\;;$ (13)

as the test quark only enters into the Lagrangian as
$\overline{\psi}\partial_{0}\psi$, it drops out of the Hamiltonian.

To ensure the conservation of color electric charge, however, it is still
necessary to impose Gauss’s law. For this it is convenient to introduce an
auxiliary field, $\chi(\vec{x})$:

${\mathcal{H}}_{\rm Gauss}(\vec{x})={\rm
tr}\left(\chi(\vec{x})\left(\vec{D}\cdot\vec{E}(\vec{x})-g\,{\mathcal{Q}}(\vec{x})\right)\right)\;,$
(14)

where ${\mathcal{Q}}^{a}(\vec{x})=\psi^{\dagger}(\vec{x})t^{a}\psi(\vec{x})$
is the color charge of the test quark and the $t^{a}$ are the generators in
the fundamental representation, $a=1\ldots(N^{2}-1)$. Since only particles
without spin enter in the effective Lagrangian, it is not necessary to bother
with Dirac matrices. The color charge ${\mathcal{Q}}$ transforms homogeneously
in the adjoint representation,
${\mathcal{Q}}(\vec{x})\rightarrow\Omega^{\dagger}(\vec{x})\,{\mathcal{Q}}(\vec{x})\,\Omega(\vec{x})$,
as does the constraint field $\chi(\vec{x})$ 444On the lattice, $\chi$ lives
on sites, not links.. For a Polyakov loop, the color charge
${\mathcal{Q}}(\vec{x})$ is a single $\delta$-function in $\vec{x}$; for a
Wilson loop, there are two $\delta$-functions, and so on for more loops.

As discussed by Gervais and Sakita Gervais and Sakita (1978), states for test
quarks must be included in the partition function. To understand how they
contribute, I ask:

How does the exponential of a trace become the trace of an exponential?

That is, how does the test charge ${\mathcal{Q}}$ in Eq. (14) transform into
the Wilson and Polyakov loops of Eqs. (2) and (6)?

The answer is an exercise in the character for a representation of a Lie group
Georgi (2000); Cvitanovic (2008); Greiner and Muller (2013); Hall (2015). This
was used originally by Susskind Susskind (1979) 555In Eq. (60) of Ref.
Susskind (1979), the sum is over the representations of the electric field, to
give the character of $\chi$, as in Eq. 16. The product of characters which
arise for the two point function of Polyakov loops, Eq. (68), is given without
comment. I show that this is due to the sum over states of the test charge.,
and is related to an analysis by Greiner and Müller Greiner and Muller (2013).

A representation ${\mathcal{R}}$ of the $SU(N)$ group is characterized by a
Young tableaux, which are $N-1$ integers, $n_{1},n_{2}\ldots n_{N-1}$, where
$n_{1}\geq n_{2}\ldots\geq n_{N-1}$ Georgi (2000); Cvitanovic (2008); Greiner
and Muller (2013); Hall (2015). For the case of $SU(2)$, there is only one
row, where $n_{1}$ equals the spin, $n_{1}=j=0,1,2\ldots$.

The representations which contribute to the states of the electric field are
denoted as $|{\mathcal{R}}(\vec{x})\rangle$, and that for the test quark as
$|\widetilde{{\mathcal{R}}}(\vec{x})\rangle$. While the test quark lies in a
fixed representation at a few points in space, all representations contribute
to the state space of the electric field at each point in space. For the
example of $SU(2)$, all $j(\vec{x})$ contribute to the electric field at each
$\vec{x}$, while only a single $\widetilde{j}(\vec{x})$ contributes to that of
the test quark, at the point where the Polyakov loop lies.

The expectation value of the Polyakov loop is given by

$\displaystyle\langle{\mathbb{P}}(\vec{y})\rangle$ $\displaystyle=$
$\displaystyle\widetilde{\sum}\;\int{\cal D}\chi(\vec{x})\;\exp\left(-\int
d^{3}x\,{\mathcal{H}}_{0}(\vec{x})/T\right)\;,$
$\displaystyle{\mathcal{H}}_{0}(\vec{x})$ $\displaystyle=$
$\displaystyle{\mathcal{H}}(\vec{x})+{\rm
tr}\;\chi(\vec{x})\,\vec{D}\cdot\vec{E}(\vec{x})+{\mathcal{H}}_{\mathcal{Q}}(\vec{x})\;,$
$\displaystyle{\mathcal{H}}_{\mathcal{Q}}(\vec{x})$ $\displaystyle=$
$\displaystyle-{\rm tr}\;\chi(\vec{x})\,{\mathcal{Q}}(\vec{x})\;.$ (15)

The sum over states, $\widetilde{\sum}$, includes those for the gauge field,
the $\vec{A}(\vec{x})$ and $\vec{E}(\vec{x})$, and the states for the test
quarks, $\psi(\vec{x})$ and $\overline{\psi}(\vec{x})$. It is also necessary
to include a path integral for the constraint field, $\chi(\vec{x})$.
Dynamical quarks can be included directly.

I note that it is meaningful to compute the expectation value of a single
Polyakov loop in a non-Abelian gauge theory, as the color charge is always
screened. At low temperature this happens either because of confinement
(without dynamical quarks), or the pair production of mesons (with dynamical
quarks). At high temperature, there is always Debye screening 666 The Abelian
theory is different, as while there is Debye screening at high temperature,
there is no screening at low temperature. Further, in a finite box the
presence of a test charge is inconsistent with periodic boundary conditions
Hilf and Polley (1983), and so open boundary conditions must be used. In the
unscreened phase of the Abelian theory it is still possible to measure the
potential between a test charge and anti-charge, ${\cal
V}_{\infty}(\vec{x})$.. Typical of a system with the spontaneous breaking of a
global symmetry, the expectation value of the Polyakov loop is only well
defined after introducing an appropriate 777The appropriate sources for
Polyakov loops must involve a sum over an infinite number of loops. This is
because the matrix for any representation of $SU(N)$ is traceless, so for a
single loop the term linear in $A_{\mu}$ vanishes at small $A_{\mu}$. This
holds for a sum over any finite number of loops, but fails if the sum is
infinite. An appropriate source is that for which an infinitesimal source
generates an expectation value which is also infinitesimal. For details, see
Refs. Korthals Altes  _et al._ (2020a, b); Hidaka and Pisarski (2021).
external source for the corresponding field, and then tuning that source to
zero.

It is easy to perform the sum over states for the Polyakov loop in Eq. 15, as
the only quantum number carried by the test quark is that for color electric
charge. Since the color charge transforms homogeneously under gauge
transformations, I can assume that it is a diagonal matrix. Then if the quark
and anti-quark have color $1$,
${\mathcal{Q}}(\vec{x})^{a}=t^{a}_{11}\delta^{3}(\vec{x}-\vec{y})$, where
$\vec{y}$ is the position of the loop, and contributes to
${\mathbb{P}}(\vec{x})$ as
$\sim\exp(ig\chi^{a}\,t^{a}_{11})\delta^{3}(\vec{x}-\vec{y})$. If the quark
and anti-quark have color $2$, the contribution to ${\mathbb{P}}(\vec{x})$ is
$\sim\exp(ig\chi^{a}\,t^{a}_{22})\delta^{3}(\vec{x}-\vec{y})$, and so on.

Thus the trace over states of the test quark is just a trace over color,

$\langle{\mathbb{P}}(\vec{y})\rangle=\sum\int{\cal D}\chi(\vec{x})\;{\rm
e}^{-\int\,{\mathcal{H}}_{\rm 0}/T}\;{\rm tr}\;{\rm e}^{ig\chi(\vec{y})}\;.$
(16)

Having summed over the states for the test quark and anti-quark, the remaining
sum, $\sum$, is only for the $\vec{A}$ and $\vec{E}$, along with the path
integral over the constraint field, $\chi(\vec{x})$.

From this derivation, it is apparent that $\chi(\vec{x})$ corresponds to the
time-like component of the gauge potential in the Lagrangian formalism,
$A_{0}(\vec{x})$ Gervais and Sakita (1978); Goldstone and Jackiw (1978);
Jackiw (1980); Christ and Lee (1980). The transformation from $A_{0}=0$ gauge
in the Lagrangian formalism is given by identifying the gauge transformation
$\Omega(\vec{x},1/T)={\bf L}(\vec{x};,1/T,0)$ with $\exp(ig\chi(\vec{x}))$.
After averaging over the spatial volume, this agrees with Eq. (9). Similarly,
the thermal Wilson loop ${\mathcal{W}}_{T}$ equals

${\mathcal{W}}_{T}={\rm tr}\;{\rm e}^{ig\chi(\vec{x})}\;{\bf
L}(\vec{x};\vec{0})\;{\rm e}^{-ig\chi(\vec{0})}\;{\bf L}(\vec{0};\vec{x})\;.$
(17)

The generalization to higher representations of the test quark is direct, with
the generator is given by a bird track diagram, Eq. (4.35) of Cvitanovic
Cvitanovic (2008). The sum over all color states is given by summing over all
of the legs of the bird track.

In the Lagrangian formalism, the holonomous potential for the eigenvalues of
the thermal Wilson line is familiar, and computed by expanding about a
constant, background field $A_{0}\neq 0$ Gross  _et al._ (1981); Weiss (1981);
Bhattacharya  _et al._ (1991); Belyaev (1991); Bhattacharya  _et al._ (1992);
Gocksch and Pisarski (1993); Korthals Altes (1994); Korthals Altes  _et al._
(1994, 1997, 2000); Giovannangeli and Korthals Altes (2001, 2005a, 2005b);
Dumitru  _et al._ (2014); Guo (2014); Nishimura  _et al._ (2018); Guo and Du
(2019); Korthals Altes  _et al._ (2020a, b); Hidaka and Pisarski (2021); Guo
and Kuang (2021). In the Hamiltonian formalism, non-trivial holonomy becomes
the factor of $\exp(ig\chi(\vec{x}))$, which enters as an imaginary chemical
potential for the color charge.

In the gauge theory without dynamical quarks, the holonomous potential
manifestly exhibits the global $Z(N)$ degeneracy for the $SU(N)/Z(N)$ theory.
With dynamical quarks, however, depending upon the representation of the
quarks and the color, it is possible to have metastable states with negative
pressure Belyaev  _et al._ (1992); Chen  _et al._ (1992); Kogan (1994); Smilga
(1994, 1997). This occurs because the zero of the potential for non-trivial
holonomy has an absolute significance, as the pressure with with vanishing
holonomy. However, while a bubble of such a metastable state can have negative
pressure, it only lasts as long as it takes the surface of the bubble to
collapse upon itself, decaying through cavitation Rajagopal and Tripuraneni
(2010); Brennan .

’t Hooft loops: Wilson loops represent the propagation of test electric
charge. ’t Hooft first constructed a dual order parameter to the Wilson loop,
which represents the propagation of a test magnetic charge ’t Hooft (1979,
1980). The Wilson and ’t Hooft loops satisfy a commutation law, which in
vacuum excludes the simultaneous confinement of electric and magnetic charges.
The commutation law follows by considering the Wilson loop as the propagator
for a test electric charge: as a tiny Wilson loop encircles a ’t Hooft loop,
by definition the phase of a test charge (in the fundamental representation)
changes by ${\rm e}^{2\pi i/N}$.

At nonzero temperature in the $SU(N)/Z(N)$ gauge theory, due to the global
$Z(N)$ symmetry there are $N$ degenerate vacua in the deconfined phase. At
high temperature, it is possible to consider a box which is long in one
spatial direction, and to compute the interface tension between a $Z(N)$
vacuum at one end of the box, and a different $Z(N)$ vacuum at the other This
interface tension can be computed semi-classically Bhattacharya  _et al._
(1991); Belyaev (1991); Bhattacharya  _et al._ (1992) from the holonomous
potential Gocksch and Pisarski (1993); Korthals Altes (1994); Korthals Altes
_et al._ (1994, 1997, 2000); Giovannangeli and Korthals Altes (2001, 2005a,
2005b); Dumitru  _et al._ (2014); Guo (2014); Nishimura  _et al._ (2018); Guo
and Du (2019); Korthals Altes  _et al._ (2020a, b); Hidaka and Pisarski
(2021); Guo and Kuang (2021). Korthals-Altes, Kovner, and Stephanov showed
that the interface tension is equivalent to an area law for the spatial ’t
Hooft loop Korthals-Altes  _et al._ (1999); Korthals-Altes and Kovner (2000).
Numerical simulations on the lattice have measured how the ’t Hooft loop
changes with temperature de Forcrand  _et al._ (2001); de Forcrand and von
Smekal (2002); de Forcrand and Noth (2005). See, also, Refs. Reinhardt (2003);
Reinhardt and Epple (2007).

In the Hamiltonian form, a domain wall can be constructed by using a constant
$\chi$ field, corresponding to constant $A_{0}$ in the Lagrangian formalism.
The simplest model to study is the Abelian theory in $1+1$ dimensions, where
the object analogous to a domain wall in higher dimensions is a soliton. Since
gauge fields have no physical degrees of freedom in two spacetime dimensions,
it is necessary to add dynamical fermions. Adding massless fields gives the
Schwinger model, but this theory behaves contrary to naive expectation, as
even fractional test charges are screened by dynamical fields with integral
charge Coleman  _et al._ (1975); Gross  _et al._ (1996); Dempsey  _et al._
(2021).

If the dynamical fields are massive, though, dynamical fermions with integral
charge do not screen fractional test charges Coleman  _et al._ (1975); Gross
_et al._ (1996); Dempsey  _et al._ (2021). Smilga first noted the existence of
thermal solitons in the massive Schwinger model Smilga (1994). In the
Euclidean Lagrangian, one takes a background, classical field $A_{0}^{\rm
cl}=2\pi T\,q/e$, where $e$ is the Abelian coupling constant; in the
Hamiltonian form, one takes a similar background for the constraint field,
$\chi$. As $q$ is a periodic variable, a thermal soliton interpolates from
$q=0$ at $x=-\infty$ to $q=1$ at $x=+\infty$. At high temperature, $T\gg m$,
the potential for $q$ generated at one loop order is $\sim T^{2}$ times a
periodic function of $q$, Eq. (3.9) of Ref. Smilga (1994). At low temperature,
the potential is Boltzmann suppresed, $\sim{\rm e}^{-m/T}$, and vanishes
smoothly as $T\rightarrow 0$.

I suggest that such solitons are stable. At $T\neq 0$ imaginary time is
topologically equivalent to a torus, $S^{1}$. As $q$ is a periodic variable,
then, mappings from space onto $q$ are determined by the first homotopy group,
$\pi_{1}(S^{1})=Z$, which is the set of the integers.

Smilga and others argued that thermal solitons are unphysical Smilga (1994);
Hansson  _et al._ (1995); Kiskis (1995); Smilga (1997), I suggest that they
represent new, collective excitations at $T\neq 0$, which evaporate smoothly
as $T\rightarrow 0$. This can be studied numerically at nonzero temperature in
real time, using either tensor networks on a classical computer Bañuls  _et
al._ (2009); Banuls  _et al._ (2020); Lerose  _et al._ (2022); Frías-Pérez and
Bañuls (2022), or even with the noisy intermediate-scale quantum computers
which are available at present. This is similar to studying the screening of
background electric fields at nonzero $\theta$ Klco  _et al._ (2018);
Chakraborty  _et al._ (2020); Kharzeev and Kikuchi (2020); Pla  _et al._
(2021); Shaw  _et al._ (2020); de Jong  _et al._ (2021); Florio and Kharzeev
(2021); Honda  _et al._ (2022, 2021); Pederiva  _et al._ (2021).

There are many other problems which can be addressed with the formalism
developed here. In particular, deep inelastic scattering is usually described
by the propagation of time-like Wilson lines Echevarria  _et al._ (2021). My
approach can be adapted to the light-front directly Lerose  _et al._ (2022);
Frías-Pérez and Bañuls (2022); Wilson  _et al._ (1994), especially using
quantum computers Kreshchuk  _et al._ (2020, 2021a, 2021b).

Lastly, if thermal solitons are stable in $1+1$ dimensions, presumably thermal
domain walls exist in $3+1$ dimensions. In the early universe, they can arise
from the $U(1)$ of electromagnetism, when regions which are causally
disconnected from one another first come in contact. They persist until the
thermal potential for the domain wall of the lightest electrically charged
particle, which is the electron, is Boltzmann suppressed. As this temperature
is presumably below that for nucleosynthesis, and as domain walls dominate the
energy density of the universe, such thermal $U(1)$ domain walls could be of
cosmological consequence.

###### Acknowledgements.

This work was inspired by a talk which Tom Cohen gave virtually at the Yukawa
Institute for Theoretical Physics, during a workshop on the “QCD phase diagram
and lattice QCD”, YITP-W-21-09, October 25-29, 2021. I thank him, M. Creutz,
A. Florio, L. Glozman, C. Korthals-Altes, S. Mukherjee, P. Petreczky, O.
Philipsen, E. Poppitz, and A. Smilga for discussions. This research was
supported by the U.S. Department of Energy under contract DE-SC0012704. I was
also led to consider this problem because of the support of the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract
number DE-SC0012704.

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