In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, the Polyakov loop is the thermal analogue of the
Wilson loop
In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representat ...
, acting as an
order parameter
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
for
confinement
Confinement may refer to
* With respect to humans:
** An old-fashioned or archaic synonym for childbirth
** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods
** Civil confi ...
in pure
gauge theories
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
at nonzero
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
s. In particular, it is a Wilson loop that winds around the
compactified Euclidean temporal direction of a
thermal quantum field theory
In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature.
...
. It indicates confinement because its
vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
must vanish in the confined phase due to its non-invariance under center gauge transformations. This also follows from the fact that the expectation value is related to the
free energy of individual
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s, which diverges in this phase. Introduced by
Alexander M. Polyakov in 1975, they can also be used to study the
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
between pairs of quarks at nonzero temperatures.
Definition
Thermal quantum field theory is formulated in Euclidean spacetime with a compactified
imaginary temporal direction of length
. This length corresponds to the inverse temperature of the field
. Compactification leads to a special class of
topologically nontrivial Wilson loops that wind around the compact direction known as Polyakov loops. In
theories a straight Polyakov loop on a spatial coordinate
is given by
where
is the
path-ordering operator and
is the Euclidean temporal component of the gauge field. In
lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretised onto a lattice.
Details
Although most lattice field theories are not exactly sol ...
this operator is reformulated in terms of temporal
link fields at a spatial position
as
:
The
continuum limit
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model (physics), lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approxi ...
of the lattice must be taken carefully to ensure that the compact direction has fixed extent. This is done by ensuring that the finite number of temporal lattice points
is such that
is constant as the lattice spacing
goes to zero.
Order parameter
Gauge fields need to satisfy the periodicity condition
in the compactified direction. Meanwhile, gauge transformations only need to satisfy this up to a
group center
In abstract algebra, the center of a group, , is the set of elements that commute with every element of . It is denoted , from German '' Zentrum,'' meaning ''center''. In set-builder notation,
:.
The center is a normal subgroup, . As a subgr ...
term
as
. A change of basis can always diagonalize this so that
for a complex number
. The Polyakov loop is topologically nontrivial in the temporal direction so unlike other Wilson loops it transforms as
under these transformations. Since this makes the loop gauge dependent for
, by
Elitzur's theorem
In quantum field theory and statistical field theory, Elitzur's theorem states that in gauge theories, the only operators that can have non-vanishing expectation values are ones that are invariant under local gauge transformations. An important i ...
non-zero expectation values of
imply that the center group must be
spontaneously broken, implying confinement in pure gauge theory. This makes the Polyakov loop an order parameter for confinement in thermal pure gauge theory, with a confining phase occurring when
and deconfining phase when
. For example,
lattice calculations of
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
with infinitely heavy quarks that decouple from the theory shows that the deconfinement phase transition occurs at around a temperature of
MeV. Meanwhile, in a gauge theory with quarks, these break the center group and so confinement must instead be deduced from the spectrum of asymptotic states, the color neutral
hadron
In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the ele ...
s.
For gauge theories that lack a nontrivial group center that could be broken in the confining phase, the Polyakov loop expectation values are nonzero even in this phase. They are however still a good indicator of confinement since they generally experience a sharp jump at the
phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
. This is the case for example in the
Higgs model with the
exceptional gauge group .
The
Nambu–Jona-Lasinio model
In quantum field theory, the Nambu–Jona-Lasinio model (or more precisely: ''the Nambu and Jona-Lasinio model'') is a complicated effective theory of nucleons and mesons constructed from interacting Dirac fermions with chiral symmetry, parall ...
lacks local color symmetry and thus cannot capture the effects of confinement. However, Polyakov loops can be used to construct the Polyakov-loop-extended Nambu–Jona-Lasinio model which treats both the
chiral condensate and the Polyakov loops as classical homogeneous
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
that couple to quarks according to the
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and symmetry breaking patters of quantum chromodynamics.
Quark free energy
The free energy
of
quarks and
antiquarks
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...
, subtracting out the
vacuum energy
Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. The vacuum energy is a special case of zero-point energy that relates to the quantum vacuum.
The effects of vacuum energy can be experiment ...
, is given in terms of the
correlation functions of Polyakov loops
:
This free energy is another way to see that the Polyakov loop acts as an order parameter for confinement since the free energy of a single quark is given by
. Confinement of quarks means that it would take an infinite amount of energy to create a configuration with a single free quark, therefore its free energy must be infinite and so the Polyakov loop expectation value must vanish in this phase, in agreement with the center symmetry breaking argument.
The formula for the free energy can also be used to calculate the potential between a pair of infinitely massive quarks spatially separated by
. Here the potential
is the first term in the free energy, so that the correlation function of two Polyakov loops is
:
where
is the energy difference between the potential and the first
excited state
In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). Excitation refers to a ...
. In the confining phase the potential is linear
, where the constant of proportionality is known as the string tension. The string tension acquired from the Polyakov loop is always bounded from above by the string tension acquired from the Wilson loop.
See also
*
Quark–gluon plasma
Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a 1 ...
*
't Hooft loop
References
{{reflist
Gauge theories
Quantum chromodynamics
Lattice field theory
Phase transitions