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quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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 representati ...
, acting as an order parameter for confinement 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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
at nonzero
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
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. I ...
. It indicates confinement because its
vacuum expectation value In quantum field theory, the vacuum expectation value (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. One of the most widely used exa ...
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 nucleus, atomic nuclei ...
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 r ...
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 \beta. This length corresponds to the inverse temperature of the field \beta \propto 1/T. Compactification leads to a special class of topologically nontrivial Wilson loops that wind around the compact direction known as Polyakov loops. In \text(N) theories a straight Polyakov loop on a spatial coordinate \boldsymbol x is given by where \mathcal P is the path-ordering operator and A_4 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. This involves studying field theory on a space or spacetime that has been discretised onto a lattice. Details Although most lattice field theories are not ...
this operator is reformulated in terms of temporal link fields U_4(\boldsymbol m, j) at a spatial position \boldsymbol m as : \Phi(\boldsymbol m) = \frac\text \bigg \prod_^ U_4(\boldsymbol m, j)\bigg The continuum limit 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 N_T is such that \beta = N_T a is constant as the lattice spacing a goes to zero.


Order parameter

Gauge fields need to satisfy the periodicity condition A_\mu(\boldsymbol x, x_4+\beta) = A_\mu(\boldsymbol x, x_4) in the compactified direction. Meanwhile, gauge transformations only need to satisfy this up to a group center term h as \Omega(\boldsymbol x, x_4+\beta) = h \Omega(\boldsymbol x, x_4). A change of basis can always diagonalize this so that h=zI for a complex number z. The Polyakov loop is topologically nontrivial in the temporal direction so unlike other Wilson loops it transforms as \Phi(\boldsymbol x) \rightarrow z \Phi(\boldsymbol x) under these transformations. Since this makes the loop gauge dependent for z \neq 1, by Elitzur's theorem non-zero expectation values of \langle \Phi\rangle 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 \langle \Phi\rangle = 0 and deconfining phase when \langle \Phi\rangle \neq 0. For example, lattice calculations of
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study 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 of ...
with infinitely heavy quarks that decouple from the theory shows that the deconfinement phase transition occurs at around a temperature of 270 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 is a composite subatomic particle made of two or more quarks held together by the strong nuclear force. Pronounced , the name is derived . They are analogous to molecules, which are held together by the electri ...
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 physics, chemistry, and other related fields like biology, 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 Sta ...
. This is the case for example in the Higgs model with the exceptional gauge group G_2. The Nambu–Jona-Lasinio model 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 by ...
that couple to quarks according to the
symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
and symmetry breaking patters of quantum chromodynamics.


Quark free energy

The free energy F of N quarks and \bar N antiquarks, 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 : e^ = \langle \Phi(\boldsymbol x_1)\dots \Phi(\boldsymbol x_)\Phi^\dagger(\boldsymbol x'_1) \cdots \Phi^\dagger(\boldsymbol x_')\rangle. 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 e^ = \langle \Phi(\boldsymbol x)\rangle. 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 r =, \boldsymbol x_1 - \boldsymbol x_2, . Here the potential V(r) is the first term in the free energy, so that the correlation function of two Polyakov loops is : \langle \Phi(\boldsymbol x_1)\Phi(\boldsymbol x_2)\rangle \propto e^(1+\mathcal O(e^)), where \Delta E is the energy difference between the potential and the first
excited state In quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Add ...
. In the confining phase the potential is linear V(r) = \sigma r, 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 Thermodynamic equilibrium#Local and global equilibrium, thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasm ...
* 't Hooft loop


References

{{reflist Gauge theories Quantum chromodynamics Lattice field theory Phase transitions