't Hooft operator
Encyclopedia
In theoretical physics
, a 't Hooft operator, introduced by Gerard 't Hooft in the 1978 paper On the phase transition towards permanent quark confinement is a dual version of the Wilson loop
in which the electromagnetic potential A is replaced by its electromagnetic dual Amag where the exterior derivative
of A is equal to the Hodge dual
of the exterior derivative of Amag. In d spacetime dimensions, Amag is a (d-3)-form
and so the 't Hooft operator is the integral of Amag over a (d-3)-dimensional surface.
While the Wilson loop is an order operator
, 't Hooft operator is an example of a disorder operator
because it creates a singularity
or a discontinuity
in the fundamental fields such as the electromagnetic potential A. For example, in an SU(N) Yang Mills gauge theory
a 't Hooft operator creates a Dirac magnetic monopole
with respect to the center
of SU(N). If a condensate
is present which transforms in a representation of SU(N) which is invariant under the action of the center, such as the adjoint representation
, then the magnetic monopole will be confined by a vortex
lying along a Dirac string
from the monopole to either an antimonopole or to infinity. This vortex is similar to a Nielsen-Olesen vortex
, but it carries a charge under the center of SU(N), and so N such vortices may annihilate.
In his 1978 paper, 't Hooft demonstrated that Wilson loops and 't Hooft operators commute up to a phase which is an nth root of unity
. The choice of root of unity depends is equal to the linking number
of the Wilson loop and the vortex. 't Hooft claimed that this apparently non-local commutation relation implies that any phase of a Yang-Mills gauge theory must either contain massless particles, responsible for the interactions between the 't Hooft operator and the Wilson loop, or else at least one of the two operators must be confined by an object one dimension higher. He identified the phase in which the 't Hooft operator is confined is the Higgs phase
, in which the confinement of magnetic monopoles by vortices was a well-known consequence of the Meissner effect
, already observed in type II superconductors. He identified the phase in which the Wilson loop is confined as the confining phase, as a Wilson loop is the action of an electric charge. Finally he left open the possibility of mixed phases, in which both are confined. Although such mixed phases had not been seen in quantum field theory at the time, they are now know to occur for example in the Argyres-Douglas conformal field theory. Therefore he argued that gauge theories are necessarily in one of these four possible phases.
't Hooft found a simple formula for the scalings of the Wilson and 't Hooft operators in the various phases. When a given operator is confined, a finite tension surface is created whose boundary is the operator. The action of the configuration, in the limit in which the configuration is large, therefore scales with the volume of this surface. In the confining phase the Wilson loops are confined by a 2-dimensional surface, and so the action of a Wilson loop scales as the area of this surface. In the Higgs phase the (d-3)-dimensional 't Hooft operator is confined, and so the action scales as the area of the (d-2)-dimensional confining surface. For example in the confining phase in 4 space-time dimensions the action of the 't Hooft operator scales as the distance squared. In the mixed phase both operators are confined, and so both obey this scaling.
On the other hand he claimed that if a given operator is Higgsed, then the corresponding gluon
s are massive and so the action falls off exponentially away from the operator. Therefore the action will be proportional to volume of the surface on which the operator is evaluated itself. For example in the Higgs phase the gauge field is Higgsed and so the Wilson loop action is proportional to the length of the loop, which scales linearly with distance. In the confining phase the 't Hooft operator is Higgsed, and so the corresponding action fails as the area of the corresponding (d-3)-dimensional surface, for example linearly in 4 spacetime dimensions. In particular 't Hooft concluded thatin 4 dimensions if both the actions of the Wilson and 't Hooft loops scale linearly then both are Higgsed and so there must be massless particles in the spectrum.
Today 't Hooft's classication of phases is the bases of the classification of QCD phase diagram
, with the Higgs phase manifested at the cold temperatures and low densities usually found on Earth, massless particles and deconfinment existing at high temperature experiments at RHIC and soon the LHC
and perhaps mixed phases existing in the cores of neutron star
s.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a 't Hooft operator, introduced by Gerard 't Hooft in the 1978 paper On the phase transition towards permanent quark confinement is a dual version of the Wilson loop
Wilson loop
In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop...
in which the electromagnetic potential A is replaced by its electromagnetic dual Amag where the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
of A is equal to the Hodge dual
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...
of the exterior derivative of Amag. In d spacetime dimensions, Amag is a (d-3)-form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
and so the 't Hooft operator is the integral of Amag over a (d-3)-dimensional surface.
While the Wilson loop is an order operator
Order operator
In quantum field theory, an order operator or an order field is a quantum field version of Landau's order parameter whose expectation value characterizes phase transitions...
, 't Hooft operator is an example of a disorder operator
Disorder operator
In theoretical physics, a disorder operator is an operator that creates a discontinuity of the ordinary order operators or a monodromy for their values.For example, a 't Hooft operator is a disorder operator. So is the Jordan–Wigner transformation....
because it creates a singularity
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
or a discontinuity
Classification of discontinuities
Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there...
in the fundamental fields such as the electromagnetic potential A. For example, in an SU(N) Yang Mills gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
a 't Hooft operator creates a Dirac magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
with respect to the center
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
of SU(N). If a condensate
Condensate
Condensate may refer to:* The liquid phase produced by the condensation of steam or any other gas* The product of a chemical condensation reaction, other than water* Natural gas condensate, in the natural gas industry- Quantum physics :...
is present which transforms in a representation of SU(N) which is invariant under the action of the center, such as the adjoint representation
Adjoint representation
In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
, then the magnetic monopole will be confined by a vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
lying along a Dirac string
Dirac string
In physics, a Dirac string is a fictitious one-dimensional curve in space, conceived of by the physicist Paul Dirac, stretching between two Dirac magnetic monopoles with opposite magnetic charges, or from one magnetic monopole out to infinity. The gauge potential cannot be defined on the Dirac...
from the monopole to either an antimonopole or to infinity. This vortex is similar to a Nielsen-Olesen vortex
Nielsen-Olesen vortex
In theoretical physics, a Nielsen–Olesen vortex is a point-like object localized in two spatial dimensions or, equivalently, a classical solution of field theory with the same property. This particular solution occurs if the configuration space of scalar fields contains non-contractible circles...
, but it carries a charge under the center of SU(N), and so N such vortices may annihilate.
In his 1978 paper, 't Hooft demonstrated that Wilson loops and 't Hooft operators commute up to a phase which is an nth root of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
. The choice of root of unity depends is equal to the linking number
Linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other...
of the Wilson loop and the vortex. 't Hooft claimed that this apparently non-local commutation relation implies that any phase of a Yang-Mills gauge theory must either contain massless particles, responsible for the interactions between the 't Hooft operator and the Wilson loop, or else at least one of the two operators must be confined by an object one dimension higher. He identified the phase in which the 't Hooft operator is confined is the Higgs phase
Higgs phase
In theoretical physics, it is often important to consider gauge theory that admits many physical phenomena and "phases", connected by phase transitions, in which the vacuum may be found.Gauge symmetry may be broken by the Higgs mechanism...
, in which the confinement of magnetic monopoles by vortices was a well-known consequence of the Meissner effect
Meissner effect
The Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. The German physicists Walther Meissner and Robert Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin...
, already observed in type II superconductors. He identified the phase in which the Wilson loop is confined as the confining phase, as a Wilson loop is the action of an electric charge. Finally he left open the possibility of mixed phases, in which both are confined. Although such mixed phases had not been seen in quantum field theory at the time, they are now know to occur for example in the Argyres-Douglas conformal field theory. Therefore he argued that gauge theories are necessarily in one of these four possible phases.
't Hooft found a simple formula for the scalings of the Wilson and 't Hooft operators in the various phases. When a given operator is confined, a finite tension surface is created whose boundary is the operator. The action of the configuration, in the limit in which the configuration is large, therefore scales with the volume of this surface. In the confining phase the Wilson loops are confined by a 2-dimensional surface, and so the action of a Wilson loop scales as the area of this surface. In the Higgs phase the (d-3)-dimensional 't Hooft operator is confined, and so the action scales as the area of the (d-2)-dimensional confining surface. For example in the confining phase in 4 space-time dimensions the action of the 't Hooft operator scales as the distance squared. In the mixed phase both operators are confined, and so both obey this scaling.
On the other hand he claimed that if a given operator is Higgsed, then the corresponding gluon
Gluon
Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....
s are massive and so the action falls off exponentially away from the operator. Therefore the action will be proportional to volume of the surface on which the operator is evaluated itself. For example in the Higgs phase the gauge field is Higgsed and so the Wilson loop action is proportional to the length of the loop, which scales linearly with distance. In the confining phase the 't Hooft operator is Higgsed, and so the corresponding action fails as the area of the corresponding (d-3)-dimensional surface, for example linearly in 4 spacetime dimensions. In particular 't Hooft concluded thatin 4 dimensions if both the actions of the Wilson and 't Hooft loops scale linearly then both are Higgsed and so there must be massless particles in the spectrum.
Today 't Hooft's classication of phases is the bases of the classification of QCD phase diagram
QCD matter
Quark matter or QCD matter refers to any of a number of theorized phases of matter whose degrees of freedom include quarks and gluons. These theoretical phases would occur at extremely high temperatures and densities, billions of times higher than can be produced in equilibrium in laboratories...
, with the Higgs phase manifested at the cold temperatures and low densities usually found on Earth, massless particles and deconfinment existing at high temperature experiments at RHIC and soon the LHC
LHC
LHC may refer to:* Large Hadron Collider, a particle accelerator and collider located on the Franco-Swiss border near Geneva, SwitzerlandLHC also may refer to:* La hora Chanante, a Spanish comedy television show...
and perhaps mixed phases existing in the cores of neutron star
Neutron star
A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...
s.