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 ...
, Wilson loops are
gauge invariant
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 group ...
operators arising from the
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
of gauge variables around closed
loops. They encode all gauge information of the theory, allowing for the construction of
loop representations which fully describe
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 ...
in terms of these loops. In pure gauge theory they play the role of
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. There exists a dual version of it, the disorder operator or disorder fiel ...
s 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 ...
, where they satisfy what is known as the area law. Originally formulated by
Kenneth G. Wilson
Kenneth Geddes "Ken" Wilson (June 8, 1936 – June 15, 2013) was an American theoretical physicist and a pioneer in leveraging computers for studying particle physics. He was awarded the 1982 Nobel Prize in Physics for his work on phase ...
in 1974, they were used to construct links and plaquettes which are the fundamental parameters in
lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice.
Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum elec ...
. Wilson loops fall into the broader class of loop
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
, with some other notable examples being the
't Hooft loops, which are magnetic duals to Wilson loops, and
Polyakov loop
In quantum field theory, the Polyakov loop is the thermal analogue of the Wilson loop, acting as an order parameter for confinement in pure gauge theories at nonzero temperatures. In particular, it is a Wilson loop that winds around the compactif ...
s, which are the thermal version of Wilson loops.
Definition
![Ehresmann connection](https://upload.wikimedia.org/wikipedia/commons/6/62/Ehresmann_connection.png)
To properly define Wilson loops in gauge theory requires considering the
fiber bundle formulation of gauge theories. Here for each point in the
-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
there is a copy of the gauge group
forming what's known as a fiber of the
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. These fiber bundles are called
principal bundles
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
. Locally the resulting space looks like
although globally it can have some twisted structure depending on how different fibers are glued together.
The issue that Wilson lines resolve is how to compare points on fibers at two different spacetime points. This is analogous to parallel transport in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
which compares
tangent vectors
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
that live in the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s at different points. For principal bundles there is a natural way to compare different fiber points through the introduction of a
connection, which is equivalent to introducing a gauge field. This is because a connection is a way to separate out the tangent space of the principal bundle into two subspaces known as the
vertical
Vertical is a geometric term of location which may refer to:
* Vertical direction, the direction aligned with the direction of the force of gravity, up or down
* Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
and horizontal subspaces. The former consists of all vectors pointing along the fiber
while the latter consists of vectors that are perpendicular to the fiber. This allows for the comparison of fiber values at different spacetime points by connecting them with curves in the principal bundle whose tangent vectors always live in the horizontal subspace, so the curve is always perpendicular to any given fiber.
If the starting fiber is at coordinate
with a starting point of the identity
, then to see how this changes when moving to another spacetime coordinate
, one needs to consider some spacetime curve
between
and
. The corresponding curve in the principal bundle, known as the
horizontal lift of
, is the curve
such that
and that its tangent vectors always lie in the horizontal subspace. The fiber bundle formulation of gauge theory reveals that the
Lie-algebra valued gauge field
is equivalent to the connection that defines the horizontal subspace, so this leads to a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
for the horizontal lift
:
This has a unique formal solution called the Wilson line between the two points
:
where
is the
path-ordering operator, which is unnecessary for
abelian theories. The horizontal lift starting at some initial fiber point other the identity merely requires multiplication by the initial element of the original horizontal lift. More generally, it holds that if
then
for all
.
Under a
local gauge transformation the Wilson line transforms as
:
This gauge transformation property is often used to directly introduce the Wilson line in the presence of
matter fields transforming in the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group
or Lie algebra whose highest weight is a fundamental weight. For example, the defini ...
of the gauge group, where the Wilson line is an operator that makes the combination
gauge invariant. It allows for the comparison of the matter field at different points in a gauge invariant way. Alternatively, the Wilson lines can also be introduced by adding an infinitely heavy
test particle In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
charged under the gauge group. Its charge forms a quantized internal
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, which can be integrated out, yielding the Wilson line.
This works whether or not there actually is any matter content in the theory.
The
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of closed Wilson lines is a gauge invariant quantity known as the Wilson loop
Mathematically the term within the trace is known as the
holonomy
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
, which describes a
mapping of the fiber into itself upon horizontal lift along a closed loop. The set of all holonomies itself forms a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, which for principal bundles must be a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the gauge group. Wilson loops satisfy the reconstruction property where knowing the set of Wilson loops for all possible loops allows for the reconstruction of all gauge invariant information about the gauge connection. Formally the set of all Wilson loops forms an
overcomplete basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of solutions to the Gauss' law constraint.
The set of all Wilson lines is in
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
with the
representations
''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of the gauge group. This can be reformulated in terms of Lie algebra language using the
weight lattice In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplic ...
of the gauge group
. In this case the types of Wilson loops are in one-to-one correspondence with
where
is the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
.
Hilbert space operators
An alternative view of Wilson loops is to consider them as operators acting on the Hilbert space of states in
Minkowski signature
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
.
Since the Hilbert space lives on a single time slice, the only Wilson loops that can act as operators on this space are ones formed using
spacelike
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
loops. Such operators