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In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. Since they encode all possible gauge information of the theory, this allows for the construciton of loop representations which fully describe gauge theories in terms of these loops. They play a key role in pure gauge theory where they act as order operators for confinement. Originally formulated by Kenneth G. Wilson in 1974, they were used to construct links and plaquettes which are the funamental parameters in lattice gauge theory.^[1] Wilson loops fall into the broader class of loop operators, with some other notable examples being the 't Hooft loops, which are in some sense dual to Wilson loops, and Polyakov loops which are the thermal version of Wilson loops.

Definition

To properly define Wilson loops in gauge theory requires considering the fibre bundle formulation of gauge theories.^[2] Here for each point in the spacetime there is a copy of the gauge group $G$ forming what's known as a fibre of the fibre bundle. These fibre bundles are called principal bundles. Locally the resulting space looks like $\mathbb {R} ^{d}\times G$ although globally it can have some twisted structure depending on how different fibres are glued together.

The issue that Wilson lines resolve is how to compare points on fibres at two different spacetime points. This is analogous to parallel transport in general relativity which compares tangent vectors that live in the tangent space at different points. For principal bundles there is a natural way to compare different fibre points to each other 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 and horizontal subspaces.^[3] The former consists of all vectors pointing along the fibre $G$ while the latter consists of vectors that are perpendicular to the fibre. This allows one to compare fibre 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 fibre.

If the starting fibre is at $x_{0}$ and takes the curve starts at the identity $g_{0}=e$ , then to see how this changes when moving to another spacetime coordinate $x_{1}$ , one needs to consider some spacetime curve $\gamma :[0,1]\rightarrow M$ between $x_{0}$ and $x_{1}$ . The corresponding curve in the principal bundle, known as the horizontal lift of $\gamma (t)$ , is the curve ${\tilde {\gamma }}(t)$ such that ${\tilde {\gamma }}(0)=g_{0}$ and that its tangent vectors always lie in the horizontal subspace. Recalling that the Lie-algebra valued gauge field $A_{\mu }(x)=A_{\mu }^{a}(x)T^{a}$ is equivalent to the connection that defines the horizontal subspace, this leads to a differential equation for the horizontal lift

i{\frac {dg(t)}{dt}}=A_{\mu }(x){\frac {dx^{\mu }}{dt}}g(t).

This has a unique formal solution called the Wilson line between the two points

g_{i}(t_{f})=W[x_{i},x_{f}]={\mathcal {P}}\exp {\bigg (}i\int _{x_{i}}^{x_{f}}A_{\mu }dx^{\mu }{\bigg )},

where ${\mathcal {P}}$ is the path-ordering operator, which is unnecessary for abelian theories. The horizontal lift starting at some initial fibre point other the identity merely requires multiplication by the initial element of the original horizontal lift. More generally, it holds that if ${\tilde {\gamma }}'(0)={\tilde {\gamma }}(0)g$ then ${\tilde {\gamma }}'(t)={\tilde {\gamma }}(t)g$ .

Under a local gauge transformation $g(x)$ the Wilson line transforms as

W[x_{i},x_{f}]\rightarrow g(x_{f})W[x_{i},x_{f}]g^{-1}(x_{i}).

This gauge transformation property is often used to directly introduce the Wilson line in the presence of matter fields $\psi (x)$ transforming in the fundamental representation of the gauge group where the Wilson line is an operator that makes the combination $\psi (x_{i})^{\dagger }W[x_{i},x_{f}]\psi (x_{f})$ gauge invariant.^[4] This allows one to compare the field at different points in a gauge invariant way. Alternatively, the Wilson lines can also be introduced by adding an infinitely heavy test particle charged under the gauge group. Its charge forms a quantized internal Hilbert space, which can be integrated out, yielding the Wilson line.^[5] This works whether or not there is any actual matter content in the theory or not.

The trace of closed Wilson lines is a gauge invariant quantity known as the \textbf{Wilson loop}

W[\gamma ]={\text{tr}}{\bigg [}{\mathcal {P}}\exp {\bigg (}i\oint _{\gamma }A_{\mu }dx^{\mu }{\bigg )}{\bigg ]}.

Mathematically the term within the trace is known as the holonomy, which describes a mapping of the fibre into itself upon horizontal lift along a closed loop. The set of all holonomies itself forms a group, which for the principal bundle must be a subgroup of the gauge group. Wilson loops satisfy the reconstruction property knowing the set of Wilson loops for all possible loops allows for the reconstruct all gauge invariant information about the gauge connection.^[6] Formally the set of all Wilson loops forms an overcomplete basis of solutions to the Gauss' law constraint.

The set of all Wilson lines is in one-to-one correspondence with the representations of the gauge group. This can be reformulated in terms of Lie algebra language using the weight lattice of the gauge group $\Lambda _{w}$ . In this case the types of Wilson loops are in one-to-one correspondence with $\Lambda _{w}^{G}/W$ where $W$ is the Weyl group.^[7]

References

^ Wilson, K.G. (1974). "Confinement of quarks". Phys. Rev. D. 10 (8): 2445–2459. doi:10.1103/PhysRevD.10.2445.
^ Nakahara, M. (2003). "10". Geometry, Topology and Physics (2 ed.). CRC Press. p. 374-418. ISBN 978-0750306065.
^ Eschrig, H. (2011). "7". Topology and Geometry for Physics. Lecture Notes in Physics. Springer. p. 220-222. ISBN 978-3-642-14699-2.
^ Schwartz, M. D. (2014). "25". Quantum Field Theory and the Standard Model (9 ed.). Cambridge University Press. p. 488-493. ISBN 9781107034730.
^ Tong, D. (2018), "2", Lecture Notes on Gauge Theory, pp. 33–38
^ Giles, R. (1981). "Reconstruction of gauge potentials from Wilson loops". Phys. Rev. D. 24 (8): 2160–2168. doi:10.1103/PhysRevD.24.2160.
^ Ofer, A.; Seiberg, N.; Tachikawa, Yuji (2013). "Reading between the lines of four-dimensional gauge theories". JHEP. 8: 115. arXiv:1305.0318. doi:10.1007/JHEP08(2013)115.

External links

Beckman, David; Gottesman, Daniel; Kitaev, Alexei; Preskill, John (2002-03-05). "Measurability of Wilson loop operators". Physical Review D. 65 (6): 065022. arXiv:hep-th/0110205. doi:10.1103/PhysRevD.65.065022. ISSN 0556-2821.

[1] Wilson, K.G. (1974). "Confinement of quarks". Phys. Rev. D. 10 (8): 2445–2459. doi:10.1103/PhysRevD.10.2445.

[2] Nakahara, M. (2003). "10". Geometry, Topology and Physics (2 ed.). CRC Press. p. 374-418. ISBN 978-0750306065.

[3] Eschrig, H. (2011). "7". Topology and Geometry for Physics. Lecture Notes in Physics. Springer. p. 220-222. ISBN 978-3-642-14699-2.

[4] Schwartz, M. D. (2014). "25". Quantum Field Theory and the Standard Model (9 ed.). Cambridge University Press. p. 488-493. ISBN 9781107034730.

[5] Tong, D. (2018), "2", Lecture Notes on Gauge Theory, pp. 33–38

[6] Giles, R. (1981). "Reconstruction of gauge potentials from Wilson loops". Phys. Rev. D. 24 (8): 2160–2168. doi:10.1103/PhysRevD.24.2160.

[7] Ofer, A.; Seiberg, N.; Tachikawa, Yuji (2013). "Reading between the lines of four-dimensional gauge theories". JHEP. 8: 115. arXiv:1305.0318. doi:10.1007/JHEP08(2013)115.

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+In [[quantum field theory]], '''Wilson loops''' are [[Introduction to gauge theory|gauge invariant]] operators arising from the [[parallel transport]] of gauge variables around closed [[loop (topology)|loops]]. Since they encode all possible gauge information of the theory, this allows for the construciton of [[loop representation in gauge theories and quantum gravity|loop representations]] which fully describe [[gauge theory|gauge theories]] in terms of these loops. They play a key role in pure gauge theory where they act as [[order operator|order operators]] for [[color confinement|confinement]]. Originally formulated by [[Kenneth G. Wilson]] in 1974, they were used to construct links and plaquettes which are the funamental parameters in [[lattice gauge theory]].<ref>{{cite journal|last1=Wilson|first1=K.G.|authorlink1=Kenneth G. Wilson|date=1974|title=Confinement of quarks|url=https://link.aps.org/doi/10.1103/PhysRevD.10.2445|journal=Phys. Rev. D|volume=10|issue=8|pages=2445-2459|doi=10.1103/PhysRevD.10.2445|pmid=|arxiv=|s2cid=|access-date=}}</ref> Wilson loops fall into the broader class of loop [[operator (physics)|operators]], with some other notable examples being the [['t Hooft operator|'t Hooft loops]], which are in some sense dual to Wilson loops, and Polyakov loops which are the thermal version of Wilson loops.
-In [[gauge theory]], a '''Wilson loop''' (named after [[Kenneth G. Wilson]]) is a [[gauge-invariant]] [[observable]] obtained from the [[holonomy]] of the [[gauge connection]] around a given loop. In the classical theory, the collection of all Wilson loops contains sufficient information to reconstruct the gauge connection, up to [[gauge transformation]].<ref name="Giles 1981">{{cite journal| authorlink=Roscoe Giles |first= R.| last= Giles | journal=[[Physical Review D]] | title=Reconstruction of Gauge Potentials from Wilson loops | volume= 24| issue=8| page= 2160| year=1981| doi=10.1103/PhysRevD.24.2160|bibcode = 1981PhRvD..24.2160G }}</ref>
-==Overview==
+==Definition==
-In [[quantum field theory]], the definition of Wilson loop observables as ''[[bona fide]]'' [[operator (mathematics)|operator]]s on [[Fock space]]s is a mathematically delicate problem and requires [[regularization (physics)|regularization]], usually by equipping each loop with a ''framing''. The action of Wilson loop operators has the interpretation of creating an elementary excitation of the quantum field which is localized on the loop. In this way, [[Michael Faraday|Faraday]]'s "flux tubes" become elementary excitations of the quantum electromagnetic field.
+To properly define Wilson loops in gauge theory requires considering the [[Gauge theory (mathematics)|fibre bundle formulation]] of gauge theories.<ref>{{cite book|last=Nakahara|first=M.|author-link=|date=2003|title=Geometry, Topology and Physics|url=|doi=|edition=2|location=|publisher=CRC Press|chapter=10|page=374-418|isbn=978-0750306065}}</ref> Here for each point in the [[spacetime]] there is a copy of the gauge group <math>G</math> forming what's known as a fibre of the [[fibre bundle]]. These fibre bundles are called [[principal bundles]]. Locally the resulting space looks like <math>\mathbb R^d \times G</math> although globally it can have some twisted structure depending on how different fibres are glued together.
-Wilson loops were introduced in 1974 in an attempt at a nonperturbative formulation of [[quantum chromodynamics]] (QCD), or at least as a convenient collection of variables for dealing with the strongly interacting regime of QCD.<ref name="Wilson 1974">{{cite journal| authorlink=Kenneth G. Wilson| first=K.| last= Wilson| journal=[[Physical Review D]] | title=Confinement of quarks | volume= 10| issue=8| page= 2445| year= 1974| doi=10.1103/PhysRevD.10.2445|bibcode = 1974PhRvD..10.2445W }}</ref> The problem of [[colour confinement|confinement]], which Wilson loops were designed to solve, remains unsolved to this day.
+The issue that Wilson lines resolve is how to compare points on fibres at two different spacetime points. This is analogous to parallel transport in [[general relativity]] which compares [[tangent vectors]] that live in the [[tangent space]] at different points. For principal bundles there is a natural way to compare different fibre points to each other through the introduction of a [[connection (mathematics)|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 and horizontal bundles|vertical]] and horizontal subspaces.<ref>{{cite book|last=Eschrig|first=H.|author-link=|date=2011|title=Topology and Geometry for Physics|url=|doi=|location=|publisher=Springer|series=Lecture Notes in Physics|chapter=7|page=220-222|isbn=978-3-642-14699-2}}</ref> The former consists of all vectors pointing along the fibre <math>G</math> while the latter consists of vectors that are perpendicular to the fibre. This allows one to compare fibre 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 fibre.
-The fact that strongly coupled quantum gauge field theories have elementary nonperturbative excitations which are loops motivated [[Alexander Markovich Polyakov|Alexander Polyakov]] to formulate the first [[string theory|string theories]], which described the propagation of an elementary quantum loop in spacetime.
+If the starting fibre is at <math>x_0</math> and takes the curve starts at the identity <math>g_0=e</math>, then to see how this changes when moving to another spacetime coordinate <math>x_1</math>, one needs to consider some spacetime curve <math>\gamma:[0,1]\rightarrow M</math> between <math>x_0</math> and <math>x_1</math>. The corresponding curve in the principal bundle, known as the [[Ehresmann connection|horizontal lift]] of <math>\gamma(t)</math>, is the curve <math>\tilde \gamma(t)</math> such that <math>\tilde \gamma(0) = g_0</math> and that its tangent vectors always lie in the horizontal subspace. Recalling that the [[Lie algebra|Lie-algebra]] valued gauge field <math>A_\mu(x) = A^a_\mu(x)T^a</math> is equivalent to the connection that defines the horizontal subspace, this leads to a [[differential equation]] for the horizontal lift
-Wilson loops played an important role in the formulation of [[loop quantum gravity]], but there they are superseded by [[spin network]]s (and, later, [[spinfoam]]s), a certain generalization of Wilson loops.
+:<math>
-In [[particle physics]] and [[string theory]], Wilson loops are often called '''Wilson lines''', especially Wilson loops around non-contractible loops of a compact manifold.
+i\frac{dg(t)}{dt} = A_\mu(x)\frac{dx^\mu}{dt} g(t).
+</math>
+This has a unique formal solution called the '''Wilson line''' between the two points
-== An equation ==
+:<math>
-The '''Wilson loop''' variable is a quantity defined by the trace of a [[path-ordered exponential]] of a [[gauge field]] <math>A_\mu</math> transported along a closed line C:
+g_i(t_f) = W[x_i, x_f] = \mathcal P\exp\bigg( i \int_{x_i}^{x_f}A_\mu dx^\mu \bigg),
+</math>
+where <math>\mathcal P</math> is the [[path-ordering|path-ordering operator]], which is unnecessary for [[abelian group|abelian]] theories. The horizontal lift starting at some initial fibre point other the identity merely requires multiplication by the initial element of the original horizontal lift. More generally, it holds that if <math>\tilde \gamma'(0) = \tilde \gamma(0)g</math> then <math>\tilde \gamma'(t) = \tilde \gamma(t)g</math>.
-:<math>W_C := \mathrm{Tr}\,(\, \mathcal{P}\exp i \oint_C A_\mu dx^\mu \,)\,.</math>
+Under a [[Symmetry_(physics)#Local_and_global|local gauge transformation]] <math>g(x)</math> the Wilson line transforms as
-Here, <math>C</math> is a closed curve in space, <math>\mathcal{P}</math> is the [[path-ordering]] operator. Under a gauge transformation
+:<math>
-:<math>\mathcal{P}e^{i \oint_C A_\mu dx^\mu} \to g(x) \mathcal{P}e^{i \oint_C A_\mu dx^\mu} g^{-1}(x)\,</math>,
+W[x_i, x_f] \rightarrow g(x_f) W[x_i, x_f] g^{-1}(x_i).
+</math>
+This gauge transformation property is often used to directly introduce the Wilson line in the presence of [[fermionic field|matter fields]] <math>\psi(x)</math> transforming in the [[fundamental representation]] of the gauge group where the Wilson line is an operator that makes the combination <math>\psi(x_i)^\dagger W[x_i,x_f]\psi(x_f)</math> gauge invariant.<ref>{{cite book|first=M. D.|last=Schwartz|title=Quantum Field Theory and the Standard Model|publisher=Cambridge University Press|date=2014|chapter=25|edition=9|page=488-493|isbn=9781107034730}}</ref> This allows one to compare the field at different points in a gauge invariant way. Alternatively, the Wilson lines can also be introduced by adding an infinitely heavy [[test particle]] charged under the gauge group. Its charge forms a quantized internal [[Hilbert space]], which can be integrated out, yielding the Wilson line.<ref>{{citation|last=Tong|first=D.|author-link=David Tong|title=Lecture Notes on Gauge Theory|chapter=2|pages=33-38|date=2018|url=https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html}}</ref> This works whether or not there is any actual matter content in the theory or not.
-where <math>x\,</math> corresponds to the initial (and end) point of the loop (only initial and end point of a line contribute, whereas gauge transformations in between cancel each other). For SU(2) gauges, for example, one has <math>g^{\pm 1}(x)\equiv\exp\{\pm i\alpha^j(x)\frac{\sigma^j}{2}\}</math>; <math>\alpha^j(x)</math> is an arbitrary real function of <math>x\,</math>, and <math>\sigma^j</math> are the three Pauli matrices; as usual, a sum over repeated indices is implied.
+The [[trace (linear algebra)|trace]] of closed Wilson lines is a gauge invariant quantity known as the \textbf{Wilson loop}
-The invariance of the [[Trace (linear algebra)|trace]] under [[cyclic permutation]]s guarantees that <math>W_C</math> is invariant under [[gauge transformation]]s. Note that the quantity being traced over is an element of the gauge [[Lie group]] and the trace is really the [[character (mathematics)|character]] of this element with respect to one of the infinitely many [[irreducible representation]]s, which implies that the operators <math> A_\mu\,dx^\mu</math>  don't need to be restricted to the "trace class" (thus with purely discrete spectrum), but can be generally hermitian (or mathematically: self-adjoint) as usual. Precisely because we're finally looking at the trace, it doesn't matter which point on the loop is chosen as the initial point. They all give the same value.
+:<math>
-Actually, if A is viewed as a [[connection form|connection]] over a [[principal bundle|principal G-bundle]], the equation above really ought to be "read" as the [[parallel transport]] of the identity around the loop which would give an element of the Lie group G.
+W[\gamma] = \text{tr}\bigg[\mathcal P \exp\bigg( i \oint_\gamma A_\mu dx^\mu\bigg)\bigg].
+</math>
+Mathematically the term within the trace is known as the [[holonomy]], which describes a [[automorphism|mapping]] of the fibre into itself upon horizontal lift along a closed loop. The set of all holonomies itself forms a [[group (mathematics)|group]], which for the principal bundle must be a [[subgroup]] of the gauge group. Wilson loops satisfy the reconstruction property knowing the set of Wilson loops for all possible loops allows for the reconstruct all gauge invariant information about the gauge connection.<ref>{{cite journal|last1=Giles|first1=R.|authorlink1=|date=1981|title=Reconstruction of gauge potentials from Wilson loops|url=https://link.aps.org/doi/10.1103/PhysRevD.24.2160|journal=Phys. Rev. D|volume=24|issue=8|pages=2160-2168|doi=10.1103/PhysRevD.24.2160|pmid=|arxiv=|s2cid=|access-date=}}</ref> Formally the set of all Wilson loops forms an [[overcompleteness|overcomplete]] [[basis (linear algebra)|basis]] of solutions to the Gauss' law constraint.
-Note that a path-ordered exponential is a convenient shorthand notation common in physics which conceals a fair number of mathematical operations. A mathematician would refer to the path-ordered exponential of the connection as "the holonomy of the connection" and characterize it by the parallel-transport differential equation that it satisfies.
+The set of all Wilson lines is in [[bijection|one-to-one correspondence]] with the [[group representation|representations]] of the gauge group. This can be reformulated in terms of Lie algebra language using the [[weight (representation theory)|weight lattice]] of the gauge group <math>\Lambda_w</math>. In this case the types of Wilson loops are in one-to-one correspondence with <math>\Lambda_w^G/W</math> where <math>W</math> is the [[Weyl group]].<ref>{{cite journal|last1=Ofer|first1=A.|authorlink1=|last2=Seiberg|first2=N.|authorlink2=Nathan Seiberg|last3=Tachikawa|first3=Yuji|authorlink3=|date=2013|title=Reading between the lines of four-dimensional gauge theories|url=|journal=JHEP|volume=8|issue=|pages=115|doi=10.1007/JHEP08(2013)115|pmid=|arxiv=1305.0318|s2cid=|access-date=}}</ref>
-At T=0, where T corresponds to temperature, the Wilson loop variable characterizes the [[color confinement|confinement]] or deconfinement of a gauge-invariant quantum-field theory, namely according to whether the variable increases with the ''area'', or alternatively with the ''circumference'' of the loop ("area law", or alternatively "circumferential law" also known as "perimeter law").
-In finite-temperature QCD, the thermal expectation value of the Wilson line distinguishes
-between the confined "hadronic" phase, and the deconfined state of the field, e.g., the [[quark–gluon plasma]].
 ==See also==