Editing Wilson loop (section)

==Definition==

[[File:Principal Bundle Path.svg|280px|thumb|right|alt=Example of a principal bundle displaying the base spacetime manifold along with its fibers. It also displays how at every point along the fiber the tangent space can be split up into a vertical subspace pointing along the fiber and a horizontal subspace orthogonal to it.|A connection on a principal bundle <math>P</math> with spacetime <math>M</math> separates out the tangent space at every point <math>x_p</math> along the fiber <math>G_p</math> into a vertical subspace <math>V_p</math> and a horizontal subspace <math>H_p</math>. Curves on the spacetime are uplifted to curves in the principal bundle whose tangent vectors lie in the horizontal subspace.]]

To properly define Wilson loops in gauge theory requires considering the [[Gauge theory (mathematics)|fiber 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|pages=374–418|isbn=978-0750306065}}</ref> Here for each point in the <math>d</math>-dimensional [[spacetime]] <math>M</math> there is a copy of the gauge group <math>G</math> forming what's known as a fiber of the [[fiber bundle]]. These fiber 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 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]] which compares [[tangent vectors]] that live in the [[tangent space]]s at different points. For principal bundles there is a natural way to compare different fiber points 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|pages=220–222|isbn=978-3-642-14699-2}}</ref> The former consists of all vectors pointing along the fiber <math>G</math> 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 <math>x_i</math> with a starting point of the identity <math>g_i=e</math>, then to see how this changes when moving to another spacetime coordinate <math>x_f</math>, one needs to consider some spacetime curve <math>\gamma:[0,1]\rightarrow M</math> between <math>x_i</math> and <math>x_f</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_i</math> and that its tangent vectors always lie in the horizontal subspace. The fiber bundle formulation of gauge theory reveals 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, so this leads to a [[differential equation]] for the horizontal lift

:<math>
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

:<math>
g_f(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 fiber point other than 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> for all <math>t\geq0</math>.

Under a [[Symmetry (physics)#Local and global|local gauge transformation]] <math>g(x)</math> the Wilson line transforms as

:<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 matter fields <math>\phi(x)</math> transforming in the [[fundamental representation]] of the gauge group, where the Wilson line is an operator that makes the combination <math>\phi(x_i)^\dagger W[x_i,x_f]\phi(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|pages=488–493|isbn=9781107034730}}</ref> 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]] charged under the gauge group. Its charge forms a quantized internal [[Hilbert space]], which can be integrated out, yielding the Wilson line as the world-line of the test particle.<ref name="tong">{{citation|last=Tong|first=D.|author-link=David Tong (physicist)|title=Lecture Notes on Gauge Theory|chapter=2|pages=|date=2018|chapter-url=https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html}}</ref> This works in quantum field theory whether or not there actually is any matter content in the theory. However, the [[swampland (physics)|swampland conjecture]] known as the completeness conjecture claims that in a consistent theory of [[quantum gravity]], every Wilson line and 't Hooft line of a particular charge consistent with the [[Dirac quantization condition]] must have a corresponding particle of that charge be present in the theory.<ref>{{cite journal|last1=Banks|first1=T.|authorlink1=Tom Banks (physicist)|last2=Seiberg|first2=N.|authorlink2=Nathan Seiberg|date=2011|title=Symmetries and Strings in Field Theory and Gravity|url=|journal=Phys. Rev. D|volume=83|issue=|pages=084019|doi=10.1103/PhysRevD.83.084019|pmid=|arxiv=1011.5120|s2cid=|access-date=}}</ref> Decoupling these particles by taking the infinite mass limit no longer works since this would form [[black hole]]s.

The [[trace (linear algebra)|trace]] of closed Wilson lines is a gauge invariant quantity known as the ''Wilson loop''

{{Equation box 1
|title=
|indent=:
|equation = <math>W[\gamma] = \text{tr}\bigg[\mathcal P \exp\bigg( i \oint_\gamma A_\mu \, dx^\mu\bigg)\bigg].</math>
|border
|border colour =#50C878
|background colour = #ECFCF4
}}

Mathematically the term within the trace is known as the [[holonomy]], which describes a [[automorphism|mapping]] of the fiber into itself upon horizontal lift along a closed loop. The set of all holonomies itself forms a [[group (mathematics)|group]], which for principal bundles must be 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.<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=|bibcode=1981PhRvD..24.2160G |s2cid=|access-date=|url-access=subscription}}</ref> Formally the set of all Wilson loops forms an [[overcompleteness|overcomplete]] [[basis (linear algebra)|basis]] of solutions to the Gauss' law constraint.

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/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=2013|issue=8|page=115|doi=10.1007/JHEP08(2013)115|pmid=|arxiv=1305.0318|bibcode=2013JHEP...08..115A |s2cid=118572353|access-date=}}</ref>

===Hilbert space operators===

An alternative view of Wilson loops is to consider them as operators acting on the Hilbert space of states in [[Minkowski space|Minkowski signature]].<ref name="tong"/> 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 [[Causal structure|spacelike]] loops. Such operators <math>W[\gamma]</math> create a closed loop of [[electric flux]], which can be seen by noting that the [[electric field]] operator <math>E^i</math> is nonzero on the loop <math>E^iW[\gamma]|0\rangle \neq 0</math> but it vanishes everywhere else. Using [[Stokes theorem]] it follows that the spatial loop measures the [[magnetic flux]] through the loop.<ref>{{cite book|last1=Peskin|first1=Michael E.|author1-link=Michael Peskin|last2=Schroeder|first2=Daniel V.|date=1995|title=An Introduction to Quantum Field Theory|publisher=Westview Press|chapter=15|page=492|isbn=9780201503975}}</ref>