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Questions tagged [wilson-loop]

In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given (closed) loop $C$. It is the trace of a path-ordered exponential of the gauge field $A_\mu$ transported along $C$, $W_C := \mathrm{Tr}(\mathcal{P}\exp i \oint_C A_\mu dx^\mu)$, where $\mathcal{P}$ is the path-ordering operator.

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Wilson loop and Polyakov loop

As I understand, the Wilson line is the operator $W(x) = P\exp(i\int_{xi}^{xf} A.dx)$, where $P$ is path ordering. The Polyakov loop $P(x)$ on the other hand is the trace of the Wilson loop $W(x)$ ...
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False solution of Landau Hamiltonian

The Landau Hamiltonian in 2D is given (in natural units $q=c=2m=1$) by $$ \hat{H} = (\hat{\vec{p}}-\vec{A}(\hat{\vec{x}}))^2 \,,$$ where $\vec{A}$ is the magnetic vector potential field. We know that ...
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Generalization of Chern-Simons Wilson Line

The Wilson line in Abelian Chern-Simons theory is $$\langle\exp\left(i\int dt\dot{x}^{\mu}A_{\mu}\right)\rangle=\int\mathcal{D}A\exp\left(i\int dt\dot{x}^{\mu}A_{\mu}+\frac{i}{4\pi}\int A\wedge F\...
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About Witten's path integral formulation of Jones polynomial

In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the ...
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Expectation value of a path-ordered exponential

Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$: $$ \overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
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Questions on how Wilson loops relate to field & charge conservation, and lattice QFT

The path-ordered exponential from which the Wilson loop is traced is, crudely, $$ \prod (I+ A_\alpha dx^\alpha) = \mathcal{P}\,\mathrm{exp}(i \oint A_\alpha dx^\alpha )$$ which returns a matrix $\...
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What is the physical meaning of Wilson loops?

I'm a mathematician trying to get some very basic physical intuition on gauge theories, so I apologize if what follows is really naive. My first super elementary question is: Am I right to think ...
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Are these the only gauge-invariant functions of $A_\mu$?

I know off course that $F_{\mu\nu}$ is a gauge invariant function of $A_\mu$ in the abelian case. Also we have $\epsilon^{\alpha\beta\mu\nu} F_{\alpha\beta}F_{\mu\nu}$ in that case. Are there any ...
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Linear Potential from Wilson Loop

It is well known that if you compute the expectation value of the Wilson loop along a suitable rectangle you can get the Coulomb potential $$\langle W(\mathcal{C})\rangle\sim e^{TV(R)} , \ \ V(R)\sim ...
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Wilson Loops and Confinement in QED

In [1] Kenneth G. Wilson proposed a mechanism for confinement using lattice paths what leds him to the concept of Wilson loop. It seems to me that he is using mainly a single abelian field. He says ...
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Polyakov Loop and Chemical Potential

I have read in a paper (http://arxiv.org/abs/1203.3556) that in a thermal field theory, the chemical potential is $\mu=T \ln P$ where $$T^{-1}=\int_{0}^{\beta} \sqrt{-\xi^2}dt,$$ $\xi$ is $\partial_t$,...
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Path-Integral of Charged Particle in Chern-Simons Gauge Fields

From the paper "Fermi-Bose Transmutations Induced by Gauge Fields" by Polyakov, http://inspirehep.net/record/22956 http://dx.doi.org/10.1142/S0217732388000398 the theory in 3D, $$\mathcal{L}=\sum_{...
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Simple explanation for what a torsor is

I am studying Chris Elliott's notes on Line and Surface Operators in Gauge Theories (available here). In the notes, there's a mention of the fact that (for $G = U(1)$), $$W_{\gamma, n}(A) = e^{in\...
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Partial derivatives of Wilson like integrals

I have a one-form field on Euclidean space. Suppose we integrate it over a loop around the specific point $x$. $$I(x)=\int_xU.$$ I want to calculate the partial derivatives of this integral respect to ...
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Difference between average effective action and Wilsonian effective action?

There is a good description about " Difference between 1PI effective action and Wilsonian effective action? " here. Now, what is the difference between average effective action, which we use that in ...
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Wilson loops, confinement and chiral symmetry relations

As per my limited understanding of confinement, I understood it as phenomenon between fermions of a gauge theory, such as $SU(3)$ gauge theory in the low energy limit (where the coupling constant ...
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Wilson loop in AdS/CFT : string interpretation

It is well known that Wilson loop is a quite hard observable to compute. In the case in which the QFT is dual to a gravitation theory in AdS space, we can use holography to compute the Wilson loop, ...
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In what types of QFTs are the Wilson loops of interest?

I have a very basic question about Wilson lines (WL). This is what I know about the WL: WL help us to learn about the important properties of gauge fields (treated as connections on the space of ...
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A Wilson line property

Consider a path, say $P$, and three points on it like $x,y,z$. If there were infinitesmally closed, then the following relation would be true. $$W_{P}(x,y) W_{P}(y,z)=W_{P}(x,z)$$ If the Lie algebra ...
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The gauge transfomation of the Wilson line

I have a question in Capter 15 of Peskin & Schroeder. The gauge transformation here in its infinitesimal form: \begin{cases} \psi(x) \to V(x)\psi(x) \quad \quad \quad \quad \quad \quad \,\,\,\, \...
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PDFs expressed through matrix elements of bi-local operators

Extracted from 'At the frontier of ParticlePhysics, handbook of QCD, volume 2', '...in the physical Bjorken $x$-space formulation, an equivalent definition of PDFs can be given in terms of matrix ...
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A derivative about chiral current in Peskin's book

In Peskin's book (an introduction to QFT), Page 655, the axial vector current is defined as follows, \begin{eqnarray*} j^{\mu5} & = & \text{symm }\lim_{\epsilon\rightarrow0}\bigg\{\bar{\psi}(x+...
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Calculating the Berry curvature in case of degenerate levels (Non abelian Berry curvature): issue

The Berry phase accumulated on a path can be described by a matrix when we look at adiabatic time evolution with a Hamiltonian with degenerate energy levels. The Berry phase matrix is given by $$ \...
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Circular Wilson Loop in AdS/CFT

I'm trying to get the AdS solution to the circular wilson loop. The standard AdS metric is: $ds^2 = \frac{L^2}{z^2}(\eta_{\mu \nu} dx^{\mu} dx^{\nu} + dz^2)$ If I take the circle of radius R at ...
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$\mathcal{N} = 4$ Super-Yang Mills propagators

In $\mathcal{N} = 4$ Super-Yang mills there are only massless particles. If one wishes to obtain a heavy quark one can see the SYM theory as a stack of (N+1)-branes in AdS$_5 \times$S$^5$ where one ...
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Wilson Loop in AdS/CFT

In AdS/CFT correspondence one can compare results in $\mathcal{N}=4$ SYM with string theory type IIB in $AdS_5 \times S^5$. One of the observables that it's possible to get non-perturbative results is ...
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Supertrace of holonomy of commutator

On page 47 of Surface operators in four-dimensional topological gauge theory and Langlands duality by Kapustin et al., the following expression is given \begin{equation} \delta\mathcal{N}=d(\omega_\...
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Coordinate variation of a Wilson loop [duplicate]

In Chern Simons Gauge Theory as a String Theory, Witten derives the general coordinate variation of a Wilson loop, i.e., equation 3.11. My question is, how does one derive this? I only managed to ...
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How to calculate thd average of a Wilson loop in LQCD?

I'm am trying to do the excercise in page 35 from Lepage's LQCD notes: http://arxiv.org/abs/hep-lat/0506036. I want to compute the thermal average of the Wilson loop of size $a\times a$, where $a$ is ...
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Are correlators constructed out of Wilson loops singular in pure Yang-Mills?

If I have some gauge invariant function of two Wilson loops (such as $\left<\text{Tr}W_1 \text{Tr}W_2\right>$) does the expectation value diverge when the loops coincide the same way $\left<\...
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How and why the phrase “quark force increases with distance”?

I have seen that phrase "force between quarks increases with distance" at many resources, some even relatively credible (albeit written for general audience). What is the reason behind that, when ...
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Gauge Field Tensor from Wilson Loop

It is possible to introduce the gauge field in a QFT purely on geometric arguments. For simplicity, consider QED, only starting with fermions, and seeing how the gauge field naturally emerges. The ...
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Expressing an adjoint representation Wilson line in terms of the fundamental representation

I'm working out some calculations with Wilson lines, defined as path-ordered exponential integrals of a gauge field: $$U = \mathcal{P}\exp\biggl(ig\int_{-\infty}^{\infty}\mathrm{d}x^\mu T^c A_{c\mu}\...
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The 6-j symbol and intersecting Wilson loops, redux

This is a quite specific question continuing the problems I have with computing the expectation value of intersecting Wilson loops I laid out here. Using the tools from the answer there, I quite ...
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Intersecting Wilson loops in 2D Yang-Mills

I am currently trying to understand 2D Yang-Mills theory, and I cannot seem to find an explanation for calculation of the expectation value of intersecting Wilson loops. In his On Quantum Gauge ...
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Why is the Yang-Mills Comparator unitary?

In chapter 15.2 of Peskin, the comparator is defined, as some object $U\left(y,\,x\right)$ which transforms as: $$ U\left(y,\,x\right) \mapsto V\left(y\right) U\left(y,\,x\right) \left[V\left(x\right)^...
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Equation 2.27 from Pachos's introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
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Does the projected spin state of the $d+id$ mean-field Hamiltonian on a triangular lattice has time-reversal(TR) symmetry?

Consider the following $d+id$ mean-field Hamiltonian for a spin-1/2 model on a triangular lattice $$H=\sum_{<ij>}(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$$, with $\chi_{ij}=\begin{pmatrix} 0 & \...
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Homeomorphism between the space of all Ashtekar connections and spacetime?

Excerpt from an essay of mine: Let $\Psi(\varsigma)$ be the wavefunction in the loop representation, where $\varsigma:[0,1]\to\mathcal{M}$, where $\mathcal{M}$ is spacetime. Then, let $\mathcal{A}$ ...
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an Abelian complex statistical phase from exchanging non-Abelian anyons?

We have some discussions in Phys.SE. about the braiding statistics of anyons from a Non-Abelian Chern-Simon theory, or non-Abelian anyons in general. May I ask: under what (physical or mathematical) ...
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High-energy scattering matrix a become wilson line

I would like to ask why the scattering matrix $S^{ab}(x)=\left\langle gluon\: b|gluon\: a\right\rangle$ can be approximated for fast particle interacting with a traget field $A^{-}$ by wilson line $S^{...
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Some question on the definition of flux in the projective construction?

Here I have some confusing points about the definition of flux in the projective construction. For example, consider the same mean-field Hamiltonian in my previous question, and assume the $2\times 2$ ...
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Some questions on the Wilson loop in the projective construction?

Based on the previous question and the comment in it, imagine two different mean-field Hamiltonians $H=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ and $H'=\sum(\psi_i^\dagger\chi_{ij}'\psi_j+H.c.)$, we ...
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Is the $SU(2)$ flux defined in the context of Projective Symmetry Group(PSG) an observable quantity?

The $SU(2)$ flux defined in the context of PSG is as follows: Consider the mean-field Hamiltonian $H_{MF}=\sum(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$ description of a 2D lattice spin-model, the ...
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Physical consequences of non-abelian non-trivial holonomy

The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#Significance) can be well described and explained in terms of holonomy of the $U(1)$ connection of the ...
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Gauge fields and strings: Loop equations

I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book: $$ \delta \Psi (C) ~=~ \int_{0}^{2\pi} {\rm P} \left(F_{\mu\nu}(x(s)) \exp \oint_C A_\mu dx^\mu \right)\dot{x}_\nu \delta x_\mu(x) \, {\...
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Intuition for gauge parallel transport (Wilson loops)

I'm looking for a geometrical interpretation of the statement that "Wilson loop is a gauge parallel transport". I have seen QFT notes describe U(x,y) as "transporting the gauge transformation", and ...
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Functional Derivation of Holonomy

I would like to know how to take the functional derivative of the holonomy, or Wilson line. I have tried it and I will show what I have done below, but before I wanted to say that I also have seen ...
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Some questions about Wilson loops

Let $G$ be the gauge group whose Yang-Mill's theory one is looking at and $A$ be its connection and $C$ be a loop in the space-time and $R$ be a finite-dimensional representation of the gauge group $G$...
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Wilson/Polyakov loops in Weinberg's QFT books

I wanted to know if the discussion on Wilson loops and Polyakov loops (and their relationship to confinement and asymptotic freedom) is present in the three volumes of Weinberg's QFT books but in some ...