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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.

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.
In the classical theory, the collection of all Wilson loops contains sufficient information to reconstruct the gauge connection, up to gauge transformation.

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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 $$ \gamma_{mn}= \int_\mathcal{C} \left\langle m(R)…
Praharsh Suryadevara
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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 $G$. Then the classical Wilson loop is defined as,…
Student
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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 other name or heading. At least I couldn't…
Student
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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 observation is that the derivative of the Dirac…
freddieknets
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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 that in magneto-static, the Wilson loop for a …
Adrien
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Berry phase and Wilson loop

According to the definition, the Wilson loop is \begin{equation} W[\mathcal{C}] =\operatorname{Tr}\left[\mathcal{P} \exp\left\{i\oint _{\mathcal{C}} A_{\mu } dx^{\mu }\right\}\right] \end{equation} where $\mathcal{P} $ is the path ordering, $A_{\mu…
Fang Lyu
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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 some other sources referred to U as "parallel…
user587155
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Gauge fields and strings: Loop equations

I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book: $$ \delta \Psi (C) ~=~ \int_{0}^{2\pi} {\rm P} \left(F_{\mu\nu}(x(s)) \exp \oint_C A_\mu dx^\mu \right)\dot{x}_\nu \delta x_\mu(x) \, {\rm d} s, \tag{7.25} $$ where the non-abelian phase…
user11881
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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 Theories in two dimensions, Witten carries out a…
ACuriousMind
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Wilson lines and Bremsstrahlung in Quantum Electrodynamics

Short: How can I use Wilson lines to compute Bremsstrahlung? I'm particularly interested in QED, as a simple example. Long: I recently learned what is a Wilson line, in the simplest sense of $$W[C] = \mathrm{Tr}_R\left[\mathcal{P}\exp\left(i \int_C…
Níck Aguiar Alves
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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 electromagnetic field. What happens physically is that after…
geodude
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Relation between sources and Wilson loops in Tong's gauge theory

In Euclidean quantum Yang-Mills with compact gauge group $G$, the VEV of a Wilson loop is: $$\tag{$\star$} \langle W[C]\rangle \equiv \int_{\mathcal{A}/G}DA e^{-S_{YM}[A]}tr\mathcal{P}\exp\left(i\int_C A\right).$$ On p87 of David Tong's notes on…
nodumbquestions
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Area and perimeter

Apparently (?), a line operator over a very large loop with length $L$ can obey either perimeter law or area law, $-\log\langle U\rangle\sim L^a$ with $a=1,2$, respectively. We call these options "deconfimenent" and "confinement", and this seems to…
AccidentalFourierTransform
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Is there an analogy for Wilson loops/lines in statistical mechanics?

When reformulated in Euclidean space, quantum field theory bears some strong resemblance to statistical mechanics: for example a scalar field $\phi$ can be seen as a spin $s$ in Landau theory, and the source $J$ in path integral formalism can be…
Pxx
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Wilson loop operator in electrodynamics

I'm trying to prove that the Wilson loop operator is well-defined in non-interacting quantum electrodynamics without matter, that is, $\hat{W}(\gamma)$ is a bounded operator on the Hilbert space. Since the Wilson loop is an exponentiation $$…
Prof. Legolasov
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