# 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.

96 questions
Filter by
Sorted by
Tagged with
85 views

• 2,348
114 views

### Modified non-abelian Wilson loop

Consider a $\mathrm{SU}(2)$ Wilson loop in some representation $R$, $$W_R= \mathrm{Tr}_R\left\{\mathcal P\exp\left(i \oint \mathrm dx^{\mu} A^a_{\mu} T^a\right)\right\}$$ where $T^a= \sigma^a/2$ is ...
• 377
1 vote
49 views

### Fourier Transforrm of a Wilson Line

Lots of times it is helpful to perform computations of Feynman Diagrams in momentum space as opposed to position space. This however appears a little tricky when it comes to Wilson Lines. Ignoring ...
• 381
157 views

### Variation of a time-ordered exponential

Consider the time-ordered exponential (Wilson line): $$U(t_{f},t_{i}) = \mathcal{T}\text{exp}\left(-i\int_{t_{i}}^{t_{f}}\mathcal{A}(t)dt\right)\tag{1}$$ Where $\mathcal{A}(t)$ is some matrix-valued ...
• 558
44 views

### Book for lattice field theory for somebody with basic understanding of Quantum Field Theory [duplicate]

I have finished a first course in Quantum Field theory and I'm looking to get into lattice field theory (mostly QCD on the lattice). What are some resources for somebody wanting to learn how things ...
112 views

### References for prerequisite material for understanding papers on Generalized Global Symmetries

I want to understand the papers https://arxiv.org/abs/1412.5148 and https://arxiv.org/abs/1703.00501. Assuming that I understand basics of gauge theories, could someone suggest some references ...
139 views

### How to prove $\Lambda_\mathrm{weight}(\mathfrak{g})/ \Lambda_\mathrm{root}(\mathfrak{g}) = \mathbb{Z}_N$ for $\mathrm {SU}(N)$?

I have a question about $\mathrm {SU}(N)$ Lie group from David Tong's Gauge Theory notes (p. 92). He considers $\Lambda_w(\mathfrak{g})$ and $\Lambda_\mathrm{root}(\mathfrak{g})$ as the weight lattice ...
• 45
59 views

### Expansion of Wilson loop operator to obtain the field tensor in non-Abelian gauge theory

I am reading Prof. Eduardo Fradkin's QFT lecture note (the link to the lecture note is gone). In his note, he considered the Wilson loop operator \begin{equation} \hat{W}_{\Gamma(x,x)}=\hat{P} \left( \...
• 691
79 views

### Missing factor 1/2 when using generalized Stokes theorem

I'm doing the following homework question: By invoking Stokes' theorem, according to which the integral of a vector field (which equals the field strength) over any two-dimension surface S that is ...
73 views

### How to calculate the chiral condensate from wilson fermions in lattice qcd?

In lattice qcd, respective more specifically in (1+1)-dimensional massless schwinger model on a lattice, iam trying to reproduce the chiral condensate by using correct constructed wilson overlap ...
55 views

### Wilson action equations of motion

Let $S_W$ be a Wilson action of $1\times 1$ plaquettes for a gauge group $G$: \begin{equation*} S_W = \beta a^4 \sum_P \left( 1-\frac{1}{N_G} \text{Re Tr}(U_P) \right), \end{equation*} where $\beta$ ...
• 1,861
150 views

### QFT/QM on loop space

I am reading "Gauging What's Real" by Richard Healey and the author argues for formulating electrodynamics/QED on loop space/the holonomy group, so that the real objects described by the ...
• 466
322 views

Let's consider free Maxwell theory: $$L = -\frac{1}{4g^2} F^{\mu\nu}F_{\mu\nu}$$ As I understand, one can describe external particles with help of Wilson lines: $$W(q,l) = e^{iq\int_l dx^\mu A_\mu} ... • 5,349 3 votes 1 answer 144 views ### Wilson loop shapes and glueball operators In the AdS/QCD correspondence, glueballs operators are given, for example, by \text{Tr}[F_{\mu \nu}F^{\mu \nu}] for 0^{++} or \text{Tr}[F_{\mu \nu}\widetilde{F}^{\mu \nu}] for 0^{-+}. However, ... • 1,861 2 votes 1 answer 286 views ### Deriving the comparator to second order in Peskin and Schroeder In section 15.1 of Peskin and Schroeder, expression (15.9) is given for the comparator U(y,x) in an infinitesimal expansion to second order:$$U(x+\epsilon n, x)=\exp \left[-i e \epsilon n^{\mu} A_{\... 134 views

### What does it mean for an extended operator to possess "local excitations"?

In the context of defect conformal field theory, we consider in operator product expansions "local excitations" of the defect (see e.g. text between eq. $(1.1)$ and $(1.2)$ in the paper ...
• 1,513
1 vote
51 views

### Wilson loop expectation value in $RP^3$ using Dehn surgery

I am currently reading Guadagnini's The link invariants of Chern-Simons field theory, the part where he computes some examples of expectation values for different spaces. For $S^2 \times S^1$, he ...
130 views

• 767
272 views

### Why don't we add Wilson loops to the SM Lagrangian?

As the title says: why don't we add Wilson loops to common Lagrangians such as the Standard Model? They're gauge invariant and (correct me if I'm wrong, not sure on that) are renormalizable. Suppose ...
• 715
1 vote
36 views

### Minimal area for circular Wilson loops in these coordinates

In all references you can see that the Poincare coordinates are used to get the minimal area for the circular wilson loop. I want to use the metric that is used also for the D3-brane (e.g. see ...
• 142
93 views

### A question on supersymmetry variation of the Wilson loop in $\mathcal{N}=4$ SYM

The Wilson loop in $\mathcal{N}=4$ SYM is $$W=\frac{1}{N}tr P \exp \int ds (i A_\mu(x) \dot{x}^\mu+\Phi_i(x)\theta^i|\dot{x}|).\tag{2.3}$$ In order to check whether this operator is supersymmetric I ...
433 views

### 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 $W(x)$ ...
• 767
1 vote
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 ...